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CHAPTER 3

Radiation effects on Unsteady

Free Convection Heat and Mass Transfer in a Walters-B Viscoelastic flow past an

Impulsively started Vertical plate.

International Journal of Scientific & Engineering Research. Vol.3, Issue 8, Aug-2012 with ISSN 2229-5518

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3.1. INTRODUCTION

The flow of a viscous incompressible fluid past an impulsively started infinite

vertical plate, moving in its own plane was first studied by Stokes [1]. It is often called

Rayleighs problem in the literature. Such a flow past an impulsively started semi

infinite horizontal plate was first presented by Stewartson [2]. Hall [3] considered the

flow past an impulsively started semi- infinite horizontal plate by a finite difference

method of a mixed explicit-implicit type. However, Hall analysed only unsteady

velocity field by neglecting the buoyancy effects. Soundalgekar [4] presented an exact

solution to the flow of a viscous fluid past an impulsively started infinite isothermal

vertical plate. The solution was derived by the Laplace transform technique and the

effect of heating or cooling of the plate on the flow field was discussed through

Grashof number ( ). Raptis and Singh [5] studied the flow past an impulsively

started infinite vertical plate in a Porous medium by a finite difference method. Many

transport processes exists in nature and in which the simultaneous heat and mass

transfer occur as a result of combined buoyancy effects of thermal diffusion and

diffusion of chemical species a few representative fields of interest in which combined

heat and mass transfer play an important role are designing of chemical processing

equipment, formation and dispersion of fog, distribution of temperature and moisture

over agricultural fields and groves of fruit trees, crop damage due to the freezing and

environmental pollution.

The Walters-B viscoelastic model [6] was developed to simulate viscous fluids

possessing short memory elastic effects and can simulate accurately many complex

polymeric, biotechnological and tribiological fluids. The Walters-B model has

therefore been studied extensively in many flow problems. Das et al [7] considered

the mass transfer effects on the flow past an impulsively started infinite vertical

65

plate with a constant mass flux and chemical reaction. Muthucumaraswamy and

Ganesan [8] studied the unsteady flow past an impulsively started isothermal vertical

plate with mass transfer by an implicit finite difference method. Muthucumaraswamy

and Ganesan [9, 10] solved the problem of unsteady flow past an impulsively started

vertical plate with uniform heat and mass flux, and variable temperature and mass flux

respectively.

In this context of space technology and in processes involving high

temperatures and the effects of radiation are of the vital importance. Recent

developments in hypersonic flights, missile re-entry, rocket combustion chambers

power plants for inter-planetary flight and gas cooled nuclear reactors, focussed

attention on thermal radiation as a mode of energy transfer, and emphasize the need

for improved understanding of radiative transfer in these processes. The interaction of

radiation with laminar free convection heat transfer from a vertical plate was

investigated by Cess [11] for an absorbing, emitting fluid in the optically thick

region , using the singular perturbation technique. Arpaci [12] considered a similar

problem in both the optically thin and optically thick regions and used approximate

integral technique and first order profiles to solve the energy equation. Cheng et al

[13] studied a related problem for an absorbing, emitting and isotropically scattering

fluid, and treated the radiation part of the problem exactly with the normal mode

expansion technique.

Raptis [14] analysed both the thermal radiation and free convection flow

through a porous medium by using a perturbation technique. Hossain and Takhar [15]

studied the radiation effects on mixed convection along a vertical plate with the

uniform surface temperature using the Keller Box finite difference method. In all these

papers, the flow taken steady. Mansour [16] studied, the radiative and free convection

66

effects on the oscillatory flow past a vertical plate. Raptis and Perdikis [17] considered

the problem of thermal radiation and free convection flow past moving plate. Das et al

[18] analysed the radiation effects on the flow past an impulsively started infinite

isothermal vertical plate, the governing equations are solved by using Laplace

transform technique. Chamkha et al [19] studied the radiation effects on the free

convection flow past a semi-infinite vertical plate. Raptis and Takhar [20] studied, Flat

plate thermal convection boundary layer flow of a Walters -B fluid using numerical

shooting quadrature.

Chang et al [21] analyzed the unsteady buoyancy-driven flow and species

diffusion in a Walters-B viscoelastic flow along a vertical plate with transpiration

effects. They showed that the flow is accelerated with a rise in viscoelasticity

parameter with both time and locations close to the plate surface and increasing

Schmidt number ( ) suppresses both velocity and concentration in time whereas

increasing species Grashof number (buoyancy parameter) accelerates flow through

time. Nanousis [22] used a Laplace transform method to study the transient rotating

hydro magnetic Walters-B viscoelastic flow regime. Hydrodynamic stability studies of

the Walters-B viscoelastic fluids were communicated by Sharma and Rana [23] for

the rotating porous media suspension regime and by Sharma et al. (2002) for Rayleigh

Taylor flow in a porous medium. Jain and Choudhury [24] studied the Hall current

and cross -flow effects on free and forced Walters-B viscoelastic convection flow with

thermal radiative flux effects. Mahapatra et al. [25] examined the steady two-

dimensional stagnation-point flow of a Walters-B fluid along a flat deformable

stretching surface. They found that a boundary layer is generated when the inviscid

freestream velocity exceeds the stretching velocity of the surface and the flow is

accelerated with increasing magnetic field. This study also identified the presence of

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an inverted boundary layer when the surface stretched velocity exceeds the velocity of

the free stream and showed that for this scenario the flow is decelerate with increasing

magnetic field.

Veena et al. [26] employed the perturbation and series expansion method and

Kummers function to examine viscoelastic hydro-magnetic boundary layer

convection flow over a porous stretching sheet with internal heat generation and stress

work, showing that an increase in viscoelasticity increases surface shear stresses.

Rajagopal et al.[27] obtained exact solutions for the combined various hydro magnetic

flow, heat and mass transfer phenomena in a conducting viscoelastic Walters-B fluid

percolating a porous regime adjacent to a stretching sheet with heat generation,

viscous dissipation and wall mass flux effects, by using confluent hyper geometric

functions for different thermal boundary conditions at the wall. Rath and Bastia [28]

used a perturbation method to analyze the steady flow and heat transfer of Walters-B

model viscoelastic liquid between two parallel uniformly porous disks rotating about a

common axis, showing that drag is enhanced with suction but reduced with injection

and that heat transfer rate is accentuated with a rise in wall suction or injection.

More recently, Beg at al [29] studied the effects of thermal radiation on

impulsively started vertical plate using network simulation method [NSM]. The above

studies did not consider combined viscoelastic momentum, heat and mass transfer

from a impulsively started vertical plate, either for the steady case or unsteady case.

Owing to the significance of this problem in chemical and medical biotechnological

processing (e.g. medical cream manufacture) we study the transient case of such a

flow in the present paper using the robust Walters-B viscoelastic rheological material

model. A Crank-Nicolson finite difference scheme is utilized to solve the unsteady

dimensionless, transformed velocity, thermal and concentration boundary layer

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equations in the vicinity of the impulsively started vertical plate. Another motive of

the study is to further investigate and observe high heat transfer performance,

commonly attributed to extensional stresses in viscoelastic boundary layers.

The aim of the present paper is to analyse the radiation effects on an unsteady

two-dimensional laminar simultaneous free convection heat and mass transfer in a

Walters-B viscoelastic flow past an impulsively started vertical plate. The equations of

continuity, linear momentum, energy and diffusion, which govern the flow field, are

solved by using an implicit finite difference method of The CrankNicolson type. The

behaviour of the velocity, temperature, concentration, skin-friction and Sherwood

number has been discussed for variations in the governing parameters. Here the results

analysed graphically how the parameters affected the flow.

3.2. MATHEMATICAL ANALYSIS

An unsteady twodimensional laminar natural convection radiative flow of a

viscoelastic fluid past an impulsively started vertical plate is considered. The x- axis

is taken along the plate in the upward direction and y-axis is taken normal to it. The

physical model is shown in fig. 1.

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Initially assumed that the plate and the fluid are at the same temperature

and concentration level everywhere in the fluid. At the time > 0 the

temperature of plate and concentration level near the plate are raised to and

respectively and are maintained constantly thereafter. It is assumed that the

concentration of the diffusing species in the mixture is very less in comparison to

the other chemical species, which are present and hence the Soret and Dufour effects

are negligible. It is also assumed that there is no chemical reaction between the

diffusing species and the fluid. Under the above assumptions, the governing boundary

layer equations with Boussinesqs approximation are:

+ = 0(3.2.1)

+ + = ( ) + ( ) + (3.2.2)

+ + = 1 (3.2.3)

+ + = (3.2.4)

The initial and boundary conditions are:

0: = 0, = 0, = , =

> 0: = , = 0, = , = = 0(3.2.5)

= 0, = , = , = 0

0, ,

By using the Rosseland approximation the radiative heat flux is given by

= 43 (3.2.6)

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Where is the Stefan -Boltzmann constant and is the mean absorption

coefficient. It should be noted that by using the Rosseland approximation, the present

analysis is limited to optically thick fluids. If temperature differences within the flow are

significantly small, then equation [11] can be linearised by expanding into the Taylor

series about , which after neglecting the higher order terms takes the form:

4 3 (3.2.7)

In view of equations (11) and (12), equation (8) reduces to

+ + = +163 (3.2.8)

On introducing the following non-dimensional quantities:

= , = , = , = , =

= , = , =

, =

( )(3.2.9)

=4

, =( )

, = , = , = ,1

= .

Equations (3.2.1)(3.2.4) are reduced under (3.2.9)to the following non-dimensional form:

+ = 0(3.2.10)

+ + = + . + . UY t (3.2.11)

+ + = 1 1 +

43 (3.2.12)

+ + = 1 (3.2.13)

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The corresponding initial and boundary conditions are:

0 = 0, = 0, = 0, = 0

> 0: = 1, = 0, = 1, = 1 = 0(3.2.14)

= 0, = 0, = 0, = 0

0, 0, 0

L is the length of the plate, - the Grashof number the Prandtl number,

- Schmidt number, - the radiation parameter, - the buoyancy ratio parameter,

- the viscoelasticity parameter.

Knowing the velocity, temperature, and concentration fields, it is interesting to

find local Skin-friction, Nusselt number and Sherwood numbers in non dimensional

equation model, computed with the following mathematical expressions.

= (3.2.15)

=

(3.2.16)

=

(3.2.17)

We note that the dimensionless model defined by equations (3.2.10) to (3.2.13)

under conditions ( 3.2.14 ) reduces to Newtonian flow in the case of vanishing

viscoelasticity, i.e. when .

3.3. NUMERICAL TECHNIQUE

In order to solve these unsteady, non-linear coupled equations (3.2.10 to3.2.13)

under conditions (3.2.14) an implicit finite difference scheme of Crank-Nicolson type

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has been employed. The region of integration is considered as a rectangle with sides

(= 1) and (= 14), where corresponds to = , which lies very well

outside the momentum, energy and concentration boundary layers .the maximum of Y

was chosen as 14 after some preliminary investigations, so that the last two of the

boundary conditions (3.2.14) are satisfied within the tolerance limit 10 . The grid

system is shown in the following figure. The finite difference equations corresponding to

equations (15) - (18) are as follow.

, , + , , + , , + , ,4

+ , , + , ,

2 = 0(3.3.1)

, ,

+ , , , + , ,

2+ ,

, , + , , 4

=

, 2 , + , + , 2 , + ,

2( )+ . ,

+ ,2

+ . ,+ ,

2

, 2 , + , + , 2 , + ,

2( ) (3.3.2)

, , + ,

, , + , . , 2 + ,

. , + , ,4 =

11 +

43

, 2 , + , + , 2 , + ,2( ) (3.3.3)

, , + ,

, , + , ,2 + ,

, , + , ,4 =

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, 2 , + , + , 2 , + ,

2 ( ) (3.3.4)

The region of integration is considered as a rectangle with side (=

1)and (= 14) Where corresponds to = , which lies very well

outside the momentum, energy and concentration boundary layers. After some

preliminary numerical experiments the mesh sizes have been fixed as =

0.05, = 0.25 with time step = 0.01. The computations are executed initially

by reducing the spatial mesh sizes by 50% in one direction ,and later in both

directions by 50%. The results are compared. It is observed that in all cases, the results

differ in only fifth decimal places. Hence, the choice of mesh sizes is verified as

extremely efficient. During any one time step, the coefficients , and ,

appearing in the difference equations are treated as constants. Here designates the grid

point along the x-direction, j-along the y-direction. The values of , , and are

known at all grid points when t = 0 from the initial conditions, the computations for

, , and at a time level ( + 1) , using the values at previous time level are

carried out as follows. The grid system is shown in the following figure.

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The finite difference equations at every internal nodal point on a particular

level constitutes a tridiagonal system of equations and is solved by using Thomas

algorithm. Thus the values of C are known at every nodal point at a particular

-level at ( + 1)th time level. In a similar way computations are carried out by moving

along -direction. After computing values corresponding to each at a time level, the

values at the next time level are determined in a similar manner. Computations are

repeated until the steady state is reached. Here the steady state solution is assumed

to have reached when the absolute difference between the values of the velocity

, temperature , as well as the concentration at two consecutive time steps are

less than 10 at all grid points. The scheme is unconditionally stable. Here the local

truncation error given by ( + + )and it tends to zero as ,

tend to zero. It follows that the CNM scheme is compatible and stable and ensures the

convergence. The derivatives involved in (15)(18) are evaluated using five-point

approximation formula and the integrals are evaluated using Newton- Cotes closed

integration formula.

3.4. RESULTS AND DISCUSSION

A representative set of numerical results is shown graphically in Figs.3.2(a)

3.9(c), to illustrate the influence of physical parameters, viz. viscoelastic parameter ()

= 0.005, thermal Grashof parameter ( ) = 1.0, species Grashof parameter ( ) = 1.0,

Schmidt number ( ) = 0.25 (oxygen diffusing in the viscoelastic fluid), radiation

parameter( ) = 0.5 and Prandtl number ( ) = 0.71 (water-based solvents). All graphs

therefore, correspond to these values unless specifically otherwise indicated. For the

special case of = 1, the species diffuse at the same rate as momentum in the

viscoelastic fluid. Both concentration and boundary layer thickness are the same for this

case. For < 1, species diffusivity exceeds momentum diffusivity, so that mass is

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diffused faster than momentum. The opposite case applies when > 1. An increase

in will suppress concentration in the boundary layer regime since higher values

will physically manifest in a decrease of molecular diffusivity ( ) of the viscoelastic

fluid, i.e. a reduction in the rate of mass diffusion.

From figs. 3.2a to 3.2c, we have presented the variation of velocity ( ),

temperature function ( ) and concentration ( ) versus ( ) with on viscoelasticity () at

X = 1.0. An increase in from 0 to 0.001, 0.003, 0.005 and the maximum value of

0.007, as depicted in figure 2a, clearly enhances the velocity, which ascends sharply

and peaks in close vicinity to the plate surface (Y = 0). With increasing distance from

the plate wall however the velocity is adversely affected by increasing

viscoelasticity, i.e. the flow is decelerated. Therefore close to the plate surface the flow

velocity is maximized for the case of a Newtonian fluid (vanishing viscoelastic effect,

i.e. =0) but this trend is reversed as we progress further into the boundary layer

regime. The switchover in behaviour corresponds to approximately Y = 1. With

increasing ( ) velocity profiles decay smoothly to zero in the free stream at the edge of

the boundary layer. The opposite effect is caused by an increase in time. A rise in t

from 3.69 through 3.54, 3.72 4.98 to 5.42 causes a decrease in flow velocity, nearer

the wall in this case the maximum velocity arises for the least time progressed. With

more passage of time (t = 5.42) the flow is decelerated. Again there is a reverse in the

response at ~1, and thereafter velocity is maximized with the greatest value of time.

In figure 3.2b increasing viscoelasticity is seen to decrease temperature throughout

the boundary layer. As we approach the free stream the effects of viscoelasticity are

negligible since the profiles are all merged together. All profiles decay from the

maximum at the wall to zero in the free stream. The graphs show therefore that

increasing viscoelasticity cools the flow. With progression of time, however the

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temperature, is consistently enhanced i.e. the fluid is heated as time progresses. A

similar response is observed for the concentration field, , in figure 3.2c. Increasing

viscoelasticity again reduces concentration, showing that species diffuses more

effectively in Newtonian fluids (=0) than in strongly viscoelastic fluids. Once again

with greater elapse in time the concentration values are reduced throughout the

boundary layer regime (0 < Y< 14).

Figs. 3.3(a) to 3.3(c) illustrate the effect of Prandtl number, and time, on

velocity, , temperature, , and concentration, C. Increasing clearly reduces strongly

velocity, (figure 3.3a) both in the near-wall regime and the far-field regime of the

boundary layer. Velocity is therefore maximized when = 0.3 (minimum) and minimized

for the largest value of (10.0). defines the ratio of momentum diffusivity () to

thermal diffusivity. < 1 physically corresponds to cases where heat diffuses faster than

momentum. = 0.71 is representative of water-based solvents and >> 1e.g. 10

corresponds to lubricating oils etc. An increase in time, , also serves to strongly retard the

flow. With increasing Pr from 0.3 through 0.7, 1.0, 5.0 to 10.0, temperature, as shown in

figure 3.3b, is markedly reduced throughout the boundary layer. The descent is increasingly

sharper near the plate surface for higher values. With lower values a more gradual

(monotonic) decay is witnessed. Smaller values in this case, cause a thinner thermal

boundary layer thickness and more uniform temperature distributions across the boundary

layer. Smaller fluids possess higher thermal conductivities so that heat can diffuse away

from the plate surface (wall) faster than for higher fluids (thicker boundary layers). Our

computations show that a rise in depresses the temperature function. For the case of

= 1, thermal and velocity boundary layer thicknesses are equal. Conversely the

concentration values (figure 3.3c) are slightly increased with a rise in from 0.3 through

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intermediate values to 10. However with progression of time the concentration is found to

be decreased in the boundary layer regime.

Figs. 3.4(a) to 3.4(c) depict the distributions of velocity ( ) and concentration

( ) versus coordinate ( ) for various Schmidt numbers ( ) and time ( ), close to the

leading edge at = 1.0, are shown. These figures again correspond to a viscoelasticity

parameter () = 0.005, i.e weak elasticity and strongly viscous effects. An increase in Sc

from 0.1 (low weight diffusing gas species) through 0.5 (oxygen diffusing) to 1.0 (denser

hydrocarbon derivatives as the diffusing species), 3.0 and 5.0, clearly strongly decelerates

the flow. In Fig. 3.4a, the maximum velocity arises at = 0.1 very close to the wall. All

profiles then descend smoothly to zero in the free stream. With higher Sc values the

gradient of velocity profiles is lesser prior to the peak velocity but greater after the peak. An

increase in time, is found to accelerate the flow. Conversely the temperature values (figure

3.4b) are slightly increased with a rise in Sc from 0.1 through intermediate values to 0.6.

However with progression of time the temperature is found to be decreased in the boundary

layer regime. Figure 3.4(c) shows that a rise in strongly suppresses concentration levels

in the boundary layer regime. defines the ratio of momentum diffusivity () to molecular

diffusivity ( ). For < 1, species will diffuse much faster than momentum so that

maximum concentrations will be associated with this case ( = 0.1 ). For > 1,

momentum will diffuse faster than species causing progressively lower concentration

values. All profiles decay monotonically from the plate surface (wall) to the free stream.

With a decrease in molecular diffusivity (rise in Sc) concentration boundary layer thickness

is therefore decreased. For the special case of = 1, the species diffuses at the same rate

as momentum in the viscoelastic fluid. Both concentration and boundary layer thicknesses

are the same for this case. Concentration values are also seen to increase continuously with

time, . An increase in Schmidt number effectively depresses concentration values in the

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boundary layer regime since higher Sc values will physically manifest in a decrease of

molecular diffusivity ( ) of the viscoelastic fluid, i.e. a reduction in the rate of mass

diffusion. Lower values will exert the reverse influence since they correspond to higher

molecular diffusivities. Concentration boundary layer thickness is therefore considerably

greater for = 0.1 than for = 5.0.

Figs. 3.5(a) to 3.5(c): It is of great interest to show how the radiation-conduction

interaction effects on velocity, temperature and concentration. It is observed that, initially

for lower values of the radiation parameter , the heat transfer is dominated by conduction,

as the values of increases the radiation absorption in boundary layer increases, i.e. an

increase in the radiation parameter from 0.1 through 0.5, 1.0, 3.0, 5.0 and 10.0 results in a

decrease in the temperature and increase in concentration shown in figs. 3.5(b) and 3.5(c).

Here the velocity profiles increases initially and then slowly decreases to zero due to

viscoelasticity shown in fig. 3.5(a), i.e. when the radiation parameter is increasing,

observed that the time progress initially increases and slowly decreases.

VELOCITY : Thermal Grashof number (Gr) versus Y values Time: 7.400 4.990 4.360 3.360 2.710 Y- axis 0.1 1.0 2.0 4.0 6.0 (Gr)

0 1 1 1 1 1 1 0.56192 0.52574 0.49936 0.46414 0.43991 2 0.29423 0.25255 0.22508 0.19171 0.17075 3 0.14602 0.11407 0.0951 0.07411 0.06204 4 0.07016 0.04979 0.03886 0.02779 0.02192 5 0.03311 0.02136 0.01564 0.01029 0.00766 6 0.01548 0.00908 0.00625 0.00379 0.00267 7 0.0072 0.00385 0.00249 0.00139 0.00093 8 0.00334 0.00163 0.00099 0.00051 0.00032 9 0.00154 0.00069 0.00039 0.00019 0.00011

10 0.00071 0.00029 0.00016 0.00007 0.00004 11 0.00032 0.00012 0.00006 0.00003 0.00001 12 0.00014 0.00005 0.00002 0.00001 0 13 0.00005 0.00002 0.00001 0 0 14 0 0 0 0 0

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CONCENTRATION : Thermal Grashof number (Gr) versus Y values Time (t) : 7.400 4.990 4.360 3.360 2.710

Y- axis 0.1 1.0 2.0 4.0 6.0 (Gr) 0 1 1 1 1 1 1 0.56192 0.52574 0.49936 0.46414 0.43991 2 0.29423 0.25255 0.22508 0.19171 0.17075 3 0.14602 0.11407 0.0951 0.07411 0.06204 4 0.07016 0.04979 0.03886 0.02779 0.02192 5 0.03311 0.02136 0.01564 0.01029 0.00766 6 0.01548 0.00908 0.00625 0.00379 0.00267 7 0.0072 0.00385 0.00249 0.00139 0.00093 8 0.00334 0.00163 0.00099 0.00051 0.00032 9 0.00154 0.00069 0.00039 0.00019 0.00011

10 0.00071 0.00029 0.00016 0.00007 0.00004 11 0.00032 0.00012 0.00006 0.00003 0.00001 12 0.00014 0.00005 0.00002 0.00001 0 13 0.00005 0.00002 0.00001 0 0 14 0 0 0 0 0

TEMPERATURE : Thermal Grashof number (Gr) versus Y values Time (t) : 7.400 4.990 4.360 3.360 2.710

Y- axis 0.1 1.0 2.0 4.0 6.0 (Gr) 0 1 1 1 1 1 1 0.6169 0.57141 0.53753 0.49117 0.4585 2 0.33331 0.27429 0.23529 0.18819 0.15905 3 0.16784 0.12057 0.09348 0.06494 0.04956 4 0.08196 0.05111 0.03579 0.02163 0.01494 5 0.03953 0.02136 0.01352 0.00712 0.00446 6 0.01896 0.00888 0.00508 0.00234 0.00133 7 0.00906 0.00368 0.00191 0.00077 0.00039 8 0.00432 0.00152 0.00071 0.00025 0.00012 9 0.00204 0.00063 0.00027 0.00008 0.00003

10 0.00095 0.00026 0.0001 0.00003 0.00001 11 0.00043 0.0001 0.00004 0.00001 0 12 0.00018 0.00004 0.00001 0 0 13 0.00006 0.00001 0 0 0 14 0 0 0 0 0

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The effects of thermal Grashof number are shown in Figs. 3.6(a)-3.6(c) for

the velocity , temperature , and species concentration distributions. Fig. 3.6(a)

indicates that an increasing from 0.1 through 1, 2, 4 to 6 strongly boosts velocity in

particular over the zone 0 < < 2. There is a rapid rise in the velocity near the wall

especially for the cases = 4 and = 6. Peak value of = 6 is about ~0.75

the profiles generally descend smoothly towards zero although the rate of descend is

greater corresponding to higher thermal Grashof numbers. defines the ratio of

thermal buoyancy force to the viscous hydrodynamic force and as expected does

accelerate the flow. Temperature distribution versus Y is plotted in figure 3.6(b) and

is seen to decrease with a rise in thermal Grashof number, results which agree with

fundamental studies on free convection (Schlichting boundary layer theory). Increasing

values figure 3.6(c) is seen to considerably reduce concentration function values (C)

throughout the boundary layer.

The effects of species Grashof number on velocity, temperature function

and concentration function are presented in Figs. 3.7(a) to 3.7(c). Velocity, , is

observed to increase considerably with rise in from 0.1 to 6. Hence species

Grashof number boosts velocity of the fluid indicating that buoyancy as an accelerating

effect on the flow field. Temperature T undergoes marked decrease in value however

with a rise in species Grashof number as illustrated in fig 3.7b. All temperatures

descend from unity at the wall to zero in the free stream. The depression in temperature

is maximized by larger species Grashof number in the vicinity ~2.1 , indicating a

similar trend to the influence of thermal Grashof number (figure 3.7b). Dimensionless

concentration, as depicted in fig. 3.7c, is also seen to be reduced by increasing the

species Grashof number. All profiles decay smoothly from unity at the vertical surface

to zero as .

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In figs. 3.8(a)-3.8(c) the variation of dimensionless local skin friction (surface

shear stress), x, Nusselt number (surface heat transfer gradient), Nux and the Sherwood

number (surface concentration gradient), Shx, versus axial coordinate (X) for various

viscoelasticity parameters () and time ( ) are illustrated. Shear stress is clearly

enhanced with increasing viscoelasticity (i.e. stronger elastic effects), i.e. the flow is

accelerated, a trend consistent with our earlier computations in figure 3.8(a). The ascent

in shear stress is very rapid from the leading edge (X = 0) but more gradual as we

progress along the plate surface away from the plane. With an increase in time, t, shear

stress, x, is however increased. Increasing viscoelasticity () is observed in figure

3.8(b) to enhance local Nusselt number, Nux values whereas they are again decreased

with greater time. Similarly in figure 3.8(c), the local Sherwood number Shx values are

elevated with an increase in elastic effects i.e. a rise in from 0 (Newtonian flow)

through 0.001, 0.003 to 0.005 but depressed finally in slightly with time.

In figs. 3.9(a) to 3.9(c), the influence of radiation parameter ( ) and time (t) on

x, Nux and Shx, versus axial coordinate (X) are depicted. An increase in F from 0.1

through 0.5, 1, 3 to 5, strongly increases both x and Nux along the entire plate surface

i.e. for all . However with an increase in time ( ) both shear stress and local Nusselt

number are enhanced. With increasing values, local Sherwood number, Shx, as shown

in figure 3.9(c), is boosted considerably along the plate surface; gradients of the profiles

are also found to diverge with increasing values. However an increase in time

serves to reduce local Sherwood numbers.

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SKIN-FRICTION( ) : Radiation parameter (F) versus Y values

Time(t): 4.850 5.140 5.840 4.530 4.450

Y-axis 0.1 0.5 1.0 3.0 5.0 (F)

0 6.46978 6.51239 6.52971 6.54809 6.55303

0.25 1.28461 1.33217 1.35134 1.37152 1.37683

0.5 0.54781 0.59942 0.62075 0.64385 0.65006

0.75 0.25792 0.31136 0.33394 0.35879 0.36557

1.0 0.0834 0.13813 0.1617 0.18794 0.19515

NUSSELT NUMBER( ) : Radiation parameter (F) versus Y values

Time(t): 4.850 5.140 5.840 4.530 4.450

Y- axis 0.1 0.5 1.0 3.0 5.0 (F)

0 0 0 0 0 0

0.25 0.03833 0.0852 0.11409 0.15758 0.17247

0.5 0.06185 0.12826 0.16811 0.226 0.24536

0.75 0.08359 0.16536 0.21458 0.28505 0.3084

1.0 0.10447 0.19905 0.25658 0.33834 0.3653

SHERWOOD NUMBER( ): Radiation parameter (F) versus Y values

Time(t): 4.850 5.140 5.840 4.530 4.450

Y- axis 0.1 0.5 1.0 3.0 5.0 (F)

0 0 0 0 0 0

0.25 0.16267 0.15837 0.15658 0.15468 0.15419

0.5 0.27819 0.27164 0.2687 0.26542 0.26454

0.75 0.39461 0.38569 0.38154 0.3768 0.3755

1.0 0.5134 0.50209 0.49669 0.49045 0.48871

83

3.5. CONCLUSION

A two-dimensional, unsteady, laminar and incompressible boundary layer

model has been presented for the external flow, heat and mass transfer in a Walters-B

viscoelastic buoyancy-driven radiative flow from an impulsively started vertical plate.

The Walters-B viscoelastic model has been employed which is valid for short memory

polymeric fluids. The dimensionless conservation equations have been solved with the

well-tested, robust, highly efficient and implicit Crank Nicolson finite difference

numerical method. The present computations have shown that increasing viscoelasticity

accelerates the velocity and enhances shear stress (local skin-friction), local Nusselt

number and local Sherwood number, but reduces temperature and concentration in the

boundary layer. The steady state velocity increases with the increase in or and

conversely temperature decreases in the fluid. It is also observed that the velocity

decreases due to increase in Schmidt number . The local heat transfer rate

decreases with increase in . Here the radiation effects on flow. It is found that as the

radiation parameter increases then the rate of heat transfer also increases. This

phenomenon is of interest in very high temperature (e.g. glass) flows in the mechanical

and chemical process industries and is currently under investigation.

84

3.6. GRAPHS

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