research article mathematical modelling of radiative...

Research ArticleMathematical Modelling of Radiative HydromagneticThermosolutal Nanofluid Convection Slip Flow in SaturatedPorous Media

Mohammed Jashim Uddin,1 Osman Anwar Bég,2 and Ahmad Izani Md. Ismail3

1 American International University-Bangladesh, Banani, Dhaka 1213, Bangladesh2 Gort Engovation Research (Biomechanics), Southmere Avenue, Bradford BD73NU, UK3Universiti Sains Malaysia, Penang 11800, Malaysia

Correspondence should be addressed to Mohammed Jashim Uddin; jashim [email protected]

Received 5 January 2014; Revised 5 March 2014; Accepted 14 April 2014; Published 13 May 2014

Academic Editor: Kuppalapalle Vajravelu

Copyright © 2014 Mohammed Jashim Uddin et al. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

High temperature thermal processing of nanomaterials is an active area of research. Many techniques are being investigated tomanipulate properties of nanomaterials for medical implementation. In this paper, we investigate thermal radiation processing of ananomaterial fluid sheet extruded in porous media. Amathematical model is developed using a Darcy drag force model. Instead ofusing linear radiative heat flux, the nonlinear radiative heat flux in the Rosseland approximation is taken into account which makesthe present study more meaningful and practically useful. Velocity slip and thermal and mass convective boundary conditionsare incorporated in the model. The Buongiornio nanofluid model is adopted wherein Brownian motion and thermophoresiseffects are present. The boundary layer conservation equations are transformed using appropriate similarity variables and theresulting nonlinear boundary value problem is solved using Maple 14 which uses the Runge-Kutta-Fehlberg fourth fifth ordernumerical method. Solutions are validated with previous nonmagnetic and nonradiative computations from the literature,demonstrating excellent agreement. The influence of Darcy number, magnetic field parameter, hydrodynamic slip parameter,convection-conduction parameter, convection-diffusion parameter, and conduction-radiation parameter on the dimensionlessvelocity, temperature, and nanoparticle concentration fields is examined in detail. Interesting patterns of relevance are observedto improve manufacturing of nanofluids.

1. Introduction

Nanofluid transport in porous media has developed into asubstantial area of research in recent years. This has beenmotivated by the thermally enhancing properties of nanoflu-ids [1] which are achieved owing to the presence of metallicnanoparticles suspended in base fluids (water, oil, etc.).Recent applications of nanofluid convection in porous mediainclude solar collectors [2, 3], microbial fuel cells [4], materi-als processing [5, 6], biological propulsion [7], and geother-mal energy systems [8] (where nanofluid injection results ingreater thermal efficiency). These investigations have builton earlier seminal theoretical works of Buongiornio [9]and Nield and Kuznetsov [10] wherein elegant formulationsfor nanofluid convection have been developed and which

prioritize Brownianmotion and thermophoresis effects. Elec-trically conducting nanofluid flows which respond to theimposition of magnetic fields have also received some atten-tion in recent years motivated by manufacture of complexfluids for aerospace and industrial systems [11–14]. In variousmanufacturing processes, high temperature effects and alsoporous media are also encountered, in some cases, simulta-neously. Thermal radiation heat transfer [15] is important inthe former. Drag forces induced by the porous material fibersexert a significant influence on fluid flow and heat transfercharacteristics in the latter. Numerous studies of radiative-convective flows in porous media have been communicated(e.g., [16]), and are relevant also to thermal insulation engi-neering and materials fabrication among other technologies.Interesting studies in this regard include Vafai and Tien [17],

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 179172, 11 pageshttp://dx.doi.org/10.1155/2014/179172

2 Mathematical Problems in Engineering

Takhar et al. [18], and Rashidi et al. [19]. In these studiesgenerally algebraic flux models have been used to solve forthe radiative contribution to heat transfer and this is gen-erally simulated via a conduction-radiation parameter. TheRosseland diffusion, non-gray Schuster-Schwartzchild two-flux and othermodels have been employed by the researchers.The Rosseland model is the easiest to implement and allowsradiative effects to be studied via a single dimensionlessparameter, for example, Rosseland number. In the contextof porous media studies, the Darcy drag force model whichrepresents the bulk matrix effect on fluid transport is themost popular although it is limited to low-speed viscous-dominated flows. Numerous articles have utilized this modelfor porous media nanofluid modeling including Nield andKuznetsov [20], Uddin et al. [21], Hady et al. [22], and others[23–26].

The objective of the present investigation is to extend thework of Makinde and Aziz [27] for MHD Navier slip flow ofa nanofluid in porous media over a radiating stretching sheetin the presence of thermal and mass convective boundaryconditions. This problem is of interest in nanomaterialmanufacturing processes. Slip effects have been shown tobe important in stretching sheet flows and recent studiesinclude Hamad et al. [28] and Prasad et al. [29–31]. Thegoverning partial differential boundary layer equations arereduced to a two-point boundary value problem with theaid of appropriate similarity variables.The reduced equationshave been numerically solved by the use of an efficientRunge-Kutta-Fehlberg fourth fifth order numerical methodwhich is available in Maple 14 [32]. The effects of keythermophysical parameters on the fluid velocity, temperature,and nanoparticle volume fraction (concentration) have beenexamined in detail. Validation of Maple solutions is includedusing earlier published results for the nonradiative case asexamined by Dayyan et al. [33]. The current study has notbeen communicated in the scientific literature to the best ofthe authors’ knowledge.

2. Governing Nanofluid Transport Model

Consider a two-dimensional regimewith a coordinate systemwith the 𝑥-axis aligned horizontally and the 𝑦-axis is normalto it. A transverse magnetic field 𝐵

0acts normal to the

bounding surface. The magnetic Reynolds number is smallso that the induced magnetic field is effectively negligiblewhen compared to the applied magnetic field. We neglectthe electric field associated with the polarization of chargesand Hall effects. It is further assumed that the left of theplate is heated by the convection from the hot fluid oftemperature 𝑇

𝑓(> 𝑇𝑤

> 𝑇∞) which provides a variable heat

transfer coefficient ℎ𝑓(𝑥). Consequently a thermal convective

boundary condition arises. It is further assumed that theconcentration in the left of the plate 𝐶

𝑓is higher than that

of the plate concentration 𝐶𝑤and free stream concentration

𝐶∞which provides a variablemass transfer coefficient ℎ

𝑚(𝑥).

As a result a mass convective boundary condition arises. TheOberbeck-Boussinesq approximation is utilized and the fourfield equations are the conservation of mass, momentum,thermal energy, and the nanoparticles volume fraction.These

equations can be written in terms of dimensional forms,extending the formulations of Buongiorno [9] and Makindeand Aziz [27]:

𝜕𝑢

𝜕𝑥

+

𝜕V𝜕𝑦

= 0, (1)

𝜌𝑓(𝑢

𝜕𝑢

𝜕𝑥

+ V𝜕𝑢

𝜕𝑦

) = 𝜇

𝜕2𝑢

𝜕𝑦2−

𝜇

𝐾𝑃

𝑢 − 𝜎𝐵2

0𝑢,

𝑢

𝜕𝑇

𝜕𝑥

+ V𝜕𝑇

𝜕𝑦

= 𝛼

𝜕2𝑇

𝜕𝑦2+ 𝜏{𝐷

𝐵

𝜕𝐶

𝜕𝑦

𝜕𝑇

𝜕𝑦

+ (

𝐷𝑇

𝑇∞

)(

𝜕𝑇

𝜕𝑦

)

2

}

−

1

𝜌𝑓𝑐𝑓

𝜕𝑞𝑟

𝜕𝑦

,

𝑢

𝜕𝐶

𝜕𝑥

+ V𝜕𝐶

𝜕𝑦

= 𝐷𝐵

𝜕2𝐶

𝜕𝑦2+ (

𝐷𝑇

𝑇∞

)

𝜕2𝑇

𝜕𝑦2.

(2)

The appropriate boundary conditions are, following Datta[34] and Karniadakis et al. [35],

𝑢 = 𝑢𝑤+ 𝑢slip, V = 0,

− 𝑘

𝜕𝑇

𝜕𝑦

= ℎ𝑓(𝑇𝑓− 𝑇) , −𝐷

𝐵

𝜕𝐶

𝜕𝑦

= ℎ𝑚(𝐶𝑓− 𝐶)

at 𝑦 = 0,

𝑢 → 0, 𝑇 → 𝑇∞, 𝐶 → 𝐶

∞as 𝑦 → ∞.

(3)

Here 𝛼 = 𝑘/(𝜌𝑐)𝑓: thermal diffusivity of the fluid, 𝜏 =

(𝜌𝑐)𝑝/(𝜌𝑐)𝑓: ratio of heat capacity of the nanoparticle and

fluid,𝐾𝑝: permeability of themedium, (𝑢, V): velocity compo-

nents along 𝑥 and 𝑦 axes, 𝑢𝑤

= 𝑈𝑟(𝑥/𝐿): velocity of the plate,

𝐿: characteristic length of the plate, 𝑢slip = 𝑁1](𝜕𝑢/𝜕𝑦): linearslip velocity, 𝑁

1: velocity slip factor with dimension s/m, 𝜌

𝑓:

density of the base fluid, 𝜎: electric conductivity, 𝜇: dynamicviscosity of the base fluid, 𝜌

𝑃: density of the nanoparticles,

(𝜌𝐶𝑃)𝑓: effective heat capacity of the fluid, (𝜌𝐶

𝑃)𝑃: effective

heat capacity of the nanoparticle material, 𝜀: porosity, 𝐷𝐵:

Brownian diffusion coefficient,𝐷𝑇: thermophoretic diffusion

coefficient, and 𝑞𝑟: radiative heat transfer in 𝑦-direction.

We consider the fluid to be a gray, absorbing-emitting butnonscattering medium. We also assume that the boundarylayer is optically thick and the Rosseland approximation ordiffusion approximation for radiation is valid [36, 37]. Thus,the radiative heat flux for an optically thick boundary layer(with intensive absorption), as elaborated by Sparrow andCess [38], is defined as 𝑞

𝑟= −(4𝜎

1/3𝑘1)(𝜕𝑇4/𝜕𝑦), where 𝜎

1

(= 5.67 × 10−8W/m2 K4) is the Stefan-Boltzmann constantand 𝑘

1(m−1) is the Rosseland mean absorption coefficient.

Purely analytical solutions to the partial differential boundaryvalue problem defined by (1)–(3) are not possible. Even anumerical solution is challenging.Hencewe aim to transform

Mathematical Problems in Engineering 3

the problem to a systemof ordinary differential equations.Wedefine the following dimensionless transformation variables:

𝜂 =

𝑦

√𝐾𝑝

, 𝜓 = 𝑈𝑟

𝑥

𝐿

√𝐾𝑝𝑓 (𝜂) ,

𝜃 (𝜂) =

𝑇 − 𝑇∞

𝑇𝑓− 𝑇∞

, 𝜙 (𝜂) =

𝐶 − 𝐶∞

𝐶𝑓− 𝐶∞

,

(4)

where 𝐿 is the characteristic length. From (4), we have 𝑇 =𝑇∞{1 + (𝑇

𝑟− 1)𝜃}, where 𝑇

𝑟= 𝑇𝑓/𝑇∞

(the wall temperatureexcess ratio parameter) and hence 𝑇4 = 𝑇4

∞{1 + (𝑇

𝑟− 1)𝜃}

4.Substitution of (4) into (2)-(3) generates the following simi-larity equations:

𝑓

+ ReDa (𝑓𝑓 − 𝑓2 − 𝑀𝑓) − 𝑓 = 0, (5)

𝜃+ RePrDa𝑓𝜃 + Pr [Nb𝜃𝜙 +Nt𝜃2]

+

4

3𝑅

[{1 + (𝑇𝑟− 1) 𝜃}

3𝜃]

= 0,

(6)

𝜙+ LeReDa𝑓𝜙 + Nt

Nb𝜃= 0. (7)

The relevant boundary conditions are

𝑓 (0) = 0, 𝑓(0) = 1 + 𝑎𝑓

(0) ,

𝜃(0) = −Nc [1 − 𝜃 (0)] , 𝜙 (0) = −Nd [1 − 𝜙 (0)] ,

𝑓(∞) = 𝜃 (∞) = 𝜙 (∞) = 0,

(8)

where primes denote differentiation with respect to 𝜂. Thethermophysical dimensionless parameters arising in (5)–(8)are defined as follows: Re = 𝑈

𝑟𝐿/] is the Reynolds number,

Da = 𝐾𝑝/𝐿2 is the Darcy number, 𝑀 = 𝜎𝐵2

0𝐿/𝑈𝑟𝜌 is the

magnetic field parameter, Pr = ]/𝛼 is the Prandtl number,Nt = 𝜏𝐷

𝑇(𝑇𝑓− 𝑇∞)/]𝑇∞

is the thermophoresis parameter,Nb = 𝜏𝐷

𝐵(𝐶𝑓− 𝐶∞)/] is the Brownian motion parameter,

𝑅 = 𝑘𝑘1/4𝜎1𝑇3

∞is the convection-radiation parameter, Le =

]/𝐷𝐵is the Lewis number, 𝑎 = 𝑁

1]/√𝐾

𝑃is the hydrody-

namic (momentum) slip parameter, Nd = ℎ𝑚√𝐾𝑃/𝐷𝐵is the

convection-diffusion parameter, and Nc = ℎ𝑓√𝐾𝑃/𝑘 is the

convection-conduction parameter.Quantities of physical interest are the local friction factor,

𝐶𝑓𝑥, the local Nusselt number, Nu

𝑥, and the local Sherwood

number, Sh𝑥. Physically, 𝐶

𝑓𝑥represents the wall shear stress,

Nu𝑥defines the heat transfer rates, and Sh

𝑥defines the mass

transfer rates:

𝐶𝑓𝑥Re−1𝑥Da0.5𝑥

= 2𝑓(0) ,

Nu𝑥Da0.5𝑥

= − [1 +

4

3𝑅

{1 + (𝑇𝑟− 1) 𝜃 (0)}

3] 𝜃(0) ,

Sh𝑥Da0.5𝑥

= −𝜙(0) ,

(9)

where Da𝑥

= 𝐾𝑝/𝑥2 is the local Darcy number for Darcian

porousmedia andRe𝑥= 𝑢𝑤𝑥/] is the local Reynolds number.

We note that, for purely hydromagnetic boundary layer (𝑀 =0) and no slip boundary condition (𝑎 = 0), the problemreduces to the problem which has been recently consideredand investigated by Dayyan et al. [33] when Nc = Nd = 𝑅 →∞, Da = 1, Nt = 0, and Nb → 0 in our model and 𝑛 = 0 intheir paper. This provides a useful benchmark for validatingthe present model.

3. Numerical Solutions

The set of nonlinear ordinary differential equations (5)–(7)subject to the boundary conditions in (8) have been solvednumerically using Maple dsolve command with numericoption. This software uses the Runge-Kutta-Fehlberg fourthfifth (RKF45) order numerical method for solving two-point boundary value problem. The Runge-Kutta-Fehlbergfourth fifth order numerical method is a well-establishedadaptive numerical method for solving system of ordinarydifferential equations with associated conditions.The Runge-Kutta-Fehlberg algorithm uses both a fifth and a fourth orderRunge-Kutta. The error of this algorithm is determined bysubtracting these two values and can be used for adaptive stepsizing. The formula for the fifth fourth order Runge-Kutta-Fehlberg algorithm is given below:

𝑘0= 𝑓 (𝑥

𝑖, 𝑦𝑖) ,

𝑘1= 𝑓(𝑥

𝑖+

1

4

ℎ, 𝑦𝑖+

1

4

ℎ𝑘0) ,

𝑘2= 𝑓(𝑥

𝑖+

3

8

ℎ, 𝑦𝑖+ (

3

32

𝑘0+

9

32

𝑘1) ℎ) ,

𝑘3= 𝑓(𝑥

𝑖+

12

13

ℎ, 𝑦𝑖+ (

1932

2197

𝑘0−

7200

2197

𝑘1+

7296

2197

𝑘2) ℎ) ,

𝑘4=𝑓(𝑥

𝑖+ ℎ, 𝑦

𝑖+ (

439

216

𝑘0− 8𝑘1+

3860

513

𝑘2−

845

4104

𝑘3) ℎ) ,

𝑘5= 𝑓(𝑥

𝑖+

1

2

ℎ, 𝑦𝑖

+(−

8

27

𝑘0+ 2𝑘1−

3544

2565

𝑘2+

1859

4104

𝑘3−

11

40

𝑘4) ℎ) ,

𝑦𝑖+1

= 𝑦𝑖+ (

25

216

𝑘0+

1408

2565

𝑘2+

2197

4104

𝑘3−

1

5

𝑘4) ℎ,

𝑧𝑖+1

= 𝑧𝑖

+ (

16

135

𝑘0+

6656

12825

𝑘2+

28561

56430

𝑘3−

9

50

𝑘4+

2

55

𝑘5) ℎ,

(10)

where 𝑦 is a fourth-order Runge-Kutta and 𝑧 is a fifth-orderRunge-Kutta. An estimate of the error can be obtained bysubtracting the two values obtained. If the error exceeds aspecified threshold, the results can be recalculated using asmaller step size.The approach to estimating the new step sizeis shown below:

ℎnew = ℎold(𝜀ℎold

2𝑧𝑖+1

− 𝑦𝑖+1

)

1/4

. (11)


Table 1: Comparison of 𝑓(𝜂), 𝑓(𝜂), and 𝜃(𝜂) when Da = 1.

𝜂

𝑓(𝜂) 𝑓(𝜂) 𝜃(𝜂)

Dayyan et al. [33](HAM)

Present(Maple RKF45)





0.0 0.00000 0.00000 0.99999 0.99999 0.99999 0.999990.2 0.17042 0.17420 0.75376 0.75364 0.89633 0.900370.4 0.30564 0.30549 0.56813 0.56797 0.80205 0.804080.6 0.40571 0.40443 0.42820 0.42804 0.71301 0.713430.8 0.47505 0.47900 0.32272 0.32259 0.62792 0.62971.0 0.53802 0.53519 0.24322 0.24312 0.55226 0.553751.2 0.58010 0.57755 0.18330 0.18322 0.48134 0.485411.4 0.61161 0.60947 0.13814 0.13808 0.42056 0.424511.6 0.64096 0.63352 0.10410 0.10406 0.36595 0.370591.8 0.65374 0.65165 0.07845 0.07843 0.31817 0.323072.0 0.66952 0.66531 0.05912 0.05911 0.27657 0.28137

Table 2: Comparison of Skin friction factor (−𝑓(0)) for severalReynolds numbers when Da = 1, 𝑎 = 𝑀 = 0, and 𝑅 → ∞.

Re Dayyan et al. [33] Present resultsRK HAM Maple RKF45

1 1.4242 1.4198 1.41981.5 1.5811 1.5799 1.58082 1.7320 1.7234 1.73195 2.4494 2.4394 2.4492

Table 3: Comparison between RKF45, HAM, and RK for the valuesof heat transfer rate (−𝜃(0)) for several values of Reynolds numberwhen Nc = Nd = 𝑅 → ∞ and Pr = Da = 1.

Re Dayyan et al. [33] Present resultRK HAM Maple RKF45

1 0.5033 0.5030 0.50381.5 0.6422 0.6456 0.64302 0.7592 0.7518 0.75395 1.2576 1.2636 1.2551

The step size is taken as Δ𝜂 = 0.001 and the convergencecriterionwas set to 10−6.The asymptotic boundary conditionsgiven by (8) were replaced by using a value of 10 for thesimilarity variable 𝜂max as follows:

𝜂max = 10, 𝑓(10) = 𝜃 (10) = 𝜙 (∞) = 0. (12)

The choice of 𝜂max = 10 ensured that all numerical solutionsapproached the asymptotic values in correct manner. Thecompilation times of this algorithm are of the order of severalminutes on personal computer.

4. Results and Discussion

To check the accuracy of the present Maple code, com-parisons of the skin friction factor and heat transfer ratehave been conducted with published results obtained by theRunge-Kutta and homotopy analysis methods. The compar-isons are presented in Tables 1, 2, and 3. Very good agreement

1086420

1

0.8

0.6

0.4

0.2

0

f (𝜂)

𝜂

M

0.0

10.0

Pr = 6.8, Le = R = 10,

Da = 0.5, Re = 1.0, Tr =

Nt = Nb = Nc = Nd = 0.1

a = 0.0, 0.3 and 0.8

2 and

Figure 1: Effect of hydrodynamic slip on velocity for variousmagnetic parameters.

between the Maple code and the other results confirms theaccuracy of the method used.

In Figures 1–10, we have examined only magnetic field,hydrodynamic slip, conduction-radiation, convection-con-duction, and convection-diffusion effects on the heat, mass,andmomentum transfer characteristics.The nanofluid (ther-mophoresis, Brownian motion), Lewis number, and Prandtlnumber effects are well known and have been elucidated indetail in other studies (see, for example, Uddin et al. [5]). Forbrevity we consider only specific parameter effects. Reynoldsnumber (Re) is fixed at unity and this is valid for low-speedDarcian transport.


1086420

𝜂

0.25

0.20

0.15

0.10

0.05

0

𝜃(𝜂)

M

0.0

10.0

Pr = 6.8, Le = R = 10,Da = 0.5, Re = 1.0, Tr =Nt = Nb = Nc = Nd = 0.1

a = 0.0, 0.3 and 0.8

2 and

Figure 2: Effect of hydrodynamic slip on temperature for variousmagnetic parameters.

1086420

𝜂

0.20

0.15

0.10

0.05

0

M

0.0

10.0

𝜙(𝜂)

Pr = 6.8, Le = R = 10,Da = 0.5, Re = 1.0, Tr =Nt = Nb = Nc = Nd = 0.1

2 and

a = 0.0, 0.3 and 0.8

Figure 3: Effect of hydrodynamic slip on concentration for variousmagnetic parameters.

1086420

𝜂M

0.0

10.0

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

−0.1

Pr = 6.8, = 10, = = = 1,

= = a = 0.1, R =

f (𝜂)

Le Re

Nt

NcNd

Nb

Da = 0.5, 1.0 and 2.0

10 and Tr = 2

Figure 4: Effect of Darcy number on temperature for variousmagnetic parameters.

1086420

𝜂

M

0.0

10.0

0.5

0.4

0.3

0.2

0.1

0

𝜙(𝜂)

Pr = 6.8, Le = 10, Re = Nd = Nc = 1,

Nt = Nb = a = 0.1, R = 10, Tr = 2

Da = 0.5, 1.0 and 2.0

Figure 5: Effect of Darcy number on concentration for variousmagnetic parameters.


1086420

𝜂

M

0.0

10.0

𝜃(𝜂)

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0

Pr = 6.8, Le = 10, a = 0.0,Re = Da = Nd = 1, Tr =

Nt = Nb = Nc = 0.1

R = 0.5,

2 and

1.0 and 2.0

(a)

0.5

0.6

0.7

0.4

0.3

0.2

0.1

0

𝜂

43210

𝜃

R = 0.1, 1, 2, 5, 10, 50, 100 and ∞

(b)Figure 6: (a) Effect of conduction-radiation parameter on temperature for various magnetic parameters. (b) Effect of conduction-radiationparameter on temperature profiles [39].

1086420

𝜂

M

0.0

5.0

𝜃(𝜂)

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

−0.1

Pr = 6.8, Le = 10, Re = 1,

Nd = 1, Da = 0.5, R =Nt = Nb = 0.1, a = 0,

Nc = 0.2, 1.5 and 3.0

10 and Tr = 2

Figure 7: Effect convection-conduction parameter on temperaturefor various magnetic parameters.

Figure 1 shows the variation of the velocity through theboundary layer with momentum slip parameter (𝑎) and

1086420

𝜂

M

0.0

5.0

0.5

0.4

0.3

0.2

0.1

0

𝜙(𝜂)

Pr = 6.8, Le = 10, Re = 1,

Da = 0.5, R =Nt = Nb = 0.1, a = 0, Nd = 1,

Nc = 0.2, 1.5 and 3.0

10 and Tr = 2

Figure 8: Effect convection-conduction parameter on concentra-tion for various magnetic parameters.

various magnetic field parameters (𝑀). It is found thatthe dimensionless velocity is decreased as the velocity slipparameter increases for both in the presence and absence


1086420

𝜂

M

0.0

10.0

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0

𝜃(𝜂)

Pr = 6.8, Le = 10, Re = 1,

Da = 0.5, R =Nc = Nt = Nb = 0.1, a = 0,

10 and Tr = 2

Nd = 0.5, 1.5 and 3.0

Figure 9: Effect convection-diffusion parameter on temperature forvarious magnetic parameters.

of magnetic field. Physically, the velocity slipping parameter𝑎 enhances the differences between the wall and the fluidvelocities adjacent to the wall increases. Further, it is seenthat, as the slipping parameter enhances, the thickness ofthe velocity boundary layer decreases. Increasing the slippingfactor may be looked at as a miscommunication betweenthe source of motion (the plate) and the fluid domain. Itis clear that hydrodynamic behavior of the problem underconsideration is more sensitive to the variations in smallvalues of 𝑎 as compared with the variations in large values of𝑎. Note that the case 𝑎 = 0 corresponds to the conventional noslip boundary condition at the wall.There is a strong decreasein velocity with magnetic filed as well as slip parameters.Clearly the presence of a magnetic field retards the flow forboth slip flow and nonslip flow. An application of a transversemagnetic field to an electrically conducting fluid gives rise toa resistive type force known as the Lorentz force. This forcehas the tendency to retard the motion of the fluid in theboundary layer and to increase its temperature and concen-tration. The presence of the Lorentzian hydromagnetic dragimpedes boundary layer flow strongly and serves as a potentcontrol mechanism. Momentum boundary layer thickness isgreatly reduced with strong magnetic field and also strongmomentum slip. Asymptotic convergence of all profiles isobserved with greater transverse coordinate, testifying to theaccurate imposition of infinity boundary conditions.

Figure 2 demonstrates that, with an increase in momen-tum slip parameter, the temperature is substantially elevatedthroughout the boundary layer. Physically, as the velocity slipparameter 𝑎 increases, less flow will be induced close to the

1086420

𝜂

M

0.0

10.0

0.5

0.6

0.7

0.4

0.3

0.2

0.1

0

𝜙(𝜂)

Pr = 6.8, Le = 10, Re = 1,

Da = 0.5, R =Nc = Nt = Nb = 0.1, a = 0,

10 and Tr = 2

Nd = 0.5, 1.5 and 3.0

Figure 10: Effect of convection-diffusion parameter on concentra-tion for various magnetic parameters.

plate layer and, hence, the hot plate heats a lesser amount offluid and this causes higher increases in the fluid temperature.Temperatures are also observed to be markedly higher withstrong magnetic field (𝑀 = 10.0) as compared with anabsence of magnetic field (𝑀 = 0.0). Greater momentum slipclearly aids in thermal diffusion from the wall to the bodyof the fluid, which results in a heating of the boundary layerand significant increase in thermal boundary layer thickness.Withmagnetic field being present, Lorentzianmagnetic bodyforce (see (5)) necessitates greater work from the fluid tosustain motion.This supplementary work needed to drag thefluid against themagnetic field is dissipated as thermal energy(heat) and thermal boundary layer thickness is thereforeelevated. With strong momentum slip and magnetic field(𝑎 = 0.8; 𝑀 = 10.0), thermal boundary layer thickness ismaximum with the converse apparent for the nonslip andnonconducting case (𝑎 = 0, 𝑀 = 0). The presence ofstrong wall slip and magnetic field therefore significantlyalters the temperature distribution in the regime and thisis important in manipulating material characteristics inprocessing operations.

Figure 3 also demonstrates that increasing momentumslip and magnetic field serves to enhance the nanoparticleconcentration. Concentration boundary layer thickness willtherefore be strongly increased with greater wall slip andmagnetic field strength. With wall slip being absent, concen-trationmagnitudes are clearly observed to be reduced.Gener-ally Figures 1–3 show that momentum diffusion is inhibitedwith hydrodynamic slip and magnetic field whereas energyand species (nanoparticle concentration) diffusion are aided.


Figures 4 and 5 illustrate the temperature and nanopar-ticle concentration response to a variation in Darcy andmagnetic field parameters. Physically, greater Darcy numberimplies a greater permeabiity in the porous medium. Thiscorresponds to a decrease in presence of solid fibers anda reduction in thermal conduction heat transfer within themedium. Increasing Da values will therefore result in a fall intemperatures in the regime, as clearly observed in Figure 4.This will be accompanied by a decrease in thermal boundarylayer thickness. Similarly Figure 5 shows that concentrationmagnitudes are depressed with greater Darcy number, andthis results in a decrease in concentration boundary layerthickness. The negative effect of increasing magnetic field onboth temperatures and concentrations is confirmed in Fig-ures 4 and 5. Effectively the presence of a porousmedium andmagnetic field may be exploited to regulate temperature andnanoparticle distributions which is of considerable advantagein the manufacture of nanomaterials where specific spatialproperties may be required for different technological appli-cations, as elaborated by Rana et al. [40].

The thermal radiation features in the energy conservationequation (6) are inversely proportional to the conduction-radiation parameter 𝑅. Thus, small 𝑅 signifies a large radia-tion effect while 𝑅 → ∞ corresponds to no radiation effect.The effect of conduction-radiation parameter (nonlinearRosseland parameter, 𝑅) on the dimensionless temperatureevolution in the boundary layer is shown in Figure 6(a). Itrepresents the relative contribution of thermal conductionheat transfer to thermal radiation heat transfer and whenit assumes unity value, both heat transfer modes contributeequally. For 𝑅 > 1, conduction will dominate and vice versafor 𝑅 < 1. Clearly as 𝑅 is increased, thermal radiationcontribution is depressed and this is manifest with a con-siderable depletion in temperatures in the boundary layer.Therefore, for low values of 𝑅, thermal radiation is strongand this will correspond to maximum values of temperatureand thicker thermal boundary layers. Evidently the presenceof strong thermal radiation flux is demonstrated to heatthe thermal boundary layer markedly and is beneficial tomaterials processing systems where higher temperatures areoften required to alter material characteristics. The trends inthe present computations have been verified by many otherstudies in the literature including Rahman and Eltayeb [39],Das [41], Pantokratoras and Fang [42], and Mushtaq et al.[43]. In order to further verify the present numerical method,we compare our Figure 6(a) with Rahman and Eltayeb [39](see Figure 6(b)). A close agreement is found between presentfigure and published figure.

Figures 7 and 8 depict the influence of the convection-conduction parameter (Nc) (essentially the Biot number)on the dimensionless temperature and nanoparticle con-centration profiles. This parameter arises only in the wallthermal boundary condition (8). It is worth mentioning thatthe convection-conduction parameter, Nc, is the ratio of theinternal thermal resistance of a solid to the boundary layerthermal resistance. When Nc = 0 (i.e., without Biot number)the left side of the plate with hot fluid is totally insulated,the internal thermal resistance of the plate is extremely high,and no convective heat transfer to the cold fluid on the right

side of the plate takes place. Here the results for constantwall temperature case 𝜃(0) = 1 can be recovered whenNc → ∞. It exerts a substantial influence on temper-ature evolution (Figure 7), in particular, as anticipated, atthe wall. As Nc is increased, there is a strong elevation inwall temperature values. The larger values of Nc accompanythe stronger convective heating at the sheet which rises thetemperature gradient at the sheet. This allows the thermaleffect to penetrate deeper into the quiescent fluid. Due tothis reason the temperature and thermal boundary layerthickness are increasing functions of Nc. The magnitudeswithmagnetic field being present are also considerably higherthan with magnetic field being absent. Thermal boundarylayer thickness is therefore minimized with low values of theconvection-conduction parameter. The concentration field isdriven by the temperature gradient and since temperature isan increasing function of Nc, one would expect an increasein 𝜙with an increase in Nc.This is what we can observe fromFigure 8. Concentration boundary layer thickness is there-fore also enhanced with increasing convection-conductionparameter.

Figures 9 and 10 display the effect of the convection-diffusion parameter (Nd) on temperature and nanoparti-cle concentration (species) distributions, respectively. Thisparameter also features only in a single boundary condition,namely, the wall concentration gradient boundary conditionin (8), and is inversely proportional to the Brownian diffusioncoefficient. A strong enhancement in both temperatures andconcentration values is generated with increasing Nd values.Both thermal and concentration boundary layer thicknessare therefore enhanced. With magnetic being field present,magnitudes of temperature and nanoparticle concentrationare always greater than for the electrically nonconductingcase (𝑀 = 0). As with the Nc parameter, the most significantresponse in temperature and nanoparticle concentrationprofiles is witnessed at the wall, since both parameters aresimulated via wall boundary conditions.

5. Conclusions

A mathematical model has been developed for steady-statetwo-dimensional incompressible laminar nanofluid mag-netohydrodynamic convective-radiative flow in a porousmedium adjacent to a vertical sheet. The left of the sheet isheated by the convection from a hot fluid which provides avariable heat transfer coefficient leading to the presenceof a thermal convective boundary condition. Hydrody-namic slip and thermal radiation heat transfer have beenincluded in the model and furthermore a mass convec-tive boundary condition is incorporated. The transformedmomentum, energy, and nanoparticle concentration bound-ary layer equations have been shown to feature a num-ber of thermophysical parameters, namely, Reynolds num-ber, Darcy number, Prandtl number, Lewis number, mag-netic field parameter, thermophoresis parameter, Brownianmotion parameter, hydrodynamic (momentum) slip param-eter, convection-conduction parameter, convection-diffusionparameter, and conduction-radiation parameter (nonlinear


Rosseland parameter). A Maple numerical solution employ-ing Runge-Kutta-Fehlberg quadrature has been obtained forthe strongly nonlinear boundary value problem. The effectsof several parameters on the momentum, heat, and masstransfer (species diffusion) characteristics of the nanofluidhave been evaluated in detail. Computations have also beenvalidated with earlier published results, demonstrating verygood correlation. The present solutions have shown thefollwoing.

(i) Increasing hydrodynamic (momentum) slip signif-icantly retards the boundary layer flow, whereas itmarkedly increases temperature and nanoparticleconcentration values.

(ii) Increasing magnetic field damps the velocity char-acteristics, that is, decelerates the flow, whereas itenhances both temperatures and nanoparticle con-centration values.

(iii) Increasing Darcy number (corresponding to greaterporous medium permeability) strongly boosts thetemperature and concentration magnitudes.

(iv) Decreasing Rosseland conduction-radiation parame-ter (which corresponds to greater thermal radiativeheat flux presence) notably elevates temperatures inthe boundary layer.

(v) Increasing convection-conduction parameter (Nc)(which is simulated via a thermal convective bound-ary condition) strongly enhances both temperatureand nanoparticle concentration values.

(vi) Increasing convection-diffusion parameter (Nd)(which is simulated via a mass convective boundarycondition) strongly enhances both temperature andnanoparticle concentration values.

The present study has considered Newtonian steady-statemagnetoconvective nanofluid flow. Transient [44] and non-Newtonian effects [45] are also relevant to magnetic nano-materials processing operations, and these will be consideredin the near future.

Nomenclature

𝑎: Slip parameter𝐵0: Constant magnetic field

𝐶: Dimensional concentration𝑐𝑝: Specific heat at constant pressure

(J/kg K)𝐶𝑓𝑥: Skin friction factor

𝐷𝐵: Brownian diffusion coefficient

𝐷𝑇: Thermophoretic diffusion coefficient

𝑓(𝜂): Dimensionless stream functionℎ𝑓(𝑥): Local heat transfer coefficient (W/m2 K)

(ℎ𝑓)0: Constant heat transfer coefficient(W/m2 K)

ℎ𝑚(𝑥): Local mass transfer coefficient

(ℎ𝑚)0: Constant mass transfer coefficient

𝑘: Thermal conductivity (m2/s)𝐿: Characteristic length of the plate (m)

Le: Lewis number𝑀: Magnetic field parameter𝑁1(𝑥): Variable velocity slip factor

Nb: Brownian motion parameterNc: Conduction-convection parameterNd: Diffusion-convection parameterNt: Thermophoresis parameterNu𝑥: Heat transfer rate

Pr: Prandtl number𝑝: Pressure (Pa)𝑞𝑚: Wall mass flux (kg/sm2)

𝑞𝑤: Wall heat flux (W/m2)

Re: Reynolds numberRe𝑥: Local Reynolds number

𝐴: Thermal conductivity parameterSh𝑥: Local Sherwood number

𝑇: Dimensional temperature𝑢, V: Velocity components along axes𝑢𝑒: External velocity (m/s)

𝑈∞: Reference velocity (m/s)

𝑥, 𝑦: Cartesian coordinates aligned along andnormal to the plate (m).

Greek Symbols

𝛼: Thermal diffusivity (m2/s)𝛼𝑖: Real numbers

𝛽: Pressure gradient parameter𝜏: Ratio of effective heat capacity of the

nanoparticle material to the heatcapacity of the fluid

𝜎: Electric conductivity𝜃(𝜂): Dimensionless temperature𝜙(𝜂): Dimensionless concentration

(nanoparticle volume fraction)𝜂: Similarity variable𝜇: Dynamic viscosity of the fluid (Ns/m2)]: Kinematic viscosity of the fluid (m2/s)𝜌𝑓: Fluid density (kg/m3)

𝜓: Stream function.

Subscripts

𝑤: Condition at the wall∞: Ambient condition.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

The authors acknowledge financial support from UniversitiSains Malaysia, RU Grant 1001/PMATHS/811252.

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