Scholarly article on topic 'Effect of thermal radiation and Hall current on heat and mass transfer of unsteady MHD flow of a viscoelastic micropolar fluid through a porous medium'

Effect of thermal radiation and Hall current on heat and mass transfer of unsteady MHD flow of a viscoelastic micropolar fluid through a porous medium Academic research paper on "Mathematics"

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Abstract of research paper on Mathematics, author of scientific article — B.I. Olajuwon, J.I. Oahimire, M. Ferdow

Abstract Heat and mass transfer effects on unsteady flow of a viscoelastic micropolar fluid over an infinite moving permeable plate in a saturated porous medium in the presence of a transverse magnetic field with Hall effect and thermal radiation are studied. The governing system of partial differential equations is transformed to dimensionless equations using dimensionless variables. The dimensionless equations are then solved analytically using perturbation technique to obtain the expressions for velocity, microrotation, temperature and concentration. With the help of graphs, the effects of magnetic field parameter M, thermal radiation parameter Nr, Hall current parameter m, K, viscoelastic parameter a, and slip parameter h on the velocity, microrotation, temperature and concentration fields within the boundary layer are discussed. The result showed that increase in Nr and m increases translational velocity across the boundary layer while (a) decreases translational velocity in the vicinity of the plate but the reverse happens when away from the plate. As h increases the translational velocity across the boundary layer increases. The higher the values of Nr, the higher the micro-rotational velocity effect while m lowers it. Also the effects n, a, m, Nr, Pr and Sc on the skin friction coefficient, Nusselt number and Sherwood numbers are presented numerically in tabular form. The result also revealed that increase in n reduces the skin friction coefficient. Pr enhances the rate of heat transfer while Sc enhances the rate of mass transfer.

Academic research paper on topic "Effect of thermal radiation and Hall current on heat and mass transfer of unsteady MHD flow of a viscoelastic micropolar fluid through a porous medium"

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Engineering Science and Technology, an International Journal

journal homepage: http://www.elsevier.com/locate/jestch

Full length article

Effect of thermal radiation and Hall current on heat and mass transfer of unsteady MHD flow of a viscoelastic micropolar fluid through a porous medium

B.I. Olajuwon a, J.I. Oahimire b' *, M. Ferdow c

a Department of Mathematics, Federal University of Agriculture, Abeokuta, Nigeria b Department of Mathematics, University Port — Harcourt, Port — Harcourt, Nigeria c Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh

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ARTICLE INFO

Article history: Received 19 March 2014 Received in revised form 20 May 2014 Accepted 20 May 2014 Available online 2 August 2014

Keywords: Micropolar fluid Perturbation technique Heat and mass transfer Hall effect Thermal radiation Porous medium

ABSTRACT

Heat and mass transfer effects on unsteady flow of a viscoelastic micropolar fluid over an infinite moving permeable plate in a saturated porous medium in the presence of a transverse magnetic field with Hall effect and thermal radiation are studied. The governing system of partial differential equations is transformed to dimensionless equations using dimensionless variables. The dimensionless equations are then solved analytically using perturbation technique to obtain the expressions for velocity, microrotation, temperature and concentration. With the help of graphs, the effects of magnetic field parameter M, thermal radiation parameter Nr, Hall current parameter m, K, viscoelastic parameter a, and slip parameter h on the velocity, microrotation, temperature and concentration fields within the boundary layer are discussed. The result showed that increase in Nr and m increases translational velocity across the boundary layer while (a) decreases translational velocity in the vicinity of the plate but the reverse happens when away from the plate. As h increases the translational velocity across the boundary layer increases. The higher the values of Nr, the higher the micro-rotational velocity effect while m lowers it. Also the effects n, a, m, Nr, Pr and Sc on the skin friction coefficient, Nusselt number and Sherwood numbers are presented numerically in tabular form. The result also revealed that increase in n reduces the skin friction coefficient. Pr enhances the rate of heat transfer while Sc enhances the rate of mass transfer.

Copyright © 2014, Karabuk University. Production and hosting by Elsevier B.V. All rights reserved.

1. Introduction

Heat and mass transfer from different geometries embedded in porous media has many engineering and geophysical applications such as drying of porous solids, thermal insulations, cooling of nuclear reactors, crude oil extraction, underground energy transport, etc. Micropolar fluids are those consisting of randomly oriented particles suspended in a viscous medium, which can undergo a rotation that can affects the hydrodynamics of the flow, making it a distinctly non-Newtonian fluid. They constitute an important branch of non-Newtonian fluid dynamics where microrotation effects as well as microinertia are exhibited. The theory of Micropolar fluids originally developed by Eringen [2] has been a popular field of research in recent years. Eringen's theory has provided a good model for studying a number of complicated fluids, such as colloidal fluids, polymeric fluids and blood. Micropolar fluid flow

* Corresponding author.

E-mail address: ishola_1@hotmail.com (J.I. Oahimire). Peer review under responsibility of Karabuk University.

induced by the simultaneous action of buoyancy forces is of great interest in nature and in many industrial applications as drying processes, solidification of binary alloy as well as in astrophysics, geophysics and oceanography.

When the strength of the magnetic field is strong, one cannot neglect the effect of Hall current. It is of considerable importance and interest to study how the results of the hydrodynamical problems get modified by the effect of Hall currents. Hall currents give rise to a cross flow making the flow three dimensional. Several authors [1,3—10] studied MHD flow of a micropolar fluid. Rakesh [11] studied effect of slip conditions and Hall current on unsteady MHD flow of a viscoelastic fluid past an infinite vertical porous plate through porous medium.

We extended the work of Rakesh [11] by incorporating angular momentum and concentration equations with thermal radiation term to study Hall current and thermal radiation effect on heat and mass transfer of unsteady MHD flow of a viscoelastic micropolar fluid through a porous medium. The governing equations are solved analytically using perturbation method and effect of various physical parameters are discussed numerically and graphically.

http://dx.doi.org/10.1016/jjestch.2014.05.004

2215-0986/Copyright © 2014, Karabuk University. Production and hosting by Elsevier B.V. All rights reserved.

Abbreviations j* microinertia per unit mass

N1, N microrotation components

B0 magnetic flux density k thermal conductivity

n parameter that relates to microgyration vector to shear V0 scale of suction velocity

stress K* permeability of the porous mediun

e* concentration x*, y* and z* distance along axes

K0 limiting viscosity M magnetic field parameter

Cf skin friction coefficient m Hall current parameter

Nu Nusselt number m1 Maxwell's reflection coefficient

Cm couple stress coefficient qr radiative heat flux

P pressure Nr radiation parameter

Cp specific heat at constant pressure

Pr Prandtl number Greek symbol

n chemical molecular diffusivity a fluid thermal diffusivity

Sc Schmidt number m fluid dynamic viscosity

g acceleration due to gravity bt coefficients of thermal expansion

Sh Sherwood number p fluid density

L* characteristic length bc coefficient of concentration expansion

t* time s electrical conductivity

Gc modified Grashof number n fluid kinematic viscosity

T* temperature y spin gradient viscosity

Gr Grashof number nr* fluid kinematic rotational viscosity

u*, v* and w* components of velocities w* frequency of oscillation

2. Mathematical formulation and method

2.1. Mathematical formulation

We consider the unsteady flow of a viscous incompressible and electrically conducting viscoelastic micropolar fluid over an infinite vertical porous plate, subjected to a constant transverse magnetic field B0 in the presence of thermal and concentration buoyancy effects. The induced magnetic field is assumed to be negligible compared to the applied magnetic field. The x*-axis is taken along the planar surface in the upward direction and the y*-axis taken to be normal to it as shown in Fig. 1. Due to the infinite plane surface assumption, the flow variables are function of y* and t* only. The plate is subjected to a constant suction velocity V0

The governing equations of flow under the usual Boussinesq approximation are given by

du* *du* . d2u* d3u* dN1 / * .

v* + vdy* = (n + nr)dy*2-Kodt*dy*2 +nrdy- + gbT{T -

B0 (V + mw) n

r___L___1

C* - - s p(1 + m2)' - Ku

d* + v d* = (n + nr)"dTT - K0^*d,*2 - nr

dy*2 0dt*dy*2

n * - —w

,*f dN* *dN*\

pJ ^âF + v ârj = g

p (1 + m2) K

*( dN* *dN*\

pJ ^sF + vW) =

dT* * —r + v ■ » dt**

dy* dy

dT* _ k d2 T* 1 dq dy* = pep dy*2 - pep dy

dC* *9C* _ n_

d* + v dy* = dy*2

(6) (7)

The appropriate boundary conditions for the problem are

du \ * ,*(du \ ,,* du

; w = Udy*J ' N* = -ndy*

N* = n dfT * = t; + (t; - t; )eiu*t*

Fig. 1. Physical model.

c* = c + c; - c; eiu f at y* = 0

* * w - mu

/0, N**/0, N2 -

at y /œ (8)

Substituting Equation (13) into Equations (2)—(8) yield the following dimensionless equations:

where u*, v* and w* are velocity components along x*, y* and z*-axis respectively, N* and N* are microrotation components along x* and z*-axis respectively, n is the kinematic viscosity, nr is the kinematic micro-rotation viscosity, K0 is the limiting viscosity, g is the acceleration due to gravity, br and bc are the coefficients of thermal expansion and concentration expansion respectively, T* is the dimensional temperature of the fluid, T^ and T* denotes the temperature at the plate and temperature far away from the plate respectively, C* is the dimensional concentration of the solute, Cw and C* are concentration of the solute at the plate and concentration of the solute far from the plate respectively, K* is the permeability of the porous medium, k is the thermal conductivity of the medium, p is the density of the fluid, j* is the micro-inertia density or micro-inertia per unit mass, g is the spin gradient viscosity, L* is the characteristic length, u* is the dimensional frequency of oscillation, a is the electrical conductivity, m is the Hall current parameter and D is the molecular diffusivity, qr is the radiative heat flux.

The constant that related to microgyration vector and shear stress is n, where 0 < n < 1. The case n = 0 represents concentrated particle flows in which the microelement close to the wall surface are unable to rotate. This case is also known as the strong concentration of microelements. The case n = 0.5 indicates the vanishing of anti-symmetric part of the stress tensor and denotes weak concentration of microelements. The case n = 1 is used for the modeling of turbulent boundary layer flows. We shall consider n = 0 and n = 0.5.

Following Rosseland approximation the radiative heat flux qr is modeled as

4s* vT*4 3 k* dy*

where s* is the Stefan—Boltzman constant and k* is the mean absorption coefficient. Assuming that the difference in temperature within the flow are such that T*4 can be expressed as a linear combination of the temperature, we expand T*4 in Taylor's series about T* as follows:

T*4 + 4T*

1 du du , d2u d3u dN1

—---=(1 + b)—T - a-T + b—1 -

Ad* dh V -T h> a._2 dtdh2 dh

4 dt d h

+ Gre + GcC -

1 + m2

(mw + u)

1 dw dw d2w d3w

—---=(1 + b)—r- a-T

4 dt dh ' dh2 dtdh2

1 dN1 4 ~W

--2 (w - mu) - —

1 + m2 K

1 dN2 dN2

4 ~dt dh 1

. d2N2 dh2

4 dt d h Pr

1 dC dC _ 1 d2C

4 dt - dh = Sc dh2

where b = nr/n is the dimensionless viscosity ratio, a = K0Vjj/4n2 is the viscoelastic parameter, M = sBjn/pVj is the magnetic field parameter, Nr = 16T*s*/3k*k is the thermal radiation parameter, Gr = nbtg(Tw - T*)/V3 is the Grashof number, Gc = nbcg (Cw - C*)/V0 is the modified Grashof number, Pr = npCp/K is the Prandtl number, Sc = n/D is the Schmidt number, K = K*Vj/n2 is the permeability of the porous medium parameter and L = gVj/pn3j is the material parameter

Also the boundary conditions becomes

, du . dw iut „ iut u = h —, w = h —, q = e , C = e , dh dh

Ni = -n—; N2 = n— at y = 0

and neglecting higher order terms beyond the first degree in (T* - T*), we have

T*4 z _3t*4 ^ 4T T*

Differentiating Equation (9) with respect to y* and using Equation (11) to obtain

dqr _ -16T3 s* d2T* dy* = 3k* dy*2

Let us introduce the following dimensionless variables:

u* v* w* V0y* .. nN* u = V, v = V, w = —, h = , N1 = —21, V0 V0 V0 n Vg

N vN2 t t*V02 m 4nu* h V0L* N2 = vg". t = u = -jf, h = —

u/0, w/0, q/0, C/0, N1/0, N2/0 at y/œ (20)

We now simplify (14)—(17) by introducing q = u + iw and p = N1 + iN2 to have

^^=(1 + b) ^ - a

1 dq dq

4 dt - dh

■ M 2 (1 - im)q + Gre + GcC - % 1 + m2 K

1 dp dp d2p

4 dt- dh = dh2

1 de de _ 1 4 dt- dh = Pr

(1 + Nr)

T^ — T*

__J œ

T * - T *. 1 -1/1/ J ^

C* C *

C* - C*

-, J =

1 dC dC_ 1 d2C (13) 4 ~dt - dh = Sc dh2

and the corresponding boundary conditions are q = h q = eiut, C = eiut, P = inat y = 0

q/0, q/0, C/0, P/0, at y/œ 2.2. Method of solution

ai = 1 + b

In order to solve Equations (21)—(24) subject to the boundary conditions (25), we assume a perturbation method of this form:

q = q0(h)eiut, P = P0(h)eiut, q = q0(h)eiut, C = Q(h)e'

3. Calculation

Substituting Equation (26) into Equations (21)—(24), we obtain the following set of equations:

(ai - ia2)q0 + q0 - (a3 + ia4)q0 = - GcC0 - №0

LP0 + P0- 4 P0 = 0

a5q0 + q0 - 4 q0 = 0

C0+ ScC0-iUScC0 = 0

where a1 = 1 + b, a2 = au, a3 = M/1 + m2 + 1/K, a4 = (u/4)-(Mm/1 + m2), a5 = (1 + Nr)/Pr.

The corresponding boundary conditions can be written as

q0 = h °q°, q0 = 1, C0 = 1, P0 = in ^h0, at y = 0 vh vh

a2 = au

M 1 1 + m2 + K

_ u Mm

a4 = 4 - TTm2

1 + Nr

1 + pi -f iuL r1 =-2L-

1 + v 1 + iua5

r2=—2a—

Sc + a/Sc2 + iuSc r3 =-2-

T1 = a1 - ia2

T2 = a3 + ia4

r 1 + y/1 -f 4^1 r4 _-

rjT1 - r2 - T2

r2T1 - r3 - T2

-r1bn[r4hr2Â2 + r4hr3A3 + r4A2 + r4A3 - (1 + hr4)(r2À2 + ^3)]

(r2T1 - r1 - T2) (1 + hr4) - bn(r2r4h + r1r4 + (1 + hr4)r2)

q0 = 0, q0 = 0, C0 = 0, P0 = 0, at y/œ (31)

The solution of (27)—(30) satisfying the boundary conditions (31) are given by

q = (A1e-r4h + A2e-r2h + A3e-r3h + A4e-r1h) eiut (32)

P = B1eiut-r1h (33)

q = eiut-r2h (34)

C = eiut-r3h (35) where

-h(r2A2 + r3A3 + riA4) - (A2 + A3 + A4)

A1 (1 + hr4)

B1 = -ni(r4Ai + r2A2 + r3A3 + riA4)

4. Results and discussion

The results are presented as velocity, microrotation, temperature and concentration profiles in Figs. 2—16 below;

The effects of magnetic field parameter on velocity distribution profiles across the boundary layer are presented in Fig. 2. It is obvious that the effect of increasing values of the magnetic field parameter M results in a decreasing velocity distribution across the boundary layer. This is due to the fact that the effect of a transverse magnetic field give rise to a resistive type force called

Fig. 2. Velocity profiles for different values of magnetic field parameter.

the Lorentz force. The force has the tendency to slow the motion of the fluid.

Fig. 3 shows the translational velocity profiles with different values of radiation parameter. And the effect of increasing the radiation parameter is to increase the translational velocity. This is because when the intensity of heat generated through thermal radiation increased, the bond holding the components of the fluid particles is easily broken and the fluid velocity will increased. Fig. 4 displays the effect of Hall current parameter on the translational velocity distribution profiles. It is noticed that the Hall current parameter increases the velocity.

Fig. 5 depicts the effects of permeability of the porous medium parameter (K) on velocity distribution profiles and it is obvious that as permeability parameter (K) increases, the velocity increases along the boundary layer thickness which is expected since when the holes of porous medium become larger, the resistivity of the medium may be neglected.

Fig. 6 shows the influence of the viscoelastic parameter on translational velocity profiles. The velocity decreases as viscoelastic parameter increases in the vicinity of the plate but the reverse happens as one moves away from it. Fig. 7 illustrates the variation

Fig. 3. Velocity profiles for different values of radiation parameter.

Distance

Fig. 4. Velocity profiles for different values of Hall current parameter.

of slip parameter with translational velocity distribution profiles. As the parameter increases the translational velocity across the boundary layer increases. It is expected since the slip parameter has the tendency to reduce the friction forces which increases fluid velocity.

Fig. 8 depicts the micro-rotational velocity profiles for different values of magnetic field parameter respectively. The micro-rotational velocity distribution profiles decreases with increase in the magnetic field parameter.

Fig. 9 illustrates the effect of radiation parameter on micro-rotational velocity profiles. The profiles increases as the parameter increases. Fig. 10 illustrates the micro-rotational velocity distribution for different values of Hall current parameter. The figure shows that as Hall current parameter increases, micro-rotational velocity decreases.

The effects due to permeability of the porous medium parameter (K) on micro-rotational velocity is shown in Fig. 11. It is observed that as the parameter increases, the micro-rotational velocity decreases in the vicinity of the plate and reverse happens far away from the plate.

Fig. 5. Velocity profiles for different values of permeability of the porous medium parameter.

0.02 0.01

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Distance

0123456789 10 Distance

Fig. 6. Velocity profiles for different values of viscoelastic parameter.

Fig. 8. Microrotation profiles for different values of magnetic field parameter.

Fig. 12 illustrates the micro-rotational velocity distribution for different values of viscoelastic parameter. (a). The figure shows that as viscoelastic parameter increases, the micro-rotational velocity decreases. Fig. 13 shows that the effect of increasing slip parameter is to increase the micro-rotational velocity in the vicinity of the plate and decrease it far away from the plate.

Fig. 14 illustrates the effect of radiation parameter on micro-rotational velocity profiles. The profiles increases as the parameter increases.

Fig. 15 presents the effect of the Prandtl number Pr on the temperature profiles. Increasing the value of Pr has the tendency to decrease the fluid temperature in the boundary layer as well as the thermal boundary layer thickness. This causes the wall slope of the temperature to decrease as Pr is increasing causing the Nusselt number to increase as can be clearly seen in Table 2. Fig. 16 shows concentration distribution profiles for different values of Sc. It can be noted from the figure that the concentration of the fluid decreases as the Sc increases.

The local skin friction coefficient, couple stress coefficient, Nusselt number and Sherwood number are important physical quantities of engineering interest. The skin friction coefficient (Cf) at the wall is given by

Cf = ~Vk = [1 + (1 - n)L]q'(0)

where tw is the wall shear stress given by

tw = -[1 + (1 - n)L] [r4A1 + r2A2 + r3A3 + M4] eiut (36)

The couple stress coefficient (CW) at the plate is written as

cw = M32 = P (0) V0

where Mw is the wall couple stress given by

Mw = -r!B1eiut) (37)

The rate of heat transfer at the surface in terms of the Nusselt number is given by

Fig. 7. Velocity profiles for different values of slip parameter.

Fig. 9. Microrotation profiles for different values of radiation parameter.

Distance

Fig. 10. Microrotation profiles for different values of Hall parameter.

Distance

Fig. 12. Microrotation for different values of viscoelastic parameter.

1 œ 1 W

and on simplification Nusselt number is given by

NuRe-1 = -q'(0) = r2eiut (38)

where Rex = xV0/n.

The rate of mass transfer at the surface in terms of the local Sherwood number is given by

and on simplification the local Sherwood number is given by ShRe-1 = -C'(0) = r3eiut (39)

The numerical result for skin friction coefficient, couple stress coefficient, Nusselt number and Sherwood number are shown in Tables 1—3 below:

Table 1 shows the effects of parameter that relates micro-gyration vector to shear stress (n),viscoelastic parameter (a), Hall current parameter (m) and radiation parameter (Nr) on skin friction coefficient and couple stress coefficient. It is observed that increase in value of n decreases skin friction coefficient which is not surprising since n = 0 represents strong concentration and n = 0.5 represents weak concentration of the microelements while couple stress coefficient increases with increase in n. Increase in visco-elastic parameter (a) decreases skin friction coefficient and increases couple stress coefficient, increase in Hall current parameter (m) and radiation parameter (Nr) increases both skin friction coefficient and couple stress coefficient.

Table 2 shows the effect of Prandtl number (Pr) and radiation parameter (Nr) on the Nusselt number. The Nusselt number increases as Pr increases and decreases as Nr increases. This shows that the surface heat transfer from the porous plate increases with the increasing values of Pr and decreases with increasing value of Nr. Table 3 shows that the effect of increasing the Sc is to increase

Fig. 11. Microrotation profiles for different values of permeability of the porous medium parameter.

Fig. 13. Microrotation profiles for different values of slip parameter.

Distance

Fig. 14. Temperature profiles for different values of radiation parameter.

Fig. 15. Temperature profiles for different values of Prandtl number.

Table 1

Effect of n, a, m and Nr parameter on Cf and CW with h = 0.2, K = 1, ut = p/20, b = 0.5, Gr = 2, Gc = 1, M = 5, Pr = 3, L = 1.

n A m Nr Cf cw

0 0.2 0.2 0.5 1.0587 0

0.5 0.2 0.2 0.5 0.8312 0.1155

1 0.2 0.2 0.5 0.5706 0.2359

0.5 0.5 0.2 0.5 0.7775 0.1471

0.5 1 0.2 0.5 0.6512 0.1763

0.5 1.5 0.2 0.5 0.5252 0.1825

0.5 0.2 0 0.5 0.8272 0.1011

0.5 0.2 0.3 0.5 0.8370 0.1227

0.5 0.2 0.4 0.5 0.8451 0.1296

0.5 0.2 0.2 0.2 0.7650 0.1076

0.5 0.2 0.2 0.4 0.8102 0.1131

0.5 0.2 0.2 0.8 0.8880 0.1219

Table 2

Effect of Pr and Nr parameter on NuRex 1 with n = 0.5, K = 1, b = 0.5, Gr = 2, Gc = 1, M = 5, Pr = 3, ut = p/20, L = 1, m = 0.2, h = 0.2, a = 0.2.

Pr Nr NuRex 1

3 0.5 1.9661

4 0.5 2.6176

5 0.5 3.2716

3 0.2 2.4544

3 0.4 2.1054

3 0.8 1.6421

Table 3

Effect of Sc and u parameter on ShRex 1

with Nr = 0.5, n = 0.5, m = 0.2, h = 0.2,

b = 0.5, Gr = 2, Gc = 1, M = 5, Pr = 3.

Sc ShRex 1

2 1.9661

3 2.9444

4 3.9271

the rate of mass transfer. These results are in good agreement with Modather [7], Roslinda [9] and Rakesh [11].

Fig. 16. Concentration profiles for different values of Schmidt number.

5. Conclusion

An analytical study of the MHD heat and mass transfer flow of an incompressible, electrically conducting viscoelastic micropolar fluid over an infinite vertical plate through porous medium was conducted. The results are discussed through graphs and tables for different values of parameters. Following conclusions can be drawn from the results obtained:

* In the presence of a uniform magnetic field, increase in the strength of the applied magnetic field decelerated the fluid motion along the wall of the plate inside the boundary layer.

* Increase in Hall current parameter increases the momentum and thermal boundary layer thickness.

* The Nusselt number increased as the Prandtl number increased and decrease as radiation parameter increased.

* The Sherwood number increased as the Schmidt number increased.

* The radiation parameter increases both skin friction coefficient and couple stress coefficient.

Acknowledgments

The authors thank the anonymous referees for their constructive comments and careful reading of the manuscript.

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