chemical reaction induced rayleigh-bÈnard convection...
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CHEMICAL REACTION INDUCED
RAYLEIGH-BÈNARD CONVECTION IN A
DENSELY PACKED POROUS MEDIUM
SATURATED WITH A COUPLE-STRESS FLUID
Dissertation submitted in partial fulfillment of the requirements for the award of the degree of
MASTER OF PHILOSOPHY IN MATHEMATICS
By
APARNA U. Register No. 0935304
Supervisor Dr. S. MARUTHAMANIKANDAN
Department of Mathematics Christ University
Bangalore-560 029
HOSUR ROAD
BANGALORE-560 029
2010
DEDICATED TO
MY BELOVED PARENTS
S. UDAYKUMAR & LALITHA
DECLARATION
I hereby declare that the dissertation entitled “Chemical Reaction
Induced Rayleigh-Bènard Convection in a Densely Packed Porous
Medium Saturated with a Couple-Stress Fluid” has been undertaken by me
for the award of M.Phil. degree in Mathematics. I have completed this under
the guidance of Dr. S. MARUTHAMANIKANDAN, Assistant Professor,
Department of Mathematics, Christ University, Bangalore – 560 029. I also
declare that this dissertation has not been submitted for the award of any
Degree, Diploma, Associateship, Fellowship or other title.
Place:
Date:
APARNA U.
Candidate
Dr. S. MARUTHAMANIKANDAN Assistant Professor Department of Mathematics Christ University Bangalore - 560 029.
CERTIFICATE This is to certify that the dissertation submitted by APARNA U. on the title “Chemical Reaction Induced Rayleigh-Bènard Convection in a
Densely Packed Porous Medium Saturated with a Couple-Stress Fluid” is
a record of research work done by her during the academic year 2009 – 2010
under my guidance and supervision in partial fulfillment of the requirements
for the award of the degree of Master of Philosophy in Mathematics. This
dissertation has not been submitted for the award of any Degree, Diploma,
Associateship, Fellowship or other title.
Place:
Date:
Dr. S. MARUTHAMANIKANDAN
Supervisor
ACKNOWLEDGEMENT
I am indebted to my supervisor, Dr. S. Maruthamanikandan, for his patience,
exceptional guidance, and continued encouragement. He has been extremely helpful
and has offered invaluable assistance and support.
I would like to thank Dr. S. Pranesh, Coordinator, Postgraduate Programme in
Mathematics, Christ University for his encouragement and support. I gratefully
acknowledge Prof. T.V. Joseph, HOD, Department of Mathematics, Christ
University and Mrs. Sangeetha George, Assistant Professor, Department of
Mathematics, Christ University for their whole-hearted support.
I extend my gratitude to Prof. K.A. Chandrasekharan, the General Research
Coordinator and Prof. Dr. Nanje Gowda, The Dean of Science of Christ
University for their valuable advice and constant support.
Special thanks to the Vice–Chancellor, Dr. (Fr.) Thomas C. Mathew,
Pro-Vice-Chancellor and Director of Centre for Research and Consultancy,
Dr. (Fr.) Abraham V. M. of Christ University for the opportunity provided to do
this course.
I really need to acknowledge my friends who provided a stimulating and
fascinating environment. I am especially thankful to Deepika, Rashmi, Shibiraj
Singh, Sanjok Lama and Chitra for sharing the good time.
Last but not least, my deep appreciation goes to my uncle T. Chandrashekar,
for his genuine concern and encouragement. APARNA U.
ABSTRACT
The problem of Rayleigh-Benard convection in a couple-stress fluid
saturated densely packed porous medium with chemical reaction is studied
within the framework of linear stability analysis. Only infinitesimal
disturbances are considered. The linear stability analysis is based on the
normal mode technique. The Darcy law is used to model the momentum
equation. Closed form solution for the basic quiescent state is first
obtained. The principle of exchange of stabilities is valid and the existence
of oscillatory instability is ruled out. The expression for the stationary
media-Darcy-Rayleigh number is obtained as a function of the governing
parameters, viz., the wave number, the couple-stress parameter and the
Frank-Kamenetskii number. The Galerkin method is used to determine the
eigenvalues. The effect of various parameters on the stability of the fluid
layer is discussed through figures.
CONTENTS
Page
No.CHAPTER I Introduction 1 1.1 Objective and Scope 1 CHAPTER II Literature Review 7
2.1 Rayleigh-Bénard Convection in Fluids 7 2.2 Convection in a Porous Medium 17 2.3 Convection with Chemical Reaction 31 2.4 Convection in Couple-Stress Fluids 35 2.5 Plan of Work 50 CHAPTER III Basic Equations, Boundary Conditions and
Dimensionless Parameters 51
3.1 Basic Equations 53 3.2 Boundary Conditions 57 3.3 Dimensionless Parameters 61 CHAPTER IV
Chemical Reaction Induced Rayleigh-Bènard Convection in a Densely Packed Porous Medium Saturated with a Couple-Stress Fluid
64
4.1 Introduction 64 4.2 Mathematical Formulation 67 4.3 Basic Quiescent State 69 4.4 Linear Stability Analysis 71 CHAPTER V
Results, Discussion and Concluding Remarks 75
5.1 Results and Discussion 75 5.2 Concluding Remarks 76 BIBLIOGRAPHY 82
1
CHAPTER I
INTRODUCTION 1.1 Objective and Scope
Fluid mechanics is the study of how fluids move and the forces on them. Fluids
include liquids and gases. Fluid mechanics can be divided into fluid statics, the
study of fluids at rest, and fluid dynamics, the study of fluids in motion. It is a
branch of continuum mechanics, a subject which models matter without using the
information that it is made out of atoms. Fluid mechanics, especially fluid
dynamics, is an active field of research with many unsolved or partly solved
problems. Fluid mechanics can be mathematically complex. Sometimes it can best
be solved by numerical methods, typically using computers. A modern discipline,
called Computational Fluid Dynamics (CFD), is devoted to this approach to solving
fluid mechanics problems. Also taking advantage of the highly visual nature of fluid
flow is Particle Image Velocimetry, an experimental method for visualizing and
analyzing fluid flow. Fluid mechanics is the branch of physics which deals with the
properties of fluids, namely liquids and gases, and their interaction with forces.
Rapid advancement in fluid mechanics began with Leonardo da Vinci
(observation and experiment), Evangelista Torricelli (barometer), Isaac Newton
(viscosity) and Blaise Pascal (hydrostatics), and was continued by Daniel Bernoulli
with the introduction of mathematical fluid dynamics in Hydrodynamica (1738).
Inviscid flow was further analyzed by various mathematicians (Leonhard Euler,
d'Alembert, Lagrange, Laplace, Poisson) and viscous flow was explored by a
multitude of engineers including Poiseuille and Gotthilf Heinrich Ludwig Hagen.
Further mathematical justification was provided by Claude-Louis Navier and
George Gabriel Stokes in the Navier-Stokes Equations, and boundary layers were
investigated (Ludwig Prandtl), while various scientists (Osborne Reynolds, Andrey
2
Kolmogorov, Geoffrey Ingram Taylor) advanced the understanding of fluid
viscosity and turbulence.
Rayleigh-Benard and Marangoni Convection Rayleigh-Benard convection (RBC) is the instability of a fluid layer which is
confined between two thermally conducting plates, and is heated from below to
produce a fixed temperature difference. Since liquids typically have positive
thermal expansion coefficient, the hot liquid at the bottom of the cell expands and
produces an unstable density gradient in the fluid layer. If the density gradient is
sufficiently strong, the hot fluid will rise, causing a convective flow which results in
enhanced transport of heat between the two plates. In order for convection to occur,
a small plume of hot fluid which begins to rise toward the top of the cell must grow
in strength, rather than fizzle out.
There are two processes which oppose this amplification. First, viscous damping
in the fluid directly opposes the fluid flow. In addition, thermal diffusion will
suppress the temperature fluctuation by causing the rising plume of hot fluid to
equilibriate with surrounding fluid, destroying the buoyant force. Convection occurs
if the amplifying effect exceeds the disippative effect of thermal diffusion and
buoyancy. This competition of forces is parameterized by the Rayleigh number,
which is the temperature difference, but appropriately normalized to take into
account the geometry of the convection cell and the physical properties of the fluid.
If the temperature difference is very large, then the fluid rises very quickly, and a
turbulent flow may be created. If the temperature difference is not far above the
onset, an organized flow resembling overturning of cylinders is formed. It is the
patterns created by these convection "rolls" that most people study.
3
Cross-sectional view of cell illustrating convection rolls
The first intensive experiments were carried out by Benard in 1900. He
experimented on a fluid of thin layer and observed appearance of hexagonal cells
when the instability in the form of convection developed. Rayleigh in 1916
developed the theory which found the condition for the instability with two free
surfaces. He showed that the instability would manifest if the temperature gradient
was large enough so that the so-called Rayleigh number exceeds a certain value
(critical value).
Experiments in the early stage were carried out with fluid heated from bottom
and the top surface is open to atmosphere. Thus the top surface is free to move and
deform. It was later (around 1960) realized that this could lead to another instability
mechanism (Benard-Marangoni convection) due to gradient in surface tension. This
mechanism coexists with the Rayleigh's mechanism but dominates in thin layer.
Most of the findings reported by Benard were actually due this latter instability
mechanism. The instabilities driven by surface tension decreases as the layer
becomes thicker. Later experiments on thermal convection (with or without free
upper surface) have obtained convective cells of many forms such as rolls, square
and hexagons.
In the case of Rayleigh-Benard convection, only buoyancy is responsible for the
appearance of convection cells. The initial movement is the upwelling of warmer
liquid from the heated bottom layer. In case of a free liquid surface in contact with
air, surface tension effect will play a role besides buoyancy. It is known that liquids
flow from places of lower surface tension to places of higher surface tension. This
is called the Marangoni effect. When applying heat from below, the temperature at
4
the top layer will show temperature fluctuations. With increasing temperature,
surface tension decreases. Thus a lateral flow of liquid at the surface will take place,
from warmer areas to cooler areas. In order to preserve a horizontal (or nearly
horizontal) liquid surface, liquid from the cooler places on the surface have to go
down into the liquid. Thus the driving force of the convection cells is the
downwelling of liquid. Marangoni convection plays an important role in Benard
convection in shallow fluid layers, in chemical engineering as well as in crystal
growth and other materials processing technologies.
Assorted Constraints In quite a few heat transfer problems, suppressing or augmenting the convection
plays a vital role. There are several mechanisms that can be used effectively to
either delay or advance the convection, namely, by applying a magnetic/electric
field externally or by Coriolis force due to rotation or by maintaining non-uniform
temperature gradient across the porous layer. A non-uniform temperature gradient
can arise in various ways, notably by (i) transient heating or cooling at a boundary
(ii) volumetric distribution of heat sources (iii) radiative heat transfer (iv) thermal
modulation (v) vertical throughflow and (vi) chemical reaction.
Chemical Reaction
When a horizontal layer of a fluid layer is heated from below, the gradient in
density leads to the onset of convection. Chemical reaction can also be the driving
force of convection. Indeed, if the density of the product is different from the
density of reactant, an isothermal reaction can result in free convection. Secondly, if
the reaction is accompanied by heat effects, the distributed heat source/sink can
cause convection. Chemically driven convective instability in a porous medium is
of practical importance because reactions of these types are common in various
5
electrochemical processes and in the oxidation of fine solids (Frank-Kamenetskii,
1969).
Couple-Stress Fluid Although the problem of Rayleigh–Benard convection has been extensively
investigated considering Newtonian fluids, relatively little attention has been
devoted to the thermal convection of non-Newtonian fluids. With the growing
importance of non-Newtonian fluids with suspended particles in modern technology
and industries, the investigation of such fluids is desirable. The study of such fluids
has applications in a number of processes that occur in industry, such as the
extrusion of polymer fluids, solidification of liquid crystals, cooling of a metallic
plate in a bath, exotic lubrication, and colloidal and suspension solutions. In the
category of non-Newtonian fluids couple-stress fluids have distinct features such as
polar effects. Couple-stress fluid theory developed by Stokes (1966, 1984) is one
among the polar fluid theories which considers couple stresses in addition to the
classical Cauchy stress. It is the simplest generalization of the classical theory of
fluids which allows for polar effects such as the presence of couple stresses and
body couples in the fluid medium.
Goal The intent of this work is to examine the problem of the effect of chemical
reaction on Rayleigh–Benard convection in a couple-stress fluid saturated densely
packed horizontal porous layer with the following scope in mind:
The study of convection in a fluid saturated porous medium has attracted
considerable interest in recent years because of its significance in many practical
fields such as chemical engineering, geothermal activities, oil recovery techniques
and biological process. It is also of practical interest in the extraction of geothermal
6
energy and more specifically in understanding the mechanism of transfer of heat
from aquifers in the deep interior of earth to shallow depths (Wooding, 1960). The
effective mixing process in petroleum reservoirs, regarded as fixed bed reactor, is
achieved by thermal convection. On the other hand, the problems concerning couple
stress fluids in porous media have been extremely important owing to several
engineering applications within fields such as chemical engineering, thermal
insulation systems, and nuclear waste management.
The problem of convection in a couple-stress fluid saturated porous medium has
been extensively studied. However, attention has not been given to the study of
Rayleigh–Benard convection in a couple-stress fluid saturating porous media with
chemical reaction. Therefore, the objective of the work is to investigate
theoretically the effect of chemical reaction on Rayleigh–Benard convection in a
densely packed porous medium saturated with a couple-stress fluid with emphasis
on how the stability criterion for the onset of convection is modified in the presence
of chemical reaction.
7
CHAPTER II
LITERATURE REVIEW
The main objective of this work is to deal with Rayleigh–Benard Convection in a
couple-stress fluid saturated densely packed horizontal porous layer in the presence
of chemical reaction. Literature pertinent to this is classified as follows.
• Rayleigh-Bénard convection (RBC) in fluids.
• Convection in a porous medium.
• Convection with chemical reaction.
• Convection in couple-stress fluids.
The relevant literature on the problem at hand is briefly discussed below in
keeping with the classifications above.
2.1 Rayleigh-Bénard Convection in Fluids
Natural convection in a horizontal layer of fluid heated from below and cooled
from above has been the subject of investigation for many decades owing to its
implications for the control and exploitation of many physical, chemical and
biological processes. We now make a brief review of the RBC problem keeping in
mind the objective and scope of the thesis.
The earliest experiment which called attention to the thermal instability was
briefly reported by Thompson (1882). Benard (1901) later presented a much more
complete description of the development of the convective flow. Lord Rayleigh
(1916) was the first to study the problem theoretically and aimed at determining the
conditions delineating the breakdown of the quiescent state. As a result, the thermal
instability situation described in the foregoing paragraph is referred to as Rayleigh-
8
Bénard convection (RBC). The Rayleigh theory was generalized and extended to
consider several boundary combinations by Jeffreys (1926), Low (1929) and
Sparrow et al. (1964). Chandra (1938) examined the RBC problem experimentally
for a gas. The most complete theory of the thermal instability problem was
presented by Pellew and Southwell (1940).
Malkus and Veronis (1958) investigated finite amplitude cellular convection
and determined the form and amplitude of convection by expanding the nonlinear
equations describing the fields of motion and temperature in a sequence of
inhomogeneous linear equations. Veronis (1959) studied finite amplitude cellular
convection in a rotating fluid and showed that the fluid becomes unstable to finite
amplitude disturbances before it becomes unstable to infinitesimal perturbations.
Palm (1960) showed that for a certain type of temperature-dependence of
viscosity, the critical Rayleigh number and the critical wavenumber are smaller than
those for constant viscosity and explained the observed fact that steady hexagonal
cells are formed frequently at the onset of convection.
Lorenz (1963) solved a simple system of deterministic ordinary nonlinear
differential equations representing cellular convection numerically. For those
systems with bounded solutions, it is found that non-periodic solutions are unstable
with respect to small modifications and that slightly differing initial states can
evolve into considerably different states.
Veronis (1966) analyzed the two-dimensional problem of finite amplitude
convection in a rotating layer of fluid by considering the boundaries to be free.
Using a minimal representation of Fourier series, he showed that, for a restricted
range of Taylor number, steady finite amplitude motions can exist for values of the
Rayleigh number smaller than the critical value required for overstability. Veronis
(1968) also examined the effect of a stabilizing gradient of solute on thermal
9
convection using both linear and finite amplitude analysis. It is found that the onset
of instability may occur as an oscillatory motion because of the stabilizing effect of
the solute in the case of linear theory and that finite amplitude instability may occur
first for fluids with a Prandtl number somewhat smaller than unity.
Krishnamurthy (1968a, b) presented a nonlinear theory of RBC problem and
discussed the formation of hexagonal cells and the existence of subcritical
instabilities. Torrance and Turcotte (1971) investigated the influence of large
variations of viscosity on convection in a layer of fluid heated from below.
Solutions for the flow and temperature fields were obtained numerically assuming
infinite Prandtl number, free-surface boundary conditions and two-dimensional
motion. The effect of temperature-dependent and depth-dependent viscosity was
studied motivated by the convective heat transport in earth’s mantle.
Busse (1975) considered the interaction between convection in a horizontal fluid
layer heated from below and an ambient vertical magnetic field. It is found that
finite amplitude onset of steady convection becomes possible at Rayleigh numbers
considerably below the values predicted by linear theory.
Booker (1976) investigated experimentally the heat transport and structure of
convection in a high Prandtl number fluid whose viscosity varies by up to a factor
of 300 between the boundary temperatures. Horne and Sullivan (1978) examined
the effect of temperature-dependent viscosity and thermal expansion coefficient on
the natural convection of water through permeable formations. They found that the
convective motion is unstable at even moderate values of the Rayleigh number and
exhibits a fluctuating convective state analogous to the case of a fluid with constant
viscosity and coefficient of thermal expansion.
Carey and Mollendorf (1980) presented a regular perturbation analysis for
several laminar natural convection flows in liquids with temperature-dependent
10
viscosity. Several interesting variable viscosity trends on flow and transport are
suggested by the results obtained. Stengel et al. (1982) obtained, using a linear
stability theory, the viscosity-ratio dependences of the critical Rayleigh number and
critical wave number for several types of temperature-dependence of viscosity.
Richter et al. (1983) showed, by an experiment with temperature-dependent
viscosity ratio as large as 106, the existence of subcritical convection of finite
amplitude near the critical Rayleigh number. Busse and Frick (1985) analyzed the
problem of RBC with linear variation of viscosity and showed an appearance of
square pattern for a viscosity ratio larger than 2.
White (1988) made an experiment for the fluid with Prandtl number of o(105)
and studied convective instability with several planforms for the Rayleigh number
up to 63000 and the temperature-dependent viscosity ratio up to 1000. He found
that if the viscosity ratio is 50 or 100 and the Rayleigh number is less than 25000,
stable hexagonal and square patterns are formed in a certain range of wavenumber
and that their wavenumbers increase with viscosity ratio. The possibility of multi-
valued solution in the thermal convection problem with temperature-dependent
viscosity has been examined numerically by Hirayama and Takaki (1993).
Tong and Shen (1992) studied high Rayleigh number turbulent convection using
the technique of photon-correlation homodyne spectroscopy to measure velocity
differences at various length scales. The measured power-law exponents are found
to be in excellent agreement with the theoretical predictions.
Massaioli et al. (1993) investigated the probability density function (pdf) of the
temperature field by numerical simulations of Rayleigh-Bénard convection in two
spatial dimensions. The pdf of the temperature has been shown to have exponential
tails, consistently with previous laboratory experiments and numerical simulations.
They also offered a new theoretical explanation for the exponential tail of the pdf.
11
Xi and Gunton (1993) presented a numerical study of the spontaneous formation
of spiral patterns in Rayleigh-Benard convection in non-Boussinesq fluids. They
solved a generalized two-dimensional Swift-Hohenberg equation that includes a
quadratic nonlinearity and coupling to mean flow. They showed that this model
predicts in quantitative detail many of the features observed experimentally in
studies of Rayleigh-Benard convection in CO2 gas. In particular, they studied the
appearance and stability of a rotating spiral state obtained during the transition from
an ordered hexagonal state to a roll state.
Mukutmoni and Yang (1994) reviewed the broad area of flow transitions of
Rayleigh-Benard convection in rectangular enclosures with sidewalls. They looked
into pattern selection for both small and intermediate enclosures.
Kafoussias and Williams (1995) studied, using an efficient numerical technique,
the effect of a temperature-dependent viscosity on an incompressible fluid in steady,
laminar, free-forced convective boundary layer flow over an isothermal vertical
semi-infinite flat plate. It is concluded that the flow field and other quantities of
physical interest are significantly influenced by the viscosity-temperature
parameter. Kafoussias et al. (1998) studied the combined free-forced convective
laminar boundary layer flow past a vertical isothermal flat plate with temperature-
dependent viscosity. The obtained results showed that the flow field is appreciably
influenced by the viscosity variation.
Severin and Herwig (1999) investigated the variable viscosity effect on the
onset of instability in the RBC problem. An asymptotic approach is considered
which provides results that are independent of specific property laws.
Kozhhoukharova et al. (1999) examined the influence of a temperature-dependent
viscosity on the axisymmetric steady thermocapillary flow and its stability with
respect to non-axisymmetric perturbations by means of a linear stability analysis.
12
The onset of oscillatory convection is studied numerically by a mixed Chebyshev-
collocation finite-difference method.
Rogers and Schatz (2000) reported the first observations of superlattices in
thermal convection. The superlattices are selected by a four-mode resonance
mechanism that is qualitatively different from the three-mode resonance responsible
for complex-ordered patterns observed previously in other nonequilibrium systems.
Numerical simulations quantitatively describe both the pattern structure and the
stability boundaries of superlattices observed in laboratory experiments. It is found
that, in the presence of inversion symmetry, superlattices numerically bifurcate
supercritically directly from conduction or from a striped base state.
Rogers et al. (2000) reported on the quantitative observations of convection in a
fluid layer driven by both heating from below and vertical sinusoidal oscillation.
Just above onset, convection patterns are modulated either harmonically or
subharmonically to the drive frequency. It is found that single frequency patterns
exhibit nearly solid-body rotations with harmonic and subharmonic states always
rotating in opposite directions. Further, flows with both harmonic and subharmonic
responses have been found near a co-dimension two point, yielding novel coexisting
patterns with symmetries not found in either single-frequency states.
You (2001) presented a simple method which can be applied to estimate the
onset of natural convection in a fluid with a temperature-dependent viscosity.
Straughan (2002) developed an unconditional nonlinear energy stability analysis for
thermal convection with temperature-dependent viscosity. The nonlinear stability
boundaries are shown to be sharp when compared with the instability thresholds of
linear theory.
Hossain et al. (2002) analyzed the effect of temperature-dependent viscosity on
natural convection flow from a vertical wavy surface using an implicit finite
13
difference method. They have focused their attention on the evaluation of local
skin-friction and the local Nusselt number. Chakraborty and Borkakati (2002)
studied the flow of a viscous incompressible electrically conducting fluid on a
continuously moving flat plate in the presence of uniform transverse magnetic field.
Assuming the fluid viscosity to be an inverse linear function of temperature, the
nature of fluid velocity and temperature is analyzed.
Abraham (2002) investigated the RBC problem in a micropolar ferromagnetic
fluid layer in the presence of a vertical uniform magnetic field analytically. It is
shown that the micropolar ferromagnetic fluid layer heated from below is more
stable as compared with the classical Newtonian ferromagnetic fluid.
Getling and Brausch (2003) studied numerically the evolution of three-
dimensional, cellular convective flows in a plane horizontal layer of Boussinesq
fluid heated from below. It is found that the flow can undergo a sequence of
transitions between various cell types. In particular, two-vortex polygonal cells may
form at some evolution stages, with an annular planform of the upflow region and
downflows localized in both central and peripheral regions of the cells. They also
showed that, if short-wave hexagons are stable, they exhibit a specific, stellate fine
structure.
Rudiger and Knobloch (2003) described the results of direct numerical
simulations of convection in a uniformly rotating vertical cylinder with no-slip
boundary conditions. They used these results to study the dynamics associated with
transitions between states with adjacent azimuthal wave numbers far from onset. In
certain regimes a novel burst-like state is identified and described.
Ma and Wang (2004) studied the bifurcation and stability of the solutions of the
Boussinesq equations, and the onset of the Rayleigh-Benard convection. A
nonlinear theory for this problem is established using a new notion of bifurcation
14
called attractor bifurcation and its corresponding theorem developed recently. This
theory includes the following three aspects. First, the problem bifurcates from the
trivial solution an attractor AR when the Rayleigh number R crosses the first critical
Rayleigh number Rc for all physically sound boundary conditions, regardless of the
multiplicity of the eigenvalue Rc for the linear problem. Second, the bifurcated
attractor AR is asymptotically stable. Third, when the spatial dimension is two, the
bifurcated solutions are also structurally stable and are classified as well. In
addition, the technical method developed provides a recipe, which can be used for
many other problems related to bifurcation and pattern formation.
Sprague et al. (2005) investigated pattern formation in a rotating Rayleigh-
Benard configuration for moderate and rapid rotation in moderate aspect-ration
cavities. While the existence of the Kuppers-Lortz rolls is predicted by the theory at
the onset of convection, square patterns have been observed in physical and
numerical experiments at relatively high rotation rates. In addition to presenting
numerical results produced from the direct numerical simulation of the full
Boussinesq equations, they derived a reduced system of nonlinear PDEs valid for
convection in a cylinder in the rapidly rotating limit.
Yanagisawa and Yamagishi (2005) carried out simulations of the Rayleigh-
Benard convection with infinite Prandtl number and high Rayleigh numbers in the
spherical shell geometry to understand the thermal structure of the mantle and the
evolution of the earth. The analysis reveals that the structural scale of convection
differs between the boundary region and the isothermal core region. The structure
near the boundary region is characterized by the cell type structure constructed by
the sheet-shaped downwelling and upwelling flows, and that of the core region by
the plume type structure which consists of the cylindrical flows.
Ma and Wang (2007) attempted at linking the dynamics of fluid flows with the
structure of these fluid flows in physical space and the transitions of this structure.
15
The two-dimensional Rayleigh-Bénard convection, which serves as a prototype
problem has been given attention and the analysis is based on two recently
developed nonlinear theories: geometric theory for incompressible flows and
bifurcation and stability theory for nonlinear dynamical systems (both finite and
infinite dimensional). They have shown that the Rayleigh-Bénard problem
bifurcates from the basic state to an attractor AR when the Rayleigh number R
crosses the first critical Rayleigh number Rc for all physically sound boundary
conditions, regardless of the multiplicity of the eigenvalue Rc for the linear
problem. In addition to a classification of the bifurcated attractor AR, the structure
of the solutions in physical space and the transitions of this structure are classified,
leading to the existence and stability of two different flows structures: pure rolls and
rolls separated by a cross the channel flow.
Zhou et al. (2007) presented an experimental study of the morphological
evolution of thermal plumes in turbulent thermal convection. They noted that as the
sheet-like plumes move across the plate, they collide and convolute into spiraling
swirls and that these swirls then spiral away from the plates to become
mushroomlike plumes which are accompanied by strong vertical vorticity. The
fluctuating vorticity is found to have the same exponential distribution and scaling
behaviour as the fluctuating temperature.
Barletta and Nield (2009) revisited the classical Rayleigh–Bénard problem in an
infinitely wide horizontal fluid layer with isothermal boundaries heated from below.
The effects of pressure work and viscous dissipation are taken into account in the
energy balance. A linear analysis is performed in order to obtain the conditions of
marginal stability and the critical values of the wave number and of the Rayleigh
number for the onset of convective rolls. Mechanical boundary conditions are
considered such that the boundaries are both rigid, or both stress-free, or the upper
stress-free and the lower rigid. It is shown that the critical value of Ra may be
16
significantly affected by the contribution of pressure work, mainly through the
functional dependence on the Gebhart number and on a thermodynamic Rayleigh
number. While the pressure work term affects the critical conditions determined
through the linear analysis, the viscous dissipation term plays no role in this
analysis being a higher order effect.
Song and Tang (2010) carried out a systematic study of turbulent Rayleigh-
Bénard convection in two horizontal cylindrical cells of different lengths filled with
water. Global heat transport and local temperature and velocity measurements are
made over varying Rayleigh numbers Ra. The scaling behavior of the measured
Nusselt number and the Reynolds number associated with the large-scale circulation
remains the same as that in the upright cylinders. The scaling exponent for the rms
value of local temperature fluctuations, however, is strongly influenced by the
aspect ratio and shape of the convection cell. The experiment clearly reveals the
important roles played by the cell geometry in determining the scaling properties of
convective turbulence.
For detailed descriptions of linear and nonlinear problems of RBC, one may refer
to the books of Chandrasekhar (1961), Gershuni and Zhukhovitsky (1976), Kays
and Crawford (1980), Ziener and Oertel (1982), Platten and Legros (1984), Gebhart
et al. (1988), Getling (1998), Colinet et al. (2001) and Straughan (2004). Chapters
on thermal convection are included in the books by Turner (1973), Joseph (1976),
Tritton (1979) and Drazin and Reid (1981). Reviews of research on convective
instability have been given by Normand et al. (1977), Davis (1987) and
Bodenschatz et al. (2000).
We have so far reviewed the literature pertaining to Rayleigh-Benard
convection. In what follows we review the literature on convective instabilities in
porous media.
17
2.2 Convection in a Porous Medium The early pioneering work concerning flow through porous media began about
one and a half centuries ago (Darcy, 1856). Later Muskat (1937) achieved a high
degree of correlation between theory and experiment and thus paved the way for a
rapid development of studies in flow through porous media. Since then extensive
investigations have been conducted by hydrologists, petroleum geologists, chemical
engineers, geologists, geophysicists, and applied mathematicians on flow
characteristics and heat transfer in a porous medium which covers a broad range of
different fields with wide applications.
The first attempt to derive Darcy’s law from the basic principles of fluid
mechanics was given by Hubbert (1956). Following Hubbert’s approach, Whittekar
(1966) attempted to derive the Darcy’s law by an integration of the Navier-Stokes’
equations about a local representative volume in a porous medium, An important
advance in this approach was made independently by Slattery (1967) and Whittekar
(1967). Both developed a theorem concerning the volume average. This average
theorem served as the basis for a rigorous derivation of macroscopic equations for a
porous medium flow from microscopic equations, which would help in gaining an
insight into the assumptions involved.
The motivation for the study of convective instability in a porous medium, a
widespread phenomenon in nature and with rich technological applications, seems
to have emerged from the similarity of this flow with the usual Rayleigh-Benard
convection. In fact, there is a fairly close analogy between these two from a
phenomenological viewpoint.
The basic assumption involved in the analytical treatment of the study of free
convection in a porous medium is that the Darcy flow model applies. Coupled with
the Boussinesq incompressible fluid model, the Darcy flow assumption leads to a
18
set of linear momentum equations. However, the mathematical problem remains
weakly nonlinear due to the convective heat transport term present in the energy
equation. The Darcy law is valid only when the permeability parameter of the
porous medium is very small ( 35 1010 −− − ). Nevertheless, for values of the
permeability parameter in the range 23 1010 −− − , Darcy model is not valid and
one has to consider the Brinkman (1947) model.
The infinitesimal convection occurring in a horizontal fluid saturated porous
layer heated from below has been extensively studied since the first papers by
Horton and Rogers (1945) and Lapwood (1948). Although a study related to the so-
called geothermal or hot spring areas was made earlier by Einarsson (1942), the
possibility of free convection in a porous medium heated uniformly from below and
its similarity with the Rayleigh-Benard problem was pointed out by Horton and
Rogers (1945) and Lapwood (1948). The much more speculative possibility is that
the earth’s mantle behaves like a porous medium. This idea has been used in a
discussion of earth-quake sources by Frank (1965) and in a model of volcanism by
Elder (1966). Lapwood (1948), using linear theory, determined the criterion for the
onset of convection.
Early experimental studies of Morrison (1947), Morrison et al. (1949), Rogers
and Schilberg (1951) and Rogers et al. (1951) exhibited quantitative disagreement
with the theoretical predictions of Horton and Rogers (1945) and Lapwood (1948)
in the sense that the observed critical temperature gradients were less, by an order
of magnitude, than the predicted results. This gap between theoretical and
experimental results was reduced to some extent by Rogers and Morrison (1950),
Rogers et al. (1951), Morrison and Rogers (1952), Elder (1958) and List (1965) by
means of some adhoc patching up of the theory to allow property variations such as
the variation of viscosity with temperature, nonlinear temperature distribution etc.
(Nield, 1968). List (1965) studied the effect of uniformly distributed heat sources
19
while Gheorghitza (1961) restricted his analysis to infinitesimal disturbances in an
inhomogeneous porous layer. Lapwood’s (1948) problem was greatly extended by
Wooding (1957, 1958, 1959, 1960, 1963) both theoretically and experimentally.
Wooding (1959) investigated the problem of the stability of a viscous liquid, the
density of which increases with height, in a vertical tube with insulating wall
containing porous material. This problem is analogous to that of Taylor (1954) and
analytically simpler than Taylor’s but many of the qualitative physical
characteristics are similar. Wooding (1959) found that the equilibrium of the liquid
is stable provided that the density gradient does not exceed certain value.
The investigation carried out by Wooding (1960) on the onset of convection in a
Hele-Shaw cell with viscous fluid is analogous to the two-dimensional convection
in a horizontal porous layer. Several other experimental studies (Schneider, 1963;
Elder, 1967; Katto and Masuoka, 1967) have been carried out as a satisfactory test
of the Horton-Rogers-Lapwood theory and a good conformity is achieved.
Elder (1967) investigated the problem of steady free convection in a porous
medium experimentally using Schmidt-Miltoverton heat transfer technique and
numerically by a method proposed by himself in an earlier study. He observed that
as in the case of ordinary viscous fluid, the motion exists only when the Rayleigh
number exceeds its critical value and is of multicellular nature which would be
considerably affected by the end effects. In fact, he studies the flow for variety of
boundary conditions. Further he modified the result when the boundary layer
thickness was comparable to the grain size and found a linear asymptotic relation
between the Nusselt number and the Rayleigh number. This relation means that the
amount of heat transferred is independent of the layer thickness and thermal
conductivity of the porous medium. Elder (1967) showed that this relation holds
only when certain condition is satisfied. As the Rayleigh number becomes large, the
sharp temperature gradient regions began to be confined to just near the boundary
20
wall, there may appear an influence of the characteristic dimension of the porous
medium as pointed out by Elder (1967).
The Benard problem from the viscous flow limit to the Darcy’s law limit has
been considered both theoretically and experimentally by Katto and Masuoka
(1967) and theoretically by Walker and Homsy (1977). Katto and Masuoka (1967)
and Mausoka (1972) pointed out that the region over which the criterion for the
onset of convection is applicable is valid up to the magnitude of 32 10−≅
dk . In
particular, they suggested that the thermal diffusivity k which appears in the
Rayleigh number should be treated as conductivity of the medium divided by the
heat capacity per unit mass of the fluid and not that of the medium. They have also
predicted the criterion for the onset of convection in a fluid layer with spherical
fillings.
Mausoka (1972) investigated the heat transfer by free convection in a horizontal
porous layer composed of glass balls and water both theoretically and
experimentally. The theoretical investigation was carried out in a region which
slightly exceeds the critical condition for the onset of convection using the
eigenfunction expansion. The experimental study covers a wide region. The heat
transfer above the critical state is divided into two regions. He showed that the low
Rayleigh number region is a process of transition to the high Rayleigh number
region.
The effect of vertical through flow on the onset of convection in a two
dimensional porous layer bounded laterally by insulated walls was studied by
Sutton (1970) using series expansion method. He showed that at large values of
aspect ratio the critical Rayleigh number approaches the value of 24π as obtained
21
by Lapwood (1948). Further, he found that at large aspect ratio, the critical
Rayleigh number increases with increasing values of dimensionless flow strength.
Beck (1972) examined the effect of lateral walls on the onset of convection in an
enclosed three-dimensional porous medium with heating from below using both the
approaches (Joseph, 1965; Westbrook, 1969) of energy method. Beck (1972)
showed that the lateral walls have little effect on the critical Rayleigh number
except for in very narrow tall boxes. Moreover, he noted that two-dimensional rolls
are invariably preferred whenever the height is not the smallest dimension.
Furthermore, when rolls do form, they are not necessarily parallel to the shortest
side in the case of viscous flow (Davis, 1967).
Natural convection is an important heat transfer mechanism in the technology of
building insulation. From the viewpoint of basic research in heat transfer, the
phenomenon is being studied mainly in terms of simple models of free convection
in rectangular enclosures filled with either Newtonian fluid or with a fluid-saturated
porous medium. The subject of free convection in enclosures is extensive and has
numerous applications to practical engineering situations. A comprehensive review
of free convection heat transfer in enclosures filled with fluid was presented by
Ostrach (1972).
The effect of inclination angle and the aspect ration on the onset of free
convection and heat transfer in an inclined porous layer with differentially heated
side walls was studied both theoretically and experimentally by Bories and
Combarnous (1973). They determined the critical conditions for the transition
between unicellular and polycellular flows.
The equivalent of Gill’s (1966) theory for convection in a vertical porous slot
was reported a few years later by Weber (1975) whose theoretical result for Nusselt
22
number agreed fairly with the experimental data reported by Schneider (1963) and
Klarsfeld (1970).
Bejan (1979) modified the Weber theory fitting the boundary layer solution with
average zero energy flux conditions along the top and bottom walls. The Nusselt
number predicted by the modified theory agreed extremely well with the
experimental data of Schneider (1963) and Klarsfeld (1970) as well as the
numerical heat transfer calculations reported by Bankvall (1974) and Burns et al.
(1977).
Heat transfer by natural convection in a vertical porous layer with horizontal
heat flow has been studied both theoretically and experimentally by Masuoka et al.
(1981). A boundary layer analysis is extended to take account of both the vertical
temperature gradient in the core of the porous layer and the apparent wall film
thermal resistance which is caused by a local increase in porosity in the vicinity of
the wall.
Griffiths (1981) studied the layered double diffusive convection in a porous
medium consisting of glass spheres experimentally. He showed that layered double
diffusive convection of a fluid within a porous medium is possible. A thin diffusive
interface was observed in a Hele-Shaw cell and in a laboratory porous medium,
despite the lack of inertial forces, with salt and sugar or heat and salt as the
diffusing components.
Finite amplitude thermohaline convection in a horizontal porous layer has been
studied by Srimani (1981) using the power integral technique which is based on the
Stuart’s shape assumption. By comparing the results of two-dimensional and three-
dimensional analysis, she concluded that the only stable finite amplitude solutions
of the infinite number of possible steady solutions is the two-dimensional rolls.
23
The stability of a fluid saturated porous layer subject to sudden rise in surface
temperature was investigated by Caltagirone (1980) using linear theory, energy
method and a two-dimensional numerical method. He concluded that linear theory
gives a sufficient condition for instability and the energy method gives a sufficient
condition for stability. Further, he found a good agreement between numerical and
energy method.
Pattern formation in convection in a porous medium is less easily visualized by
shadowgraph techniques because of the difficulties of transmitting light through the
porous medium. Howle et al. (1993) showed how these difficulties can be overcome
by constructing porous media in which the interfaces between solid and liquid are
either parallel or perpendicular to the confining boundaries of the experimental
system. Convection in such a medium can be visualized using conventional
shadowgraph methods, and they compared the stationary flow patterns observed
against measurements of heat transport.
Herron (2000) treated the problem of Rayleigh-Benard convection with internal
heat source and a variable gravity field. For the case of stress-free boundary
conditions, it is proved that the principle of exchange of stabilities holds as long as
the product of gravity field and the integral of the heat source is nonnegative
throughout the layer.
Siddheshwar and Sri Krishna (2001) studied the qualitative effect of nonuniform
temperature gradient on the linear stability analysis of the Rayleigh-Benard
convection problem in a Boussinesquian, viscoelastic fluid-filled, high-porosity
medium numerically using the single-term Galerkin technique. The eigenvalue is
obtained for free-free, free-rigid, and rigid-rigid boundary combinations with
isothermal temperature conditions. Thermodynamics and also the present stability
analysis dictates the strain retardation time to be less than the stress relaxation time
for convection to set in as oscillatory motions in a high-porosity medium.
24
Malashetty and Basavaraja (2002) investigated the effect of time-periodic
temperature/gravity modulation at the onset of convection in a Boussinesq fluid-
saturated anisotropic porous medium using a linear stability analysis. Brinkman
flow model with effective viscosity larger than the viscosity of the fluid is
considered to give a more general theoretical result. The perturbation method is
applied for computing the critical Rayleigh and wave numbers for small amplitude
temperature/gravity modulation. The shift in the critical Rayleigh number is
calculated as a function of frequency of the modulation, viscosity ratio, anisotropy
parameter and porous parameter.
Degan and Vasseur (2003) investigated the effect of anisotropy on the onset of
natural convection heat transfer in a fluid saturated porous horizontal cavity
subjected to nonuniform thermal gradients analytically. The porous layer is heated
from the bottom by a constant heat flux while the other surfaces are being insulated.
The horizontal boundaries are either rigid/rigid or stress-free/stress-free. The
hydrodynamic anisotropy of the porous matrix is considered. The principal
directions of the permeability are oriented in a direction that is oblique to the
gravity. Based on a parallel flow assumption, closed-form solution for the flow and
heat transfer variables, valid for the onset of convection corresponding to
vanishingly small wave number, is obtained in terms of the Darcy–Rayleigh number
Ra, the Darcy number Da, and the anisotropic parameters K* and θ.
Aniss et al. (2005) investigated the convective instability of a horizontal Hele–
Shaw liquid layer subject to a time-varying gradient of temperature. The stationary
component of the temperature gradient is considered either different or equal to
zero. The aspect ratio of the cell is considered smaller than unity. They examined
the effects of temperature oscillations on the onset of convective instability for these
two asymptotic cases. They have shown that for the first regime, modulation of
temperature has no effect on the convective threshold and that in contrast, the
25
second regime presents a competition between the harmonic and subharmonic
modes at the onset of convection.
Malashetty et al. (2005) examined analytically the stability of a horizontal fluid
saturated sparsely packed porous layer heated from below and cooled form above
when the solid and fluid phases are not in local thermal equilibrium. The Lapwood–
Brinkman model is used for the momentum equation and a two-field model is used
for energy equation each representing the solid and fluid phases separately.
Although the inertia term is included in the general formulation, it does not affect
the stability condition since the basic state is motionless. The linear stability theory
is employed to obtain the condition for the onset of convection. The effect of
thermal non-equilibrium on the onset of convection is discussed. It is shown that the
results of Darcy model for the non-equilibrium case can be recovered in the limit as
Darcy number Da → 0. Asymptotic analysis for both small and large values of the
inter phase heat transfer coefficient H is also presented. An excellent agreement is
found between the exact solutions and asymptotic solutions when H is very small.
Hill (2005) performed linear and nonlinear stability analyses of double-diffusive
convection in a fluid saturated porous layer with a concentration based internal heat
source using Darcy’s law.
Zhao and Bau (2006) studied theoretically the ability of linear controllers to
stabilize the conduction state of a saturated porous layer heated from below and
cooled from above. Proportional, suboptimal robust and linear quadratic Gaussian
controllers are considered. As a model system, they examined two-dimensional
convection in a box containing a saturated porous medium, heated from below and
cooled from above. The heating is provided by a large number of individually
controlled heaters. It is found that, by appropriate selection of a controller, one can
minimize, but not eliminate, the controlled linear system’s non-normality.
26
Nield and Kuznetsov (2006) applied the classical Rayleigh–Bénard theory to a
bidisperse porous medium. The linear stability analysis leads to an expression for
the critical Rayleigh number as a function of a Darcy number, two volume
fractions, a permeability ratio, a thermal capacity ratio, a thermal conductivity ratio,
an inter-phase heat transfer parameter and an inter-phase momentum transfer
parameter.
Straughan (2006) showed that the global nonlinear stability threshold for
convection with a thermal nonequilibrium model is exactly the same as the linear
instability boundary. This result is shown to hold for the porous medium equations
of Darcy, Forchheimer or Brinkman. This optimal result is important because it
shows that linearized instability theory has captured completely the physics of the
onset of convection. The equivalence of the linear instability and nonlinear stability
boundaries is also demonstrated for thermal convection in a non-equilibrium model
with the Darcy law, when the layer rotates with a constant angular velocity about an
axis in the same direction as gravity.
Nield (2007) examined the impracticality of MHD convection in a porous
medium. Nield and Kuznetsov (2007) investigated the effects of both horizontal and
vertical hydrodynamic and thermal heterogeneity on the onset of convection in a
horizontal layer of a saturated bidisperse porous medium uniformly heated from
below using linear stability theory for the case of weak heterogeneity. It is found
that the effect of such heterogeneity on the critical value of the Rayleigh number Ra
based on mean properties is of second order if the properties vary in a piecewise
constant or linear fashion.
Bhadauria (2007) studied the effect of temperature modulation on the onset of
double diffusive convection in a sparsely packed porous medium using linear
stability analysis and Brinkman-Forchheimer extended Darcy model. The effect of
permeability and thermal modulation on the onset of double diffusive convection
27
has been studied using Galerkin method and Floquet theory. The critical Rayleigh
number is calculated as a function of frequency and amplitude of modulation,
Vadasz number, Darcy number, diffusivity ratio, and solute Rayleigh number.
Bhadauria and Sherani (2008) studied the effect of temperature modulation on
the onset of thermal convection in an electrically conducting-fluid-saturated porous
medium, which is heated from below and cooled from above. The correction in the
value of the critical Darcy Rayleigh number is calculated as function of amplitude
and frequency of modulation, Darcy Chandrasekhar number, thermal Prandtl
number, magnetic Prandtl number and the Vadasz number Va. It is found that the
effect of temperature modulation on the onset of convection is to advance or delay
the convection, depending on the proper tuning of the frequency of modulation.
Rapaka et al. (2008) described and used the recently developed non-modal
stability theory to compute maximum amplifications possible, optimized over all
possible initial perturbations. The details of three-dimensional spectral calculations
of the governing equations are presented.
Motsa (2008) addressed the problem of double-diffusive convection in a
horizontal layer filled with a fluid in the presence of temperature gradients (Soret
effects) and concentration gradients (Dufour effects). The onset of convection is
studied using linear stability analysis. The critical Rayleigh numbers for the onset of
convection are determined in terms of the governing parameters.
Malashetty et al. (2008) studied double diffusive convection in a fluid-saturated
porous layer heated from below and cooled from above when the fluid and solid
phases are not in local thermal equilibrium, using both linear and nonlinear stability
analyses. The Darcy model with time derivative term is employed as momentum
equation. A two-field model that represents the fluid and solid phase temperature
fields separately is used for energy equation. The onset criterion for stationary,
28
oscillatory and finite amplitude convection is derived analytically. It is found that
small inter-phase heat transfer coefficient has significant effect on the stability of
the system. There is a competition between the processes of thermal and solute
diffusion that causes the convection to set in through either oscillatory or finite
amplitude mode rather than stationary. The nonlinear theory based on the truncated
representation of Fourier series method predicts the occurrence of subcritical
instability in the form of finite amplitude motions. The effect of thermal non-
equilibrium on heat and mass transfer is also brought out.
Malashetty and Heera (2008) studied double diffusive convection in a fluid-
saturated rotating porous layer heated from below and cooled from above, when the
fluid and solid phases are not in local thermal equilibrium, using both linear and
non-linear stability analyses. A two-field model that represents the fluid and solid
phase temperature fields separately is used for energy equation. The onset criterion
for stationary, oscillatory and finite amplitude convection is derived analytically. It
is found that small inter-phase heat transfer coefficient has significant effect on the
stability of the system. There is a competition between the processes of thermal and
solute diffusions that causes the convection to set in through either oscillatory or
finite amplitude mode rather than stationary. The effect of solute Rayleigh number,
porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz
number and Taylor number on the stability of the system is investigated. The non-
linear theory, based on the truncated representation of Fourier series method,
predicts the occurrence of subcritical instability in the form of finite amplitude
motions. The effect of thermal non-equilibrium on heat and mass transfer is also
brought out.
Postelnicu (2008) performed a linear stability analysis in order to analyze the
onset of Darcy–Brinkman convection in a fluid-saturated porous layer heated from
below, by considering the case when the fluid and solid phases are not in local
29
thermal equilibrium. The problem is transformed into an eigenvalue equation which
is solved in a first step by using a one-term Galerkin approach: an explicit
relationship between the Darcy–Rayleigh number based on the fluid properties R
and the horizontal wave number k is obtained. Minimization of R over k is
performed analytically and finally, critical values for R and k are obtained for
various values of the three parameters of the problem, namely the Darcy number D,
the porosity-scaled conductivity ratio γ and the scaled inter-phase heat transfer
coefficient H. In a second step, a general N-terms Galerkin approach is used and
finally comparisons are performed between the results given by these two
approaches.
Zeng et al. (2009) investigated numerically the problem of natural convection in
an enclosure filled with a diamagnetic fluid-saturated porous medium under strong
magnetic field. The Brinkman-Forchheimer extended Darcy model is used to solve
the momentum equations, and the energy equations for fluid and solid are solved
with the Local Thermal Non-Equilibrium (LTNE) model. The results show that the
magnetic force has significant effect on the flow field and heat transfer in a
diamagnetic fluid-saturated porous medium.
Wang and Tan (2009) studied on the basis of Brinkman model, the onset of
double-diffusive (thermosolutal) convection with a reaction term in a horizontal
sparsely packed porous media using the normal mode analysis. Some results of
Darcy model have been recovered as limiting cases.
Mokhtar et al. (2009) studied the problem of Bénard convection in a fluid
saturated porous medium heated from below with non-uniform temperature gradient
under magnetic field. A linear stability analysis is performed to undertake a detailed
investigation. We found that it is possible to delay the onset of Bénard convection
in saturated porous medium with the effect of a cubic state temperature profile and
also by increasing the magnetic field.
30
Falsaperla et al. (2010) considered the problem of thermal convection in a
rotating horizontal layer of porous medium. The porous medium is described by the
equations of Darcy. A novel aspect of this work is to consider boundary conditions
for the temperature of Newton–Robin type with heat flux prescribed as a limiting
case. The effect of rotation is found to be crucial. For the Taylor number small
enough the critical wave number is zero, but it is found that a threshold such that for
Taylor numbers beyond this non-zero critical wave numbers are found. The
threshold is verified via a weakly nonlinear analysis. Finally, a sharp global
nonlinear stability analysis is given.
Vanishree and Siddheshwar (2010) performed a linear stability analysis for
mono-diffusive convection in an anisotropic rotating porous medium with
temperature-dependent viscosity. The Galerkin variant of the weighted residual
technique is used to obtain the eigenvalue of the problem. The effect of Taylor–
Vadasz number and the other parameters of the problem are considered for
stationary convection in the absence or presence of rotation. Oscillatory convection
seems highly improbable. Some new results on the parameters’ influence on
convection in the presence of rotation, for both high and low rotation rates, are
presented.
A comprehensive review of the literature concerning convection in a fluid
saturated porous medium can be found in the books of Ingham and Pop (1998),
Nield and Bejan (2006) and Vafai (2000; 2005).
In what follows we review the literature pertaining to convection in fluids with
chemical reaction.
31
2.3 Convection with Chemical Reaction
Considering two infinite horizontal plates kept at a constant temperature, Jones
(1973) performed numerical stability analysis of a zero order exothermic reaction
and free convection.
Kordylewski and Krajewski (1984) paid attention to the interaction of chemical
reaction and natural convection in a porous medium. Assuming that a zero order
exothermic reaction occurs in the fluid phase and that local thermal equilibrium
exists between the fluid and solid phases, they formulated the problem based on
Darcy’s law along with the Boussinesq approximation. They found that a
sufficiently high Rayleigh number can prevent the system against thermal ignition.
They also concluded that interference between chemical reaction and natural
convection may lead to irregular oscillations similar to those observed in the
classical Darcy-Boussinesq problem.
Gatica et al. (1987) performed stability analysis of an isothermal first order and
non-isothermal zero order reaction in the presence of free convection. Critical
values of the Rayleigh number for both cases were analytically calculated. They
found that the calculated values compare favorably with the numerical simulation of
the full governing equations.
Viljoen and Hlavacek (1987) focused on the analysis of interaction of free
convection and exothermic chemical reaction. Making use of the Boussinesq and
the Darcy approximations, they considered a two-dimensional cavity with insulated
sidewalls and bottom, while the top being kept at a fixed temperature. With the help
of the Fourier expansion combined with a Galerkin approximation and the
continuation algorithm, they determined different branches of stability. They also
found that results of the approximate analysis are supported by the numerical
integration of the full governing nonlinear equations.
32
Farr et al. (1991) investigated the onset of three-dimensional reaction-driven
convection in a porous medium using linear stability theory. Considering cylindrical
and three-dimensional rectangular containers of arbitrary aspect ratios, they solved
numerically the linear stability problem for various parameter values by a
combination of the method of separation of variables and the shooting method.
Important qualitative differences have been pointed out between reaction-driven
convection and the standard Lapwood or Benard convection. They also presented
numerical study of reaction-driven convection in a porous two-dimensional box.
Using the orthogonal collocation and continuation techniques, they determined the
conduction and convective branches.
Vafai et al. (1993) obtained a numerical solution for chemically driven
convection in a porous cavity with isothermal walls at the top and bottom surfaces
and thermally insulated sidewalls. Both the inertia and the viscous forces have been
taken into consideration in the momentum equation.
Malashetty et al. (1994) performed a linear stability analysis to study the onset of
convective instability in a horizontal inert porous layer saturated with a fluid
undergoing zero order exothermic chemical reactions. Assuming two different
thermal boundary conditions at the lower boundary (i.e., an isothermal and adiabatic
conditions), they solved the resulting eigenvalue problem approximately using the
Galerkin method. They have found the critical Rayleigh number and the associate
wave number at a given Frank-Kamenetskii number. It was found that, with
chemical reactions, the fluid in the porous medium is more prone to instability as
compared to the case in which chemical reactions are absent.
Merkin and Chaudhary (1994) discussed the free-convection boundary-layer
flow on a vertical surface which results when there is an exothermic catalytic
chemical reaction on that surface. The system is governed by the two dimensionless
33
chemical parameters and which are measures of the activation energy and heat of
reaction respectively, as well as the Prandtl and Schmidt numbers. A series solution
is obtained valid near the leading edge of the plate and this is continued downstream
by numerical solutions of the full equations.
Churchill and Yu (2006) investigated the effect of the rate of convective heat
transfer on an energetic chemical reaction numerically and coherently. The
combination of the thermicity (the fractional increase in temperature due to the
reaction) and of uniform heating at the wall is shown to produce chaotic variations
of as much as an order of magnitude in the Nusselt number. A theoretically based
expression has been devised for the prediction of this behavior.
Patil and Kulkarni (2008) focused on the study of combined effects of free
convective heat and mass transfer on the steady two-dimensional, laminar, polar
fluid flow through a porous medium in the presence of internal heat generation and
chemical reaction of the first order. The highly nonlinear coupled differential
equations governing the boundary layer flow, heat and mass transfer are solved by
using two-term perturbation method with Eckert number E as perturbation
parameter. The velocity distribution of polar fluids is compared with the
corresponding flow problems for a viscous (Newtonian) fluid and found that the
polar fluid velocity is decreasing.
Alam et al. (2009) carried out an analysis to investigate the effects of variable
chemical reaction, thermophoresis, temperature-dependent viscosity and thermal
radiation on an unsteady MHD free convective heat and mass transfer flow of a
viscous, incompressible, electrically conducting fluid past an impulsively started
infinite inclined porous plate. The governing nonlinear partial differential equations
are transformed into a system of ordinary differential equations, which are solved
numerically using a sixth-order Runge-Kutta integration scheme with Nachtsheim-
Swigert shooting method. Numerical results for the non-dimensional velocity,
34
temperature and concentration profiles as well as the local skin-friction coefficient,
the local Nusselt number and the local Stanton number are presented for different
physical parameters. The results show that variable viscosity significantly increases
viscous drag and rate of heat transfer. The results also show that higher order
chemical reaction induces the concentration of the particles for a destructive
reaction and reduces for a generative reaction.
Sobel et al. (2009) described a simpler model system for visualizing density-
driven pattern formation in an essentially unmixed chemical system: the reaction of
pale yellow Fe3+ with colorless SCN− to form the blood-red Fe(SCN)2+ complex
ion in aqueous solution. Careful addition of one drop of Fe(NO3)3 to KSCN yields
striped patterns after several minutes. The patterns appear reminiscent of Rayleigh-
Taylor instabilities and convection rolls, arguing that pattern formation is caused by
density-driven mixing.
Mahapatra et al. (2010) studied the effect of a chemical reaction on a free
convection flow through a porous medium bounded by a vertical infinite surface.
Velocity, temperature, and concentration profiles have been obtained for different
values of parameters such as the Grashof number, Prandtl number, and the chemical
reaction parameter in the presence of homogeneous chemical reaction of first order.
It is observed that the velocity and concentration increase during a generative
reaction and decrease in a destructive reaction. The same is true for the behavior of
the fluid temperature. The presence of the porous media diminishes the temperature.
In what follows we review the literature concerning the convective instability
problems of couple-stress fluids.
35
2.4 Convection in Couple-Stress Fluids Couple-stress fluid theory developed by Stokes (1966, 1984) is one among the
polar fluid theories which considers couple stresses in addition to the classical
Cauchy stress. It is the simplest generalization of the classical theory of fluids
which allows for polar effects such as the presence of couple stresses and body
couples in the fluid medium. Srivastava (1986) investigated the problem of peristaltic transport of a couple-
stress fluid under a zero Reynolds number and long wavelength approximation. A
comparison of the results with those for a Newtonian fluid model shows that the
magnitude of the pressure rise under a given set of conditions is greater in the case
of the couple-stress fluid. The pressure rise increases as the couple-stress parameter
decreases. The difference between the pressure rise for a Newtonian and a couple-
stress fluid increases with increasing amplitude ratio at zero flow rate. However, it
is found that increasing the flow rate reduces this difference.
Jaw-Ren Lin (1998) presented a theoretical study of squeeze film behaviour for a
finite journal bearing lubricated with couple stress fluids. On the basis of the
microcontinuum theory, the modified Reynolds equation is obtained by using the
Stokes equations of motion to account for the couple stress effects due to the
lubricant blended with various additives. With the Conjugate Gradient Method of
iteration the built-up pressure is calculated, and then applied to predict the squeeze
film characteristics of the system. Compared with the case of a Newtonian
lubricant, the couple stress effects increase the load-carrying capacity significantly
and lengthen the response time of the squeeze film behaviour. On the whole, the
presence of couple stresses improves the characteristics of finite journal bearings
operating under pure squeeze film motion. It is found that the rheological effects of
couple stress fluids agree with previous works.
36
Jaw-Ren Lin (2000) presented, on the basis of the microcontinuum theory, a
theoretical analysis of the effects of couple stresses on the squeeze film behavior
between a sphere and a flat plate. The modified Reynolds equation governing the
squeeze film pressure is derived by using the Stokes constitutive equations to take
an account of the couple stress effects due to the lubricant blended with various
additives. According to the results obtained, the influence of couple stresses
signifies an enhancement in the film pressure. On the whole, the couple stress
effects characterized by the couple stress parameter produce an increase in value of
the load-carrying capacity and the response time as compared to the classical
Newtonian-lubricant case. It improves the squeeze film characteristics of the
system.
Sharma et al. (2000) considered a layer of a couple-stress fluid heated from
below in a porous medium to include the effect of uniform rotation. For the case of
stationary convection, the couple stress may hasten the onset of convection in the
presence of rotation, while in the absence of rotation, it always postpones the onset
of convection. For the case of stationary convection, rotation postpones the onset of
convection. Graphs have been plotted by giving numerical values to the parameters
to depict the stability characteristics. Rotation is found to introduce oscillatory
modes in the system, which were non-existent in its absence. A sufficient condition
for the non-existence of overstability is obtained.
Sharma and Thakur (2000) considered a layer of electrically conducting couple-
stress fluid heated from below in a porous medium in the presence of magnetic
field. For stationary convection, the couple-stress and magnetic field postpone the
onset of convection, whereas the medium permeability hastens the onset of
convection. The magnetic field introduces oscillatory modes in the system, which
were non-existent in its absence. A sufficient condition for the non-existence of
overstability is obtained.
37
Sharma and Shivani (2001a) considered a layer of couple-stress fluid heated
from below in a porous medium. Using linearized stability theory and normal mode
analysis, the dispersion relation is obtained. For stationary convection, the couple-
stress postpones the onset of convection, whereas the medium permeability hastens
the onset of convection. The principle of exchange of stabilities is valid for the
couple-stress fluid heated from below in porous medium.
Sharma and Shivani (2001b) considered a layer of electrically conducting
couple-stress fluid heated from below in porous medium in the presence of uniform
horizontal magnetic field. They found that the medium permeability hastens the
onset of convection, whereas the magnetic field and couple-stress postpone the
onset of convection for the case of stationary convection. The oscillatory modes are
introduced by the magnetic fields which were not present in the absence of the
magnetic field. The overstable case has been considered and a sufficient condition
for the non-existence of overstability is obtained.
Sunil et al. (2002) considered a layer of couple-stress fluid heated from below in
a porous medium in the presence of a uniform vertical magnetic field and uniform
vertical rotation. For the case of stationary convection, the rotation postpones the
onset of convection. The magnetic field and couple-stress may hasten the onset of
convection in the presence of rotation, while in the absence of rotation, they always
postpone the onset of convection. The medium permeability hastens the onset of
convection in the absence of rotation, while in the presence of rotation, it may
postpone the onset of convection. Graphs have been plotted by giving numerical
values to the parameters to depict the stability characteristics. The rotation and
magnetic field are found to introduce oscillatory modes in the system, which were
nonexistent in their absence. A sufficient condition for the nonexistence of
ovestabllity is also obtained.
38
Zakaria (2002) cast the equations of a polar fluid of hydromagnetic fluctuating
through a porous medium into matrix form using the state space and Laplace
transform techniques. The resulting formulation is applied to a variety of problems.
The solution to a problem of an electrically conducting polar fluid in the presence of
a transverse magnetic field and to a problem for the flow between two parallel fixed
plates is obtained. The inversion of the Laplace transforms is carried out using a
numerical approach. Numerical results for the velocity, angular velocity distribution
and the induced magnetic field are illustrated graphically for each problem.
Elsharkawy (2004) presented a mathematical model for the hydrodynamic
lubrication of misaligned journal bearings with couple stress lubricants. A modified
form for Reynolds equation in which the effects of couple stresses arising from the
lubricant blended with various additives is used. The journal misalignment is
allowed to vary in magnitudes as well as in direction with respect to the bearing
boundaries. The flexibility of the bearing liner was incorporated into the analysis by
using the thin liner model. A numerical solution for the mathematical model using a
finite difference scheme is introduced. The predicted performance characteristics
are compared with available theoretical and experimental results. The effects of the
degree and angle of misalignment, the length-to-diameter ratio, and the couple
stress parameter on static performance such as pressure distribution, load-carrying
capacity, friction coefficient, and side leakage flow and misalignment moment are
presented and discussed.
Sharma and Sharma (2004a) considered the thermal instability of a couple-stress
fluid with suspended particles. Following the linear stability analysis and normal
mode analysis, the dispersion relation is obtained. For the case of stationary
convection, couple-stress is found to postpone the onset of convection, whereas
suspended particles hasten it. It is found that the principle of exchange of stabilities
is valid. The thermal instability of a couple-stress fluid with suspended particles, in
39
the presence of rotation and magnetic field, is also considered. The magnetic field
and rotation are found to have stabilizing effects on the stationary convection and
introduce oscillatory modes in the system. A sufficient condition for the
nonexistence of ovestabllity is also obtained.
Sharma and Sharma (2004b) considered a layer of couple-stress fluid, permeated
with suspended particles, heated and soluted from below in a porous medium. The
couple-stress and stable solute gradient postpone the onset of convection, whereas
the medium permeability and suspended particles hasten the onset of convection.
The principle of exchange of stabilities is valid for the couple-stress fluid permeated
with suspended particles heated from below in porous medium. The oscillatory
modes are introduced due to the presence of stable solute gradient.
Siddheshwar and Pranesh (2004) investigated the effect of Raleigh–Benard
situation in Boussinesq–Stokes suspensions using both linear and nonlinear stability
analyses. The linear and nonlinear analyses are based on a normal mode solution
and minimal representation of double Fourier series respectively. The effect of
suspended particles on convection is delineated against the background of the
results of the clean fluid. The realm of nonlinear convection warrants the
quantification of heat transfer and this has been achieved on the Rayleigh–Nusselt
plane. Possibility of aperiodic convection is also discussed.
Sunil et al. (2004) considered a layer of couple-stress fluid permeated with
suspended particles, heated and soluted from below in a porous medium. For the
case of stationary convection, the stable solute gradient and couple-stress have
stabilizing effect on the onset of convection, whereas the suspended particles and
medium permeability have destabilizing effect on the couple-stress fluid permeated
with suspended particles. Graphs have been plotted by giving numerical values to
the parameters to depict the stability characteristics. The stable solute gradient is
40
found to introduce oscillatory modes in the system, which are nonexistent in its
absence. A sufficient condition for the nonexistence of overstability is obtained.
Liao et al. (2005) attempted to provide the dynamic characteristics of long
journal bearings lubricated with couple stress fluids. Based upon the micro-
continuum theory generated by Stokes, the dynamic Reynolds-type equation
governing the film pressure is derived to account for the couple stress effects
resulting from the non-Newtonian behavior of complex fluids. By applying the
linear stability theory to the non-linear equations of motion the journal rotor, the
equilibrium positions and dynamic characteristics of the system are evaluated. As
compared to the classical Newtonian model, the effects of couple stresses signify
enhanced stiffness and damping coefficients at moderate values of the steady
eccentricity ratio. It is found that long bearings lubricated with couple stress fluids
under small disturbance results in a higher stability threshold speed than that of the
Newtonian-lubricant case.
Sarangi et al. (2005a) extended conventional elastohydrodynamic lubrication
(EHL) analysis of point contacts to include couple-stress effects in lubricants
blended with polymer additives. A transient pressure differential equation, generally
referred to as a modified Reynolds equation, is derived from the Stokes
microcontinuum theory and solved using the finite difference method with a
successive over-relaxation scheme. The solution is obtained under isothermal
conditions, assuming a suitable exponential relation of pressure-viscosity variation.
A non-dimensional couple-stress parameter, which can be considered the molecular
length of the additives in the lubricant, is used in the analysis. From the results
obtained, the influence of the couple-stress parameter on the EHL point contacts is
apparent and cannot be neglected. Lubricants with couple stresses provide an
increase in the load-carrying capacity and reduction in friction coefficient as
compared to Newtonian lubricants. Empirical formulas for the calculation of central
and minimum film thicknesses of lubricated point contacts with couple-stress fluids
41
are derived with the nonlinear least-squares curve-fitting technique using different
numerically evaluated data.
Sarangi et al. (2005b) evaluated numerically stiffness and damping coefficients
of isothermal elastohydrodynamically lubricated point-contact problems
numerically with couple-stress fluids. A set of equations under steady-state and
dynamic conditions is derived from the modified Reynolds equation using a
linearized perturbation method. This paper is the second part of the present study;
the modified Reynolds equation derived from the Stokes micro-continuum theory is
used in the previous article. Dynamic pressures are found after solving the set of
perturbed equations using the previously obtained steady-state pressure from the
modified Reynolds equation. The stiffness and damping coefficients of the film are
determined using the dynamic pressures. Then the overall stiffness and damping
matrices of the ball bearing are obtained from load distribution, coordinate
transformation, and compatibility relations. The bearing coefficients are introduced
into a rotor system to simulate the response. It has been observed that the influence
of couple-stress fluids on the dynamics of a rotor supported on lubricated ball
bearings is marginal; hence, Newtonian theory can be used instead for simplicity.
However, with increasing content of polymer additives in lubricant, for an accurate
analysis the effect of couple stresses in a fluid should not be neglected.
Sharma and Mehta (2005) paid attention to a layer of compressible, rotating,
couple-stress fluid heated and soluted from below. For the case of stationary
convection, the compressibility, stable solute gradient and rotation postpone the
onset of convection, whereas the couple-stress viscosity postpones as well as
hastens the onset of convection depending on rotation parameter. The case of
overstability is also studied wherein a sufficient condition for the non-existence of
overstability is found.
42
Ezzat et al. (2006) introduced a magnetohydrodynamic model of boundary-layer
equations for a perfectly conducting couple-stress fluid. This model is applied to
study the effects of free convection currents with thermal relaxation on the flow of a
polar fluid through a porous medium, which is bounded by a vertical plane surface.
The state space formulation is introduced. The resulting formulation, together with
the Laplace transform technique, are applied to a variety of problems. The solution
to a thermal shock problem and to the problem of the flow in the whole space with a
plane distribution of heat sources are obtained. It is also applied to a semi-space
problem with a plane distribution of heat sources located inside the fluid. A
numerical method is employed for the inversion of the Laplace transforms. The
effects of Grashof number, material parameters, Alfven velocity, relaxation time,
Prandtl number and the permeability parameter on the velocity, the temperature and
the angular velocity distributions are discussed. The effects of cooling and heating
of a couple-stress fluid have also been discussed.
Malashetty et al. (2006) examined the double diffusive convection in a two-
component couple stress liquid layer with Soret effect using both linear and non-
linear stability analyses. The linear theory is based on normal mode technique and
the non-linear analysis is based on a minimal representation of double Fourier
series. The effect of couple stress parameter, the Soret parameter, the solute
Rayleigh number, the Prandtl number and the diffusivity ratio on the stationary,
oscillatory and finite amplitude convection are shown graphically. It is found that
the effects of couple stress are quite large and the positive Soret number enhances
the stability, while the negative Soret number enhances the instability. The
nonlinear theory predicts that finite amplitude motions are possible only for
negative Soret parameter. The transient behaviour of thermal and solute Nusselt
numbers has been investigated by solving numerically a fifth order Lorenz model
using Runge–Kutta method.
43
Naduvinamani and Kashinath (2006) studied the effect of surface roughness on
the performance of curved pivoted slider bearings. A more general type of surface
roughness is mathematically modelled by a stochastic random variable with nonzero
mean, variance and skewness. The averaged modified Reynolds type equation is
derived on the basis of Stokes microcontinuum theory for couple stress fluids. The
closed-form expressions for the mean pressure, load-carrying capacity, frictional
force and the centre of pressure are obtained. Numerical computations show that the
performance of the slider bearing is improved by the use of lubricants with
additives (couple stress fluid) as compared to Newtonian lubricants. Further, it is
observed that the negatively skewed surface roughness increases the load-carrying
capacity and frictional force and reduces the coefficient of friction, whereas the
positively skewed surface roughness on the bearing surface adversely affects the
performance of the pivoted slider bearings.
Gaikwad et al. (2007) studied the onset of double diffusive convection in a two
component couple stress fluid layer with Soret and Dufour effects using both linear
and non-linear stability analysis. The linear theory depends on normal mode
technique and non-linear analysis depends on a minimal representation of double
Fourier series. The effect of couple stress parameter, the Soret and Dufour
parameters, and the Prandtl number on the stationary and oscillatory convection are
presented graphically. The Dufour parameter enhances the stability of the couple
stress fluid system in case of both stationary and oscillatory mode. The effect of
positive Soret parameter is to destabilize the system in case of stationary mode
while it stabilizes the system in case of oscillatory mode. The negative Soret
parameter enhances the stability in both stationary and oscillatory mode. The couple
stress parameter enhances the stability of the system in both stationary and
oscillatory modes. The Dufour parameter increases the heat transfer, while the
couple stress parameter has reverse effect. The Soret parameter has negligible
influence on heat transfer. Both Dufour and Soret parameters increase the mass
44
transfer, while the couple stress parameter has dual effect depending on the value of
the Rayleigh number.
Lewicka (2007) investigated the Stokes–Boussinesq equations in a slanted (that
is, not aligned with gravity's direction) cylinder of any dimension and with an
arbitrary Rayleigh number. The author proved the existence of a non-planar
traveling wave solution, propagating at a constant speed, and satisfying the
Dirichlet boundary condition in the velocity and the Neumann condition in the
temperature.
Jaw-Ren Lin and Chi-Ren Hung (2007) presented on the basis of the Stokes
micro-continuum theory together with the averaged inertia principle, the combined
effects of non-Newtonian couple stresses and convective fluid inertia forces on the
squeeze film motion between a long cylinder and an infinite plate. A closed-form
solution has been derived for squeeze film characteristics including the film
pressure, the load capacity and the response time. Comparing with the Newtonian-
lubricant non-inertia case, the combined effects of couple stresses and convective
inertia forces provide an increase in the film pressure, the load capacity and the
response time. In addition, the quantitative effects of couple stresses and convective
inertia forces are more pronounced for cylinder–plate system operating at a larger
couple stress parameter and film Reynolds number, as well as a smaller squeeze
film height. To guide the use of the present study, a numerical example is also
illustrated for engineers when considering both the effects of non-Newtonian couple
stresses and fluid convective inertia forces.
Naduvinamani and Siddangouda (2007) presented the theoretical study of the
effect of couple stresses on the optimum lubrication characteristics of porous
Rayleigh step bearings. The lubricant with additives in the film region and also in
the porous region is modelled as the Stokes couple-stress fluid. Modified Darcy
type equations accounting for the polar effects in the porous region is considered
45
and the generalized Reynolds type equation is derived for the lubrication of the
porous Rayleigh step bearings. Exact solution of the generalized Reynolds type
equation is obtained and the closed form expressions for the bearing characteristics
are presented. Performance characteristics of the porous Rayleigh step bearing are
presented for the various values of the non-dimensional parameters such as the
couple-stress parameter and the permeability parameter. It is observed that the
couple-stress fluid lubricants provide the increased load carrying capacity and the
decreased coefficient of friction as compared to the corresponding Newtonian case.
This theory suggests that, adverse effects of the presence of porous facing on the
stator could be compensated with proper selection of lubricants with proper
additives.
Devakar and Iyengar (2008) considered Stokes’ first and second problems for an
incompressible couple-stress fluid under isothermal conditions. The problems are
solved through the use of Laplace transform technique. Inversion of the Laplace
transform of the velocity component in each case is carried out using a standard
numerical approach. Velocity profiles are plotted and studied for different times and
different values of couple stress Reynolds number. The results are presented
through graphs in each case.
Yan-yan (2008) derived, to take into account the couple stress effects, a
modified Reynolds equation for dynamically loaded journal bearings with the
consideration of the elasticity of the liner. The numerical results show that the
influence of couple stresses on the bearing characteristics is significant. Compared
with Newtonian lubricants, lubricants with couple stresses increase the fluid film
pressure, as a result enhance the load-carrying capacity and reduce the friction
coefficient. However, since the elasticity of the liner weakens the couple stress
effect, elastic liners yield a reduction in the load-carrying capacity and an increase
in the friction coefficient. The elastic deformation of the bearing liner should be
considered in an accurate performance evaluation of the journal bearing.
46
Indira et al. (2008) considered flow of couple-stress fluid flowing in an eccentric
annulus. This study has its importance whenever simultaneous flow of two fluids
has to be considered. The eccentric annulus in the domain D bounded internally by
C1 and externally by C2 is mapped onto a concentric annulus bounded internally by
1 and externally by 2 using a conformal mapping. A closed form solution is
obtained. Two dimensional velocity profile is plotted for different couple-stress
parameters, area of cross section and eccentricity parameter. Numerical
computation reveals that the use of eccentric annulus facilitates transport of more
fluid. The rate of flow increases as eccentricity increases. Rate of flow increases
with decrease in the couple stress parameter.
Aggarwal and Suman Makhija (2009) examined theoretically the thermal
stability of a couple-stress fluid in the presence of magnetic field and rotation.
Following the linear stability theory and normal mode analysis, the dispersion
relation is obtained. For stationary convection, rotation has stabilizing effect
whereas couple stresses in fluid and magnetic field have stabilizing effect under
certain conditions. It is found that principle of exchange of stabilities is satisfied in
the absence of magnetic field and rotation. The sufficient conditions for the non
existence of overstability are also obtained.
Crosby and Chetti (2009) studied static and dynamic characteristics of two-lobe
journal bearings lubricated with couple-stress fluids. The load-carrying capacity, the
stiffness and damping coefficients, the non-dimensional critical mass, and the whirl
ratio are determined for various values of the couple stress parameter l. The results
obtained are compared with the characteristics of two-lobe bearings lubricated with
Newtonian fluids. It is found that the effect of the couple stress parameter is very
significant on the performance of the journal bearing and that the stability is
improved compared to bearings lubricated with Newtonian fluids.
47
Malashetty et al. (2009) investigated the stability of a couple stress fluid
saturated horizontal porous layer heated from below and cooled from above when
the fluid and solid phases are not in local thermal equilibrium. The Darcy model is
used for the momentum equation and a two-field model is used for energy equation
each representing the solid and fluid phases separately. The linear stability theory is
employed to obtain the condition for the onset of convection. The effect of thermal
non-equilibrium on the onset of convection is discussed. It is shown that the results
of the thermal non-equilibrium Darcy model for the Newtonian fluid case can be
recovered in the limit as couple stress parameter C→0. They also presented
asymptotic analysis for both small and large values of the inter phase heat transfer
coefficient H. They found an excellent agreement between the exact solutions and
asymptotic solutions when H is very small.
Pardeep Kumar and Mahinder Singh (2009) considered the thermosolutal
instability of couple-stress fluid in the presence of uniform vertical rotation.
Following the linear stability theory and normal mode analysis, the dispersion is
obtained. For the case of stationary convection, the stable solute gradient and
rotation have stabilizing effects on the system, whereas the couple-stress has both
stabilizing and destabilizing effects. The dispersion relation is also analyzed
numerically. The stable solute gradient and the rotation introduce oscillatory modes
in the system, which did not occur in their absence. The sufficient conditions for the
non-existence of overstability are also obtained.
Srinivasacharya et al. (2009) considered an incompressible laminar flow of a
couple-stress fluid in a porous channel with expanding or contracting walls.
Assuming symmetric injection or suction along the uniformly expanding porous
walls and using similarity transformations, the governing equations are reduced to
nonlinear ordinary differential equations. The resulting equations are then solved
numerically using quasi-linearization technique. The graphs for velocity
48
components and temperature distribution are presented for different values of the
fluid and geometric parameters.
Umavathi et al. (2009) presented an analytical solution for fully developed
laminar flow between vertical parallel plates filled with 2 immiscible viscous and
couple stress fluids in a composite porous medium. The flow in the porous medium
is modeled using the Brinkman equation. The viscous and Darcy dissipation terms
are included in the energy equation. The transport properties of the fluids in both
regions are assumed to be constant. The continuity conditions for the velocity,
temperature, shear stress, and the heat flux at the interfaces between the couple
stress permeable fluid layer and the viscous fluid layer are assumed. The influence
of physical parameters on the flow, such as the couple stress parameter, porous
parameter, Grashof number, viscosity ratio and conductivity ratio, are evaluated,
and a set of graphical results is presented. An interesting and new approach is
incorporated to analyze the flow for strong, weak, and comparable porosity
conditions with the couple stress fluid parameter.
Devakar and Iyengar (2010) considered the flow of an incompressible fluid
between two parallel plates, initially induced by a constant pressure gradient. After
steady state is attained, the pressure gradient is suddenly withdrawn, while the
plates are impulsively started simultaneously. The arising flow is referred to as run
up flow and the present paper aims at studying this flow in the context of a couple
stress fluid. Using Laplace transform technique, the expression for velocity is
obtained in Laplace transform domain which is later inverted to the space time
domain using a numerical approach. The variation of velocity with respect to
various flow parameters is presented through graphs.
Mahinder Singh and Pardeep Kumar (2010) considered the problem of thermal
instability of compressible, electrically conducting couple-stress fluids in the
presence of a uniform magnetic field. Following the linear stability theory and
49
normal mode analysis, the dispersion relation is obtained. For stationary convection,
the compressibility, couple-stress, and magnetic field postpone the onset of
convection. Graphs have been plotted by giving numerical values of the parameters
to depict the stability characteristics. The principle of exchange of stabilities is
found to be satisfied. The magnetic field introduces oscillatory modes in the system
that were non-existent in its absence. The case of overstability is also studied
wherein a sufficient condition for the non-existence of overstability is obtained.
Malashetty et al. (2010) studied the onset of double-diffusive convection in a
couple-stress fluid-saturated horizontal porous layer using linear and weak
nonlinear stability analyses. The modified Darcy equation that includes the time
derivative term and the inertia term is used to model the momentum equation. The
expressions for stationary, oscillatory and finite-amplitude Rayleigh number are
obtained as a function of the governing parameters. The effect of couple-stress
parameter, solute Rayleigh number, Vadasz number and diffusivity ratio on
stationary, oscillatory and finite-amplitude convection is shown graphically. It is
found that the couple-stress parameter and the solute Rayleigh number have a
stabilizing effect on stationary, oscillatory and finite-amplitude convection. The
diffusivity ratio has a destabilizing effect in the case of stationary and finite-
amplitude modes, with a dual effect in the case of oscillatory convection. The
Vadasz number advances the onset of oscillatory convection. The heat and mass
transfer decreases with an increase in the values of couple-stress parameter and
diffusivity ratio, while both increase with an increase in the value of the solute
Rayleigh number.
50
2.5 Plan of Work The dissertation is organized as follows:
The Third Chapter consists of basic equations, approximations, boundary
conditions and a discussion of the dimensionless parameters. In Chapter IV, linear
stability analysis of Rayleigh–Benard convection in a couple-stress fluid saturated
densely packed horizontal porous layer in the presence of chemical reaction is
studied. The linear stability analysis is based on a normal mode technique. The
Galerkin technique is used to solve the variable coefficient system of differential
equations. The results are discussed in Chapter V with the help of figures. An
exhaustive bibliography follows this last chapter.
51
CHAPTER III
BASIC EQUATIONS, BOUNDARY CONDITIONS AND DIMENSIONLESS PARAMETERS
Like any mathematical model of the real world, fluid mechanics makes some
basic assumptions about the materials being studied. These assumptions are turned
into equations that must be satisfied if the assumptions are to hold true. For
example, consider an incompressible fluid in three dimensions. The assumption that
mass is conserved means that for any fixed closed surface (such as a sphere) the rate
of mass passing from outside to inside the surface must be the same as rate of mass
passing the other way. (Alternatively, the mass inside remains constant, as does the
mass outside). This can be turned into an integral equation over the surface.
Fluid mechanics assumes that every fluid obeys the following:
• Conservation of mass
• Conservation of momentum
• The continuum hypothesis, detailed below.
Further, it is often useful (and realistic) to assume a fluid is incompressible - that
is, the density of the fluid does not change. Liquids can often be modelled as
incompressible fluids, whereas gases cannot. Similarly, it can sometimes be
assumed that the viscosity of the fluid is zero (the fluid is inviscid). Gases can often
be assumed to be inviscid. If a fluid is viscous, and its flow contained in some way
(e.g. in a pipe), then the flow at the boundary must have zero velocity. For a viscous
fluid, if the boundary is not porous, the shear forces between the fluid and the
boundary results also in a zero velocity for the fluid at the boundary. This is called
the no-slip condition. For a porous media otherwise, in the frontier of the containing
vessel, the slip condition is not zero velocity, and the fluid has a discontinuous
52
velocity field between the free fluid and the fluid in the porous media (this is related
to the Beavers and Joseph condition).
Fluids are composed of molecules that collide with one another and solid
objects. The continuum assumption, however, considers fluids to be continuous.
That is, properties such as density, pressure, temperature, and velocity are taken to
be well-defined at "infinitely" small points, defining a REV (Reference Element of
Volume), at the geometric order of the distance between two adjacent molecules of
fluid. Properties are assumed to vary continuously from one point to another, and
are averaged values in the REV. The fact that the fluid is made up of discrete
molecules is ignored.
The continuum hypothesis is basically an approximation, in the same way
planets are approximated by point particles when dealing with celestial mechanics,
and therefore results in approximate solutions. Consequently, assumption of the
continuum hypothesis can lead to results which are not of desired accuracy. That
said, under the right circumstances, the continuum hypothesis produces extremely
accurate results.
Those problems for which the continuum hypothesis does not allow solutions of
desired accuracy are solved using statistical mechanics. To determine whether or
not to use conventional fluid dynamics or statistical mechanics, the Knudsen
number is evaluated for the problem. The Knudsen number is defined as the ratio of
the molecular mean free path length to a certain representative physical length
scale. This length scale could be, for example, the radius of a body in a fluid. (More
simply, the Knudsen number is how many times its own diameter a particle will
travel on average before hitting another particle). Problems with Knudsen numbers
at or above unity are best evaluated using statistical mechanics for reliable
solutions.
53
3.1 Basic Equations
When a fluid permeates a porous material, the actual path of an individual fluid
particle cannot be followed analytically. The gross effect, as the fluid slowly
percolates the pores of the medium, must be represented by a macroscopic law
which is applicable to masses of fluid compared with the dimension of the porous
structure of a medium and this is the basis for the Darcy’s law. The original from of
Darcy law states that the mean filter velocity q is proportional to the sum of the
gradient of pressure and the gravitational force, that is,
( )constantq p gρµ
= −∇ + (3.1)
. The modified Darcy law due to Muskat (1937) is
( )kq p gρµ
= −∇ + (3.2)
where k is the permeability of the porous medium which determines the intrinsic
properties of the medium. This law is generally accepted as the macroscopic
equation for Newtonian fluids. The flow governed by this law, in the case of
homogeneous isotropic porous medium, is of potential type rather than boundary
layer nature. In other words, Darcy model takes into account only the frictional
force offered by the solid particles to the fluid rather than usual viscous shear.
To study convection in a porous medium, in addition to Darcy law, we have to
use the continuity and energy equations. As shown by Lapwood (1948), the set of
basic equations results in a fourth order differential equation governing the onset of
convection. However, using physical arguments, six boundary conditions based on
no-slip condition can be defined. In mathematical sense, the problem is therefore
not properly posed. All the earlier works have sidestepped the problem by simply
54
ignoring the no-slip conditions. However, from the physical point of view, the no-
slip boundary conditions are as much valid as other conditions and there appears no
a priori reason to reject them. In fact, Morels et al. (1951) have shown that
boundary conditions resembling no-slip do exist in a packed bed reactor, which is a
porous medium device. Beavers and Joseph (1967) postulated that a boundary layer
exists in a porous medium. This is further corroborated by Brenner (1970). Taylor
(1971) and Vafai and Tien (1981) and their work suggest that the boundary layer in
a porous medium is of order k . On the other hand, the experiments of Elder
(1967) and Katto ans Masuoka (1967) suggest that the boundary layer in a
horizontal porous layer is not so important in determining the point of the onset of
convection. However, it must be borne in mind that free convective heat transfer is
essentially a boundary value problem and hence the boundary layer must play a role
in governing the magnitude of the heat transfer after the onset of convection.
Therefore, one of the approximate boundary layer types of equation in a porous
medium is the Brinkman model. This model, first postulated by Brinkman (1947)
and later used by many, consists of viscous force term q2∇µ in addition to the
Darcy resistance term qkµ
−
in the momentum equation. Thermal convective
instability in a layer of porous medium has received extensive attention over the
years and has now emerged as an important field of study in the broader area of
fluid dynamics and in the area of heat transfer in particular. The growing volume of
work involving this field is well documented by Ingham and Pop (1998), Vafai
(2000, 2005) and Nield and Bejan (2006).
On the other hand, albeit numerous studies have been undertaken in the past to
understand convective instability of fluids, most of the investigations have been
limited to Newtonian fluids. Nevertheless, the growing importance of non-
Newtonian fluids in modern technology has impressed researchers because the
55
conventional Newtonian fluids cannot precisely describe the characteristics of the
fluid flow encountered in many practical situations such as the extrusion of polymer
fluids, solidification of liquid crystals, cooling of metallic plates in a bath, exotic
lubricants and colloidal fluids to mention a few.
These fluids deform and produce a spin field due to the microrotation of
suspended particles forming a micropolar fluid. The theory of micropolar fluids was
developed by Eringen (1966) which takes care of local effects arising from the
microstructure and as well as the intrinsic motions of microfluidics. The spin field
due to microrotation of freely suspended particles sets up an anti-symmetric stress,
known as couple stress, and thus forming a couple stress fluid. Thus couple-stress
fluid, according to Eringen (1966), is a variant of micropolar fluid when
microrotation balances with the natural vorticity of the fluid. The couple-stress fluid
has distinct features such as polar effects and whose microstructure is mechanically
significant. The constitutive equations for couple-stress fluids are proposed by
Stokes (1966, 1984). The theory proposed by Stokes is the simplest one, which
allows for polar effects such as the presence of couple stress and body couple. The
theory of couple stress fluids has several industrial and scientific applications,
which comprise pumping fluids such as synthetic fluids, polymer thickened oils,
liquid crystal, animal blood, synovial fluid present in synovial joints and the theory
of lubrication (Naduvinamani et al. 2001; 2002; 2003a; 2003b; 2005).
To obtain the basic equations the following approximations have been made use of:
1. The saturated fluid and the porous layer are as if incompressible except that
variations in density, brought about by heating, is taken into account only in the
term gρ of the momentum equation. This is valid only when the speed of the
fluid is much less than that of sound and all accelerations are slow compared
with those associated with sound waves. This is the well-known Boussinesq
approximation.
56
2. At any point in the fluid, the temperature of the solid ( )ST and that of the fluid
( )FT are the same. That is, the thermal behaviour of the medium is described by
a single condition for the average temperature FS TTT == . This most
commonly used approach is valid when the flow velocity is not too high and if
both phases (solid and fluid) are well dispersed.
3. The physical properties namely, thermal conductivity, viscosity and permeability
are assumed to be constants.
4. The viscous dissipation and radiation effects are neglected.
Under these approximations, the basic equations are the following:
Conservation of momentum
( ) ( )2o 1 1. cρ q q q p ρ g µ µ qε t ε k
∂+ = −∇ + − − ∇ ∂
∇ (3.3)
Conservation of energy
( ) 2.E
a b RTF o
Tγ q T χ T Q BY Y et
−∂+ = ∇ +
∂∇ (3.4)
Conservation of mass
0. =∇ q (3.5) Equation of state
( )o o1ρ ρ α T T = − − (3.6)
57
where ( )wvuq ,,= is the mean filter velocity, t is the time, p is the pressure, ρ is
the fluid density, oρ is the reference density, g is the acceleration due to gravity,
µ is the fluid viscosity, cµ is the couple-stress viscosity, ε is the porosity, k is the
permeability of the porous medium, χ is the effective thermal diffusivity, γ is the
ratio of the specific heat of the solid due to porous medium and that of the fluid at
constant pressure, α is the thermal expansion coefficient, T is the temperature,
oT is the reference temperature, Q is the heat of reaction, B is the pre-exponential
factor, FY and Yo are mass fractions of fuel and oxidizer, and a and b are their
respective reaction order, E is the activation energy, R is the universal gas constant,
∇ is the vector differential operator and ( )zyx ,, are the spatial coordinates.
3.2 Boundary Conditions
(i) Boundary conditions on velocity The boundary conditions on velocity are obtained from mass balance, the no-slip
condition and the stress principle of Cauchy depending on the fact that the fluid
layer is bounded by free or rigid surfaces.
If Darcy law is used, the normal component of velocity must vanish at the
impermeable surfaces while slip boundary conditions are allowed. This is because
Darcy’s equation is of lower order than Navier-Stokes’ equations. If Brinkman
equation is used, then no-slip conditions at the impermeable surfaces can be
imposed because a thin boundary layer inevitably arises at the boundaries. As noted
by Beck (1972) convective term cannot be added to Darcy law simply when the
basic state is not quiescent because the addition of convective term would raise the
order of the equation and the additional boundary conditions required are presently
unknown. In the case of quiescent state such difficulty will not arise (Beck, 1972).
Thus for Darcy equation
58
0ˆ.q n = (3.7) where n̂ is the unit vector normal to the surface. For Brinkman equation
0=== wvu (3.8) at the impermeable surfaces. The following combinations of boundary surfaces are
considered in the convective instability problems:
(i) Both lower and upper boundary surfaces are rigid.
(ii) Both lower and upper boundary surfaces are free.
(iii) Lower surface is rigid and upper surface is free.
a) Rigid surfaces
If the fluid layer is bounded above and below by rigid surfaces, then the viscous
fluid adheres to its bounding surface; hence the velocity of the fluid at a rigid
boundary surface is that of the boundary. This is known as the no-slip condition and
it indicates that the tangential components of velocity in the x and y directions are
zero, i.e. u = 0, v = 0. If the boundary surface is fixed or stationary, then in addition
to u = 0, v = 0, the normal component of velocity .q n∧→ is also zero, i.e., w = 0.
Hence at the rigid boundary we have
u = v = w = 0. (3.9)
Since u = v = 0 for all values of x and y at the boundary, we have 0ux∂
=∂
and
0vx
∂=
∂, and hence from the continuity equation subject to the Boussinesq
approximation, it follows that
59
0wz
∂=
∂
at the boundaries. Thus, in the case of rigid boundaries, the boundary conditions for
the z-component of velocity are
0wwz
∂= =∂
. (3. 10)
b) Free surfaces
In the case of a free surface the boundary conditions for velocity depend on
whether we consider the surface-tension or not. If there is no surface-tension at the
boundary, i.e., the free surface does not deform in the direction normal to itself, we
must require that
w = 0. (3.11)
We have taken the z-axis perpendicular to the xy plane, therefore w does not vary
with respect to x and y, i.e.
0wx
∂=
∂ and 0w
y∂
=∂
. (3.12)
In the absence of surface tension, the non-deformable free surface (assumed
flat) is free from shear stresses so that
0u vz z
∂ ∂= =
∂ ∂. (3.13)
From the equation of continuity subject to the Boussinesq approximation, we have
0u v wx y z
∂ ∂ ∂+ + =
∂ ∂ ∂. (3.14)
Differentiating this equation with respect to ‘z’ and using Eq. (3.13) yields
60
2
2 0wz
∂=
∂. (3.15)
Thus, in the absence of surface-tension, the conditions for the z-component of
velocity at the free surfaces are
2
2 0wwz
∂= =∂
. (3.16)
This condition is the stress-free condition.
(ii) Thermal boundary conditions
The thermal boundary conditions depend on the nature of the boundaries
(Sparrow et al., 1964). Four different types of thermal boundary conditions are
discussed below.
(a) Fixed surface temperature If the bounding wall of the fluid layer has high heat conductivity and large heat
capacity, the temperature in this case would be spatially uniform and independent of
time, i.e. the boundary temperature would be unperturbed by any flow or
temperature perturbation in the fluid. Thus
T = 0 (3.17)
at the boundaries. The effect is to maintain the temperature and this boundary
condition is known as isothermal boundary condition or boundary condition of the
first kind which is the Dirichlet type boundary condition.
61
(b) Fixed surface heat flux Heat exchange between the free surface and the environment takes place in the
case of free surfaces. According to Fourier’s law, the heat flux TQ passing through
the boundary per unit time and area is
1TTQ kz
∂= −
∂ (3.18)
where Tz
∂∂
is the temperature gradient of the fluid at the boundary. If TQ is
unperturbed by thermal or flow perturbations in the fluid, it follows that
Tz
∂∂
= 0 (3.19)
at the boundaries. This thermal boundary condition is known as adiabatic boundary
condition or insulating boundary condition or boundary condition of the second
kind which is the Neumann type boundary condition.
(c) Boundary condition of the third kind This is a general type of boundary condition on temperature which is given by
T Bi Tz
∂= −
∂. (3.20)
When Bi→ ∞ , we are led to the isothermal boundary condition T = 0 and when
0Bi → , we obtain the adiabatic boundary condition 0Tz
∂=
∂.
3.3 Dimensionless Parameters Exact solutions are rare in many branches of fluid mechanics because of
nonlinearities and general boundary conditions. Hence to determine approximate
solutions of the problem, numerical techniques or analytical techniques or a
62
combination of both are used. The key to tackle modern problems is mathematical
modelling. This process involves keeping certain elements, neglecting some, and
approximating yet others. To accomplish this important step one needs to decide the
order of magnitude, i.e., smallness or largeness of the different elements of the
system by comparing them with one another as well as with the basic elements of
the system. This process is called non-dimensionalization or making the variables
dimensionless. Expressing the equations in dimensionless form brings out the
important dimensionless parameters that govern the behaviour of the system. The
first method used to make the equations dimensionless is by introducing the
characteristic quantities and the other is by comparing similar terms. We use the
former method of introducing characteristic quantities. The following are the
important dimensionless parameters arising in the present study.
(i) Rayleigh number The thermal Rayleigh number is defined as
3
o g T dRa αρµ χ∆
= ,
where T∆ is the temperature difference between the boundaries and d is the
thickness of the fluid layer. The thermal Rayleigh number plays a significant role in
fluid layers where the buoyancy forces are predominant. Physically it represents the
balance of energy released by the buoyancy force and the energy dissipation by
viscous and thermal effects. We observe from the expression of R that the terms in
the numerator drive the motion and the terms in the denominator oppose the motion.
Mathematically, this number denotes the eigenvalue in the study of stability of
thermal convection. The critical thermal Rayleigh number is the value of the
eigenvalue at which the conduction state breaks down and convection sets in.
63
(ii) Couple-stress parameter
The couple-stress parameter Γ is defined as
2cµΓ
µ d= (0 )mΓ≤ ≤ ,
where m is a finite, positive real number according to the Clasusius-Duhem
inequality. As →∞µ , we find that 0Γ → . This is the Stokesian description of
suspension.
(iii) Frank-Kamenetskii number This number is defined to be
2
2c
ERTa b
F o
c
Q BY Y E d eFKRT χ
−
= ,
where Q is the heat of reaction, B is the pre-exponential factor, FY and Yo are
mass fractions of fuel and oxidizer, and a and b are their respective reaction order,
E is the activation energy, R is the universal gas constant. The Frank-Kamenetskii
number is commonly called the reduced Damkohler number in the combustion
literature. Physically, the FK number is a ratio of the characteristic flow time to the
characteristic reaction time.
64
CHAPTER IV
CHEMICAL REACTION INDUCED
RAYLEIGH-BÈNARD CONVECTION IN A
DENSELY PACKED POROUS MEDIUM
SATURATED WITH A COUPLE-STRESS FLUID
4.1 Introduction
During the past couple of decades, a great deal of effort has been devoted to the
study of free convection in a fluid-saturated porous medium with and without a
uniformly distributed heat source with applications to nuclear reactor safety and
geothermal reservoir engineering. When an exothermic reaction takes place in a
fluid saturated porous medium and if the reaction is accompanied by heat effects,
the distributed heat source / sink can cause convection. Examples of the interaction
of chemical reaction and free convection occur in tubular laboratory reactors,
chemical vapour deposition systems, oxidation of solid materials in large containers
and others. Little work has been performed on the effect of chemical reaction on
convection in porous media until recently. We consider the flow in a porous
medium as reactions of this type are common in various electrochemical processes
(Kloesnikov, 1979) and in the oxidation of fine solids (Kordylewski and Krajewski,
1984).
Considering two infinite horizontal plates kept at a constant temperature, Jones
(1973) performed numerical stability analysis of a zero order exothermic reaction
and free convection. Kordylewski and Krajewski (1984) paid attention to the
interaction of chemical reaction and natural convection in a porous medium. They
found that a sufficiently high Rayleigh number can prevent the system against
thermal ignition. They also asserted that interference between chemical reaction and
65
natural convection may lead to irregular oscillations similar to those observed in the
classical Darcy-Boussinesq problem.
Gatica et al. (1987) performed stability analysis of an isothermal first order and
non-isothermal zero order reaction in the presence of free convection. They found
that the calculated values compare favorably with the numerical simulation of the
full governing equations. Farr et al. (1991) investigated the onset of three-
dimensional reaction-driven convection in a porous medium using linear stability
theory. They also presented numerical study of reaction-driven convection in a
porous two-dimensional box. Using the orthogonal collocation and continuation
techniques, they determined the conduction and convective branches.
Vafai et al. (1993) obtained a numerical solution for chemically driven
convection in a porous cavity with isothermal walls at the top and bottom surfaces
and thermally insulated sidewalls. Both the inertia and the viscous forces have been
taken into consideration in the momentum equation.
Malashetty et al. (1994) performed a linear stability analysis to study the onset of
convective instability in a horizontal inert porous layer saturated with a fluid
undergoing zero order exothermic chemical reactions. It was found that, with
chemical reactions, the fluid in the porous medium is more prone to instability as
compared to the case in which chemical reactions are absent.
Churchill and Yu (2006) investigated the effect of the rate of convective heat
transfer on an energetic chemical reaction numerically and coherently. The
combination of the thermicity (the fractional increase in temperature due to the
reaction) and of uniform heating at the wall is shown to produce chaotic variations
of as much as an order of magnitude in the Nusselt number.
Patil and Kulkarni (2008) focused on the study of combined effects of free
convective heat and mass transfer on the steady two-dimensional, laminar, polar
66
fluid flow through a porous medium in the presence of internal heat generation and
chemical reaction of the first order. The highly nonlinear coupled differential
equations governing the boundary layer flow, heat and mass transfer are solved by
using two-term perturbation method with Eckert number E as perturbation
parameter. The velocity distribution of polar fluids is compared with the
corresponding flow problems for a viscous (Newtonian) fluid and found that the
polar fluid velocity is decreasing.
Wang and Tan (2009) studied on the basis of Brinkman model, the onset of
double-diffusive (thermosolutal) convection with a reaction term in a horizontal
sparsely packed porous media using the normal mode analysis. More recently,
Mahapatra et al. (2010) studied the effect of a chemical reaction on a free
convection flow through a porous medium bounded by a vertical infinite surface.
On the other hand, the problem of convection in a couple-stress fluid saturated
porous medium has attracted considerable interest over the past few decades
because of its wide range of applications, which comprise pumping fluids such as
synthetic fluids, polymer thickened oils, liquid crystal, animal blood, synovial fluid
present in synovial joints and the theory of lubrication (Naduvinamani et al. 2005).
To the best of our knowledge the influence of chemical reaction on Rayleigh-
Benard convection in a couple-stress fluid saturated densely packed porous medium
has not been investigated by the researchers. In this chapter we shall study the effect
of chemical reaction on the onset of Rayleigh-Benard convection in a horizontal
inert porous layer saturated with a couple-stress fluid heated from below and cooled
from above. The Darcy law is assumed to be valid and the normal mode technique
will be used to find the criterion for the onset of convection. Only infinitesimal
disturbances are to be considered. Galerkin technique will be employed to find the
eigenvalues marking the onset of convection.
67
4.2 Mathematical Formulation
Consider a horizontal constant porosity layer of finite thickness bounded
between z = 0 and z = d (with z-axis directed vertically upward) and of infinite
extent in the horizontal xy-plane. The inert porous layer is saturated with a
chemically reactive couples-stress fluid subject to weakly exothermic chemical
reactions and is cooled from the top at a temperature of cT . If the temperature in
the whole domain of interest varies slightly from cT , a zero order reaction can be
assumed. Moreover, it is assumed that local thermal equilibrium exists between the
solid matrix and the saturated fluid. The system of equations describing the problem
under consideration is the following:
0. =∇ q , (4.1)
( ) ( )2o 1 1. cq q q p g qt k
∂+ = −∇ + − − ∇ ∂
∇ρ ρ µ µε ε
, (4.2)
( ) 2.E
a b RTF o
Tγ q T χ T Q BY Y et
−∂+ = ∇ +
∂∇ , (4.3)
( )o c1 T Tρ ρ α = − − , (4.4) where ( )wvuq ,,= is the mean filter velocity, t is the time, p is the pressure, ρ is
the fluid density, oρ is the reference density, g is the acceleration due to gravity,
µ is the fluid viscosity, cµ is the couple-stress viscosity, ε is the porosity, k is the
permeability of the porous medium, χ is the effective thermal diffusivity, γ is the
ratio of the specific heat of the solid due to porous medium and that of the fluid at
constant pressure, α is the thermal expansion coefficient, T is the temperature,
Q is the heat of reaction, B is the pre-exponential factor, FY and Yo are mass
68
fractions of fuel and oxidizer, and a and b are their respective reaction order, E is
the activation energy, R is the universal gas constant, ∇ is the vector differential
operator and ( )zyx ,, are the spatial coordinates.
The thermal boundary conditions are given by
( ), , cT x y d T= (4.5)
and
( ), ,0 hT x y T= , (4.6)
where h cT T> .
We next assume that the fluid in the porous medium is subject to a high-
activation energy such that cRTE
<< 1. With this approximation, Eq. (4.3) can be
simplified to
( ) 2.q C et
θθγ θ χ θ∂+ ∇ = ∇ +
∂, (4.7)
where c
ERTa b
F o
r
Q BY Y eCT
−
= , ( )c
r
T TT
θ−
= and 2
cr
RTTE
= . Here rT is the
prescribed reference temperature for a reacting gas.
Eqs. (4.5) and (4.6) in terms of θ reduces to
0θ = at * 1z = (4.8)
and
hθ θ= at * 0z = (4.9)
69
where h ch
r
T TT
θ −= . In Eqs. (4.8) and (4.9), the symbol ‘*’ denotes a
dimensionless quantity.
4.3 Basic Quiescent State
At an undisturbed state, we have
( ) ( ), , 0,0,0q u v w= = ; ( )b zθ θ= ; ( )bp p z= ; ( )b zρ ρ= . (4.10)
Thus the quiescent state solutions are given by the following equations
0bb
dp gd z
ρ+ = , (4.11)
[ ]1b o r bTρ ρ α θ= − , (4.12)
2
2 0bbd C edz
θθχ + = (4.13)
Eq. (4.13) can be rewritten in the dimensionless form
2
2*bbd FK e
dzθθ
= − (4.14)
where * zzd
= and 2
2c
ERTa b
F o
c
Q BY Y E d eFKRT χ
−
= . The dimensionless number
FK is the Frank-Kamenetskii number.
70
On integration, Eq. (4.14) leads to (after dropping the asterisks) the following
general solution:
1
1
21 2
2
1log log 12 1
C z
b C zC C eFK C e
θ−
−
− = + − +
(4.15)
where 1C and 2C are the integration constants to be determined.
Application of the following boundary conditions
b hθ θ= at 0z = and 0bθ = at 1z = (4.16)
gives 1C implicitly through the following equation:
1 1 1
1 1
2 21 1 1 1
2 21 1 1 1
h
h
C
FK FK e
C Ce
FK FK e
C C
θ
θ
− − − − = + − + −
(4.17)
and 2C by the relation
1 12
1
21 1
21 1
− −
= + −
C
FKC
C eFKC
. (4.18)
71
4.4 Linear Stability Analysis
We now perform a linear stability analysis by letting
( ), ,
( ) ( , , , )
( ) ( , , , )
( ) ( , , , )
b
b
b
q u v w
p p z p x y z t
z p x y z t
z x y z t
ρ ρ
θ θ θ
′ ′ ′ =
′= +
′= + ′= +
, (4.19)
where the primes indicate infinitesimally small perturbations from the undisturbed
state. On substituting Eq. (4.19) into Eqs. (4.1), (4.2), (4.4) and (4.7), neglecting the
nonlinear terms, incorporating the quiescent state solutions and eliminating the
pressure term, we obtain the following equations
( )2 2 2 4o 1
crw g T w w
t k kρ µµα ρ θεο ∂ ′ ′ ′ ′∇ = ∇ − ∇ + ∇∂
, (4.20)
and
2 bbdw C et dz
θθθγ χ θ θ′∂ ′ ′ ′+ = ∇ +
∂ (4.21)
where 2 2
21 2 2x y
∂ ∂∇ = +
∂ ∂. We now assume that the solutions of Eqs. (4.20) and
(4.21) have the form
( )
( )( )
( )
i l x m y t
i l x m y t
w W z e
z e
σ
σθ
+ +
+ +
′ =
′ = Θ , (4.22)
where l and m are the horizontal wave numbers in the x and y directions
respectively. The quantity σ refers to the growth rate. Substitution of (4.22) into
Eqs. (4.20) and (4.21) yields the following equations
72
( ) ( ) ( )22 2 2 2 2 2 2co 0o
h h h r hD k W D k W D k W gT kk k
ρ µ µσ αρε
− − − + − + Θ =
(4.23)
and
( )2 2 bbh
d W D k C edz
θθγ σ χΘ + = − Θ + Θ , (4.24)
where 2 2 2hk l m= + . Using the transformations
2* ; * ; ; *h
W d d zW a k d zd
σσχ χ
= = = = (4.25)
Eqs. (4.23) and (4.24) can be expressed (after dropping the asterisks) in the
following dimensionless form
( ) ( ) ( )22 2 2 2 2 2 2 0D a W D a W D a W a RaVaσ Γγ
− − − + − + Θ =
(4.26)
and
( )2 2 bbd W D a FK edz
θθσ Θ + = − Θ + Θ . (4.27)
Here 2
o
dVak
ε µρ χ
= is the Vadasz number, o rg kT dRa αρµ χ
= is the media
Darcy-Rayleigh number and 2cµΓ
µ d= is the couple-stress parameter.
Eqs. (4.26) and (4.27) are solved subject to the following stress-free, isothermal,
vanishing couple stress boundary conditions
2 0 at 0, 1W D W z= = Θ = = (4.28)
73
For the marginal stability (for which 0σ = ), Eqs. (4.26) and (4.27) take the
form
( ) ( )2 2 2 2 21 0D a D a W a RaΓ − − − + Θ = (4.29)
and
( )2 2 0bbdD a W FK edz
θθ− Θ − + Θ = . (4.30)
The system comprising Eqs. (4.29) and (4.30) and the homogeneous boundary
conditions (4.28) is an eigenvalue problem, with Ra being the eigenvalue. An
approximate solution of the foregoing eigenvalue problem can be obtained by the
well-known Galerkin method (Finlayson, 1972). To this end, we let
1 1W A W= and 1 1BΘ = Θ , (4.31)
where 1W and 1Θ are the trial functions which must satisfy the boundary
conditions (4.28). Substituting the expressions in Eq. (4.31) into Eqs. (4.29) and
(4.30), multiplying the resulting equations by 1W and 1Θ respectively, integrating
each equation between 0z = and 1z = , and performing some integration by parts,
we obtain
( ) 21 2 1 1 1 0X X A a Ra Y BΓ+ − = (4.32)
and
( )3 1 2 3 1 0X A Y FK Y B+ − = , (4.33)
where
( )2 2 21 1 1X DW a W= + ,
( ) ( )2 22 4 2 2
2 1 1 12X D W a W a DW= + + ,
74
3 1 1bdX W
dzθ
= Θ , 1 1 1Y W= Θ ,
( )2 2 22 1 1Y D a= Θ + Θ , 3 1 1bY eθ= Θ Θ ,
and 1
0( )f f z dz= ∫ .
On using the criterion for the existence of the unique solution of the system of
Eqs. (4.32) and (4.33), we obtain the following eigenvalue expression
( ) ( )1 2 3 2
23 1
X X Y FK YRa
a X Y
Γ+ −= . (4.34)
Keeping in mind the chosen boundary conditions (4.28), we deal with the
following trial functions
1
1
( ) sin( ) sin
W z zz z
ππ
= Θ =
.
The trial functions for the z-component of velocity and temperature satisfy the
given boundary conditions, but may not exactly satisfy the differential equations.
This results in residuals when the trial functions are substituted into the differential
equations. The Galerkin method warrants the residuals be orthogonal to each trial
function.
75
CHAPTER V
RESULTS, DISCUSSION AND
CONCLUDING REMARKS
5.1 Results and Discussion
The problem of Rayleigh-Benard convection in a couple-stress fluid saturated
densely packed porous medium with chemical reaction is studied using linear
stability analysis. Only infinitesimal disturbances are considered. The linear
stability analysis is based on the normal mode technique. The Darcy law is used to
model the momentum equation. Closed form solution for the basic quiescent state is
first obtained. It is well known that the principle of exchange of stabilities is valid
for the problem at hand, meaning the stationary instability is preferred to oscillatory
instability. In view of this, the expression for the stationary media-Darcy-Rayleigh
number Ra is obtained as a function of the governing parameters, namely, the wave
number a , the couple-stress parameter Γ and the Frank-Kamenetskii number FK.
The Galerkin invariant of the weighted-residual method is used to determine the
eigenvalues. Neutral stability curves in the ( ),Ra a plane are plotted for different
values of various parameters involved in the problem. The coordinates of the lowest
point on these curves designate the critical values cRa and ca .
Computations for 1C and 2C were performed for selected values of hθ and
for different values of FK up to its ignition value (Malashetty et al., 1994). The
results of these computations for 1hθ = are presented in Fig. 5.2. It is seen that, for
small values of FK, the basic temperature profile is nearly linear. However, the
basic temperature profile turns out to be more and more nonlinear as the value of
FK is increased.
76
Fig. 5.3 depicts the variation of Ra as a function of the wave number a for
different values of the Frank-Kamenetskii number FK and for a fixed value of Γ .
We observe from this figure that Ra decreases with an increase in FK, indicating
that the effect of chemical reaction is to advance the onset of convection. The effect
of the couple-stress parameter Γ on the onset of convection for a fixed value of FK
is shown in Fig. 5.4. The couple-stress parameter Γ is indicative of the
concentration of suspended particles. We find that Ra increases with an increase
in Γ . Evidently, the effect of Γ is to stabilize the system.
A close examination of Figs. 5.3 and 5.4 reveals that the critical wave number
ca is sensitive to the variation in both FK and Γ . This means that both FK and Γ
have a say on the size of the convection cells. As a particular case, when 0Γ = , we
recover the results of the problem dealt with by Malashetty et al. (1994).
5.2 Concluding Remarks
The effect of chemical reaction on Rayleigh-Benard convection in a densely
packed porous medium saturated with a couple-stress fluid is investigated. The
eigenvalue problem is solved numerically using the Galerkin method. The
following conclusions are drawn:
1. The principle of exchange of stabilities is valid and the existence of oscillatory
instability is ruled out.
2. Chemical reaction, giving rise to a nonlinear basic temperature distribution,
destabilizes the system.
77
3. The presence of couple stresses is to delay the onset of convection. In other
words, the effect of suspended particles whose spin matches with the vorticity of
the fluid is to enhance the stability of the system.
4. The dimension of the convection cells is influenced by the presence of both
chemical reaction and couple stresses.
Figure 5.1: Configuration of the problem.
z = d
Chemically reacting couple-stress
fluid in a porous medium
x
z
T = T0
T = T0 + ∆T
y
z = 0
g
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Figure 5.2: Basic temperature profiles for different values of FK.
bθ
1=hθ
0.1,0.3,0.5,0.7,0.9=FK
z
0 1 2 3 4 5300
400
500
600
700
800
Figure 5.3: Plot of media-Darcy-Rayleigh number Ra as a function of the wave
number a for Γ = 0.5 and for different values of the Frank-Kamenetski number FK.
FK = 0.1, 0.5, 0.9
Ra
a
0 1 2 3 4 5 6
0
300
600
900
1200
1500
1800
Figure 5.4: Plot of media-Darcy-Rayleigh number Ra as a function of the wave
number a for FK = 0.5 and for different values of the couple-stress parameter Γ .
Ra
a
Γ = 0
Γ = 0.5
Γ = 1
82
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