Scholarly article on topic 'Effect of magnetic field on unsteady natural convective flow of a micropolar fluid between two vertical walls'

Effect of magnetic field on unsteady natural convective flow of a micropolar fluid between two vertical walls Academic research paper on "Mathematics"

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Abstract of research paper on Mathematics, author of scientific article — Hari R. Kataria, Harshad R. Patel, Rajiv Singh

Abstract We study theoretically the boundary layer flow of an incompressible micropolar fluid under uniform magnetic field and motion takes place due to the buoyancy force between vertical walls. The governing unsteady boundary layer momentum, angular momentum and energy equations of micropolar fluid are nondimensionalized and solved numerically. Analytic result for steady state case is also discussed. The effects of magnetic parameter (M), vortex viscosity parameter (R), Prandtl number (Pr) and material parameter (b) on velocity, micro-rotation and Temperature profiles are discussed through several figures.

Academic research paper on topic "Effect of magnetic field on unsteady natural convective flow of a micropolar fluid between two vertical walls"

Ain Shams Engineering Journal (2015) xxx, xxx-xxx

Ain Shams University Ain Shams Engineering Journal

www.elsevier.com/locate/asej www.sciencedirect.com

MECHANICAL ENGINEERING

Effect of magnetic field on unsteady natural convective flow of a micropolar fluid between two vertical walls

Hari R. Kataria a, Harshad R. Patelb'*, Rajiv Singh a

a Department of Mathematics, Faculty of Science, The M.S. University of Baroda, Vadodara, India b Applied Science & Humanities Department, Sardar Vallabhbhai Patel Institute of Technology, Vasad, India

Received 28 March 2015; revised 2 August 2015; accepted 28 August 2015

KEYWORDS

Micropolar fluid; Vertical walls; Unsteady flow; Magnetic field

Abstract We study theoretically the boundary layer flow of an incompressible micropolar fluid under uniform magnetic field and motion takes place due to the buoyancy force between vertical walls. The governing unsteady boundary layer momentum, angular momentum and energy equations of micropolar fluid are nondimensionalized and solved numerically. Analytic result for steady state case is also discussed. The effects of magnetic parameter (M), vortex viscosity parameter (R), Prandtl number (Pr) and material parameter (b) on velocity, micro-rotation and Temperature profiles are discussed through several figures.

© 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Recently, the study of micropolar fluid has attracted many scholars as Navier-Stokes equations of Newtonian fluids cannot effectively describe the characteristics of fluid with suspended particles. Erigen [1] first introduced the theory of micropolar fluid and this theory is useful in explaining the characteristics of certain fluids such as liquid crystals,

Corresponding author. Tel.: +91 9727413159. E-mail addresses: hrkrmaths@yahoo.com (H.R. Kataria), harshadpatel2@gmail.com (H.R. Patel), Singhrajiv.bhu@gmail.com (R. Singh).

Peer review under responsibility of Ain Shams University.

suspensions and animal blood. Chamkha [2] and Abdulaziz [3], studied the fully developed free convection of a micropolar fluid in vertical channel. Si [4] and Beg [5] have studied homotopy analysis solution for micropolar fluid flow through a porous medium. Rashidi [6], has studied Heat Convection in Magnetized Micropolar Fluid by Using Modified Differential Transform Method. Sadri [7] has discussed about Semi Analytical Solution of Boundary-Layer Flow of a Micropolar Fluid through a Porous Channel. Siddangoudaa [8] studied Squeezing Film Characteristics for Micropolar Fluid between Porous Parallel Stepped Plates.

Convection flow arises in many physical situations such as in the cooling of nuclear reactors and environmental heat transfer processes amongst others. It is of three types namely free, mixed and force. Amongst them, the problems of magneto hydrodynamic free convective flow in a porous medium have drawn considerable attentions of several researchers in various scientific and technological applications such as

http://dx.doi.org/10.1016/j.asej.2015.08.013

2090-4479 © 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Nomenclature

b material parameter y dimensionless co-ordinate perpendicular to the

g acceleration due to gravity walls

L distance between two vertical walls y co-ordinate perpendicular of the walls

m temperature ratio x dimensionless angular velocity

M magnetic parameter j micro-inertia density

Pr Prandtl number K vortex viscosity

R vortex viscosity parameter l dynamic viscosity

t time in non-dimensional form

t time Greek symbols

Tc temperature of the wall at y' — L 0 temperature of the fluid in non-dimensional form

T„ temperature of the wall at y' — 0 V kinematic viscosity of the fluid

T m ill initial temperature of the fluid

u fluid velocity in non-dimensional form

u velocity of fluid

pumps, flow meters, generators, accelerators, plasma jet engines, and magnetic control of molten iron flow in steel industry and industrial processes in metallurgy and material processing, in chemical industry, industrial power engineering and nuclear engineering. Special mention can be made, for instance, to the experiments on liquid metal flows in MHD channels performed by Hartmann [9]. Rashidi [10] has studied Unsteady Two-Dimensional and Axisymmetric Squeezing flows between Parallel Plates. Siddiqui [11] considered Homot-opy Analysis Method to the Unsteady Squeezing Flow between Circular Plates. Singh [12] discussed on MHD Free Convective Flow Past a Semi-infinite Vertical Permeable wall. Hamad [13], Uddin [14] and Khan [15] have studied MHD natural convection flow on a nanofluid in different physical conditions then Khan [16] obtained the solution of unsteady two-dimensional and axisymmetric squeezing flow between parallel plates. Kataria [17] has studied induced magnetic field effects on mixed convection. Narayana [18], Oahimire [19], Olajuwon [20] and Prakash [21] have studied effect of hall current and radiation on MHD flow of a micropolar fluid. Mahmoud [22] has obtained the solution of MHD flow of a micropolar fluid over a stretching surface with heat generation and slip velocity. Beg [23] has studied nanofluid convection boundary layers from an isothermal spherical body in a permeable regime. Freidoonimehr [24] and Vendabai [25] have studied free convective MHD flow past a permeable stretching vertical surface in a nano-fluid then Borrelli [26] obtained the solution of Magneto convection of a micropolar fluid in a vertical channel.

In most of the research works, case of asymmetric or symmetric thermal condition is studied in the absence of magnetic field. In this paper, we have analysed effect of magnetic field on unsteady natural convective flow and micro rotation between infinitely long vertical walls for asymmetric/symmetric wall temperatures.

2. Basic equations and description of the problem

Consider the unsteady free-convective flow of an electric conductive micro-polar fluid between two insulated vertical walls separated by a distance L apart subjected to a uniform transverse magnetic field. The coordinate system is chosen such that

x' measures the distance along the walls and y' measures the distance normal to it. Initially, the temperatures of walls and the fluid are same says Tf. When time t > 0, the temperature of the walls at y' — 0 and y' — L is instantaneously raised and lowered to Th and Tc respectively such that Th > Tc which is there after maintained constant. A constant uniformly distributed transverse magnetic field of strength B0 is applied in the y'-direction. Physical model and coordinate system are shown below:

t > 0, T = Th,

ÎÎÎ

The transversely applied magnetic field and magnetic Reynolds number are very small and hence the induce magnetic field is negligible. No electrical field is assumed to exist and both viscous and magnetic dissipations are neglected. The hall effects, the viscous dissipation and the joule heating terms are also neglected. Under above assumptions and taking into account the Boussinesq and boundary layer approximations, momentum, angular momentum and energy equations of micropolar fluid can be expressed as follows:

Table 1 Numerical and analytic values of steady state velocity, micro rotation and temperature profile.

M Pr m R b y Analytic Numerical Analytic solution Numerical Analytic Numerical

solution u solution u w micro-rotation solution w micro solution d solution d

velocity velocity rotation temperature temperature

0.1 1.0 0 0.5 0.1 0.2 0.0320 0.0319 -0.0000783 -0.0000784 0.8000 0.8000

0.6 0.0373 0.0373 0.0001167 0.0001164 0.4000 0.4000

5 1.0 0 0.5 0.1 0.2 0.0143 0.0144 -0.00001497 -0.00001517 0.8000 0.8000

0.6 0.0127 0.0127 0.00006198 0.00006189 0.4000 0.4000

0.1 1.0 1 0.5 0.1 0.2 0.0533 0.0533 -0.0002128 -0.0002128 1 1

0.6 0.0800 0.0800 0.0001064 0.0001064 1 1

5 1.0 1 0.5 0.1 0.2 0.0211 0.0211 -0.00007158 -0.00007175 1 1

0.6 0.0289 0.0290 0.00003370 0.00003377 1 1

5 0.71 0 0.5 0.1 0.2 0.0143 0.0144 -0.00001497 -0.00001518 0.8000 0.8000

0.6 0.0127 0.0127 0.00006198 0.00006189 0.4000 0.4000

5 2 0 0.5 0.1 0.2 0.0143 0.0142 -0.00001497 -0.00001516 0.8000 0.7999

0.6 0.0127 0.0127 0.00006198 0.00006189 0.4000 0.3993

5 0.71 1 0.5 0.1 0.2 0.0211 0.0211 -0.00007158 -0.00007176 1 1

0.6 0.0289 0.0290 0.00003370 0.00003377 1 1

5 2 1 0.5 0.1 0.2 0.0211 0.0211 -0.00007158 -0.00007172 1 0.9999

0.6 0.0289 0.0289 0.00003370 0.00003376 1 0.9999

5 1 0 0.5 0.5 0.2 0.0143 0.0144 -0.0000774 -0.0000784 0.8000 0.8000

0.6 0.0127 0.0127 0.0003028 0.0003024 0.4000 0.4000

5 1 0 0.5 1.5 0.2 0.0144 0.0144 -0.0002489 -0.0002518 0.8000 0.8000

0.6 0.0127 0.0127 0.0008599 0.0008589 0.4000 0.4000

5 1 1 0.5 0.5 0.2 0.0211 0.0211 -0.0003553 -0.0003562 1 1

0.6 0.0289 0.0290 0.0001670 0.0001674 1 1

5 1 1 0.5 1.5 0.2 0.0211 0.0212 -0.0010 -0.0011 1 1

0.6 0.0290 0.0290 0.0005 0.0005 1 1

5 1 0 0.4 0.1 0.2 0.0148 0.0149 -0.00001211 -0.00001227 0.8000 0.8000

0.6 0.0129 0.0129 0.00005379 0.00005371 0.4000 0.4000

5 1 0 1.2 0.1 0.2 0.0117 0.0131 -0.00003013 -0.00002278 0.8000 0.8000

0.6 0.0111 0.0120 0.00009133 0.00007909 0.4000 0.4000

5 1 1 0.4 0.1 0.2 0.0217 0.0218 -0.00006079 -0.00006092 1 1

0.6 0.0296 0.0296 0.00002851 0.00002857 1 1

5 1 1 1.2 0.1 0.2 0.0177 0.0177 -0.0001177 -0.0001179 1 1

0.6 0.0249 0.0249 0.0000564 0.0000565 1 1

Table 2 Velocity, micro rotation and temperature profile for different step size at M t = 0.5. = 0.1, R = 0.5, b = 0.1, Pr = 1, m = 0 and

y Velocity profile for step size 31 on [0 1] Velocity profile for step size 21 on [0 1] Velocity profile for step size 11 on [0 1] Micro rotation profile for step size 31 on [0 1] Micro rotation profile for step size 21 on [0 1] Micro rotation profile for step size 11 on [0 1] Temperature profile for step size 31 on [0 1] Temperature profile for step size 21 on [0 1] Temperature profile for step size 11 on [0 1]

0.2 0.4 0.6 0.8 0.0315 0.0418 0.0365 0.0208 0.0315 0.0418 0.0365 0.0208 0.0315 0.0418 0.0365 0.0208 -0.0000757 0.0000114 0.0001147 0.0001313 -0.0000759 0.0000110 0.0001143 0.0001311 -0.0000772 0.0000091 0.0001123 0.0001297 0.7972 0.4953 0.3955 0.1972 0.7972 0.5955 0.3955 0.1972 0.7972 0.5954 0.3954 0.1972

du' ^d2il , dm' ^ x -, ,

q+k) w2 + w+pgß( Tm) ~Bl '

dm' , d2m' ( . du'

pJ~dF ={ß + °-5k)jW2 - \2m + ^

dT _ d2T

W ~a dy2 '

(i) (2) (3)

Initial and boundary conditions are

t 6 0 u = m' = 0' T = T' 0

t > 0 u = m' = 0' T = Th y

u = m' = 0' T = Tc y'

y = 0;

Introducing the following similarity transformations in Eqs.

(1)-(3),

Table 3 Velocity, micro rotation and temperature profile for different step size at M = 0.1, R = 0.5, b = 0.1, Pr = 1, m =1 and t = 0.5.

Velocity profile for step size 31 on [0 1]

Velocity profile for step size 21 on [0 1]

Velocity profile for step size 11 on [0 1]

Micro rotation profile for step size 31 on [0 1]

Micro rotation profile for step size 21 on [0 1]

Micro rotation profile for step size 11 on [0 1]

Temperature profile for step size 31 on [0 1]

Temperature profile for step size 21 on [0 1]

Temperature profile for step size 11 on [0 1]

0.2 0.0523

0.4 0.0783

0.6 0.0783

0.8 0.0523

0.0522 0.0782 0.0782 0.0522

0.0522 0.0782 0.0782 0.0522

-0.0002070 -0.0001032 0.0001032 0.0002070

-0.0002069 -0.0001032 0.0001032 0.0002069

-0.0002069 -0.0001032 0.0001032 0.0002069

0.9944 0.9910 0.9910 0.9944

0.9944 0.9907 0.9907 0.9944

0.9942 0.9907 0.9907 0.9942

curves

M=0.1 1 2 3

M=1 4 5

0.25 0.5

0.85(S.S)

0.25 0.5

0.75(S.S)

0.25 0.5

0.65(S.S)

Figure 1 Velocity profile u for different values of y at m = 0, R = 0.5, b = 0.1 and Pr = 1.

curves

0.25 0.5

0.85(S.S)

0.25 0.5

0.75(S.S)

0.25 0.5

0.65(S.S)

0.8 0.9

Figure 2 Velocity profile u for different values of y at m = 1, R = 0.5, b = 0.1 and Pr = 1.

1&4&7, 2&5&8, 3&6&9

curves

b = 0.5 1 2 3

0.25 0.5

0.8(S.S)

Figure 3 Velocity profile u for different values of y at m = 0, R = 0.5, M =5 and Pr =1.

curves 1 2

1&4&7, 2&5&8, 3&6&9

0.25 0.5

0.75(S.S)

0.25 0.5

0.7(S.S)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4 Velocity profile u for different values of y at m = 1, R = 0.5, M =5 and Pr = 1.

y = У/L, t — vt' /L2

U = u'v/bgL2(Th - Tm) x = x'v/ßgL(Th - Tm) m — (T - Tm)/{Th - Tm) R = k/i

h =(T - Tm)/(Th - Tm), Pr — v/a,

b — L /j,

M — aB2L2/ß.

du , du „dx . ,2

~dt — (1 + R) дуй + h + R - M U,

dx „ „ „ d2x , ( du

dï —(1 + °.5R) @yx - Rb( dy + 2x dt — РГ ду22 '

we get the following linear system of differential equations:

The corresponding initial boundary conditions (4) to the considered model are reduced as follows:

b = 0.5

0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0

curves ■ R = 0.4 1 2 3

----R = 0.8 4

0.25 0.5

0.75(S.S)

0.25 0.5

0.7(S.S)

0.25 0.5

0.65(S.S)

Figure 5 Velocity profile u for different values of y at m = 0, b = 0.1, Pr =1 and M =5.

curves 1 2 3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.25 0.5

0.7(S.S)

0.25 0.5

0.65(S.S)

0.25 0.5

0.6(S.S)

R = 0.4

Figure 6 Velocity profile u for different values of y at m = 1, b = 0.1, Pr =1 and M =5.

t 6 0 u — x — h = 0, 0 6 y 6 1; t > 0 u — x — 0 h — 1, y = 0;

u — x — 0, h — my — 1. (9)

The physical quantities used in the above equations are defined in the nomenclature.

taken increment step along t as 0.05 and y directions as 0.0323 in entire numerical computations. In present problem, the cost and the accuracy of the solution depend strongly on length of the vector y.This attentive problem requests the solution on mesh produced by spaced points from the spatial interval 31 values of y from the space interval [0,1] and 40 values of t from the time interval [0, 2].

3. Numerical solution

The governing linear parabolic partial differential Eqs. (6)-(8) with initial and boundary conditions are solved numerically by using Matlab software (finite difference method). We have

4. Steady state analytic solution

In order to check the accuracy of the numerical solution obtained with Matlab software, we compare the steady-state numerical solution with the analytic solution of the corre-

0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0

curves 1 2

Pr 0.71 1.0 2.0 3.0 5.0 7.0

Figure 7 Velocity profile u for different values of y at m = 0, b = 0.1, R = 0.5 and M =5.

Pr 0.71 1.0

2.0 3.0 5.0 7.0

Figure 8 Velocity profile u for different values of y at m = 1, b = 0.1, R = 0.5 and M =5.

w — c1eppffiy + + + c4e~pp2ffiy + A (10)

sponding study flow. As it is steady state solution, left hand side of Eqs. (6)-(8) becomes zero. Then there is only one independent variable and thus problem is a system of ordinary differential equation.

From Appendix A we obtain the steady state analytic result as follows:

Case-I

a1 — 0, a2 — 0

k > 0, k — p2

u — bi°epiy - bne-piy + съЬбер2у - cAe-^ + Лу + A

(ii) p1 > 0 & p2 < 0

w — C5ev/pTy + C6e-vSy + C7 cos ppffi + c8 sin pffiy + A

(i) p1 > 0 & p2 > 0

u — bi5epiy - bi6e piy + cybi2 sinР2У - c8bi2 x cosp2у + АбУ + A7 (iii) p1 < 0 & p2 > 0

(12) (13)

curves

- M=0.1 1

- M=1 4

0.25 0.5

0.95(S.S)

0.25 0.5

0.85(S.S)

0.25 0.5

0.8(S.S)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 9 Micro-rotation profile w for different values of y at m = 0, R = 0.5, b = 0.1 and Pr = 1.

curves M=0.1 1

3 0.95(S.S)

----M=1

. x 10-

0.25 0.5

0.9(S.S)

0.25 0.5

0.85(S.S)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y —>

Figure 10 Micro-rotation profile w for different values of y at m = 1, R = 0.5, b = 0.1 and Pr = 1.

w — C9 cos ypy + cw sin ypy + cne ^

+ c ! 2e-pp-y + A ( 14)

u — big sinp1y + bi9 cosp1y + спЬбеР1У

- спЬбе-РгУ + Абу + A7 (15)

(iv) p1 < 0 & p2 < 0

w — C13 COS^Jp[y + C14 sin^Jp[y + C15 COS ^/p2y

+ C16 sin ppffiy + A (16)

u — Ь20 sinp1 y + Ь21 cosp1y + c15b12 sinp2yeP2y - С1бЬ12 cosp2y + A6y + A7

k — 0

(i) a1 > 0

w — (C17 + C1gy)e^iy + (C19 + C20y)e^v^1y + A

u — (Ь29 + Ьз0у)691У + (Ь29 + ЬЗ'0 y)e-q1y + A6y + A7

10 8 6 4

га 2 g

Ь 0 -2 -4

b = 0.5 1 2 3

b =1. 5 7

curves t

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.25 0.5

1.0(S.S)

0.25 0.5

0.95(S.S)

0.25 0.5

0.9(S.S)

Figure 11 Micro-rotation profile w for different values of y at m = 0, R = 0.5, M =5 and Pr = 1.

b = 0.5 1

b = 1 4

curves t

1.5 ,x 10

0.25 0.5

1.0(S.S)

0.25 0.5

0.95(S.S)

0.25 0.5

0.9(S.S)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 12 Micro-rotation profile w for different values of y at m = 1, R = 0.5, M =5 and Pr = 1.

(ii) a < 0

W — (c2i + c22y) cos < + (c23 + c24y) sin < fëy + — v 2 V 2 Ü2

u — (b34 + b35y) sin q1y + (b36 + b37y) cos q1y + A6y + A7

k < 0, k — -p2

W — ev2y((c25 + c26y) cos Viy + (c27 + c2gy) sin Viy) + —

■3 Ï2 (22)

u — (b72 + b73y)eV2y cos Viy + (b36 + b37y)eV2y

x sin Viy + A6y + A7 (23)

1 -2 -4

curves — R = 0.4 1

-R = 0.8 4

■■ R = 1.2 7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.Î

0.25 0.5

1.0(S.S)

0.25 0.5

0.95(S.S)

0.25 0.5

0.9(S.S)

Figure 13 Micro-rotation profile w for different values of y at m = 0, b = 0.1, Pr =1 and M =5.

9,8,7,6,5,4,3,2,1

-R = 0.4

----R = 0.8

curves 1 2

R = 1.2 7

0.25 0.5

1.0(S.S)

0.25 0.5

0.95(S.S)

0.25 0.5

0.9(S.S)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 14 Micro-rotation profile w for different values of y at m = 1, b = 0.1, Pr =1 and M =5.

Case-II

a1 — 0, a2 — 0

' — (C29 + C30y) cos y'yy + (C31 + C32y) sin yyyj + a3

u — (Ь82У + Ь8з)вЧгУ cos q2y + (Ь84У + sin q2y

+ A6y + A7

Case-III

a1 — 0, a2 — 0

w — C33e^y + C34e^^/ary + C35 cos Pay + C36 sin Pay -

u — (Ь91 — C35)ery + Ь91 e-ry + Ь92 sin ry + Ь93 cos ry + Ь94у + A7

7 6 5 4 3 2

g 1 0 -1 -2 -3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 15 Micro-rotation profile w for different values of y at m = 0, b = 0.1, R = 0.5 and M =5.

. x 10 '

curves Pr

1 0.71

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 16 Micro-rotation profile w for different values of y at m = 1, b = 0.1, R = 0.5 and M =5.

Case-IV a1 — 0, a2 — 0

2 3 y a3

w — c37 + c38y + c39y + c40y +

u = b95y3 + b96y2 + b97y + b99

The steady state solution of Eq. (8) is

h = 1 + (m - 1)y

In Table 1, we have compared numerical and analytical solutions of steady-state velocity, micro rotation and Temperature

profiles for different values of the magnetic parameter M, vortex viscosity R, material parameter b and Prandtl number Pr for both the cases asymmetric and symmetric. We can see that the numerical and analytic results agree very well. Tables 2 and 3 verify that our solution is independent of step size for asymmetric and symmetric cases respectively.

5. Result and discussion

The physical parameters appearing in the model are vortex viscosity parameter R, material parameter b, Prandtl number Pr and temperature ratio m having values 0 and 1 for asymmetric and symmetric heating respectively. Several researchers have

curves 1 2

Pr 0.71 1.0 2.0 3.0 5.0 7.0

0.4 0.5 0.6

Figure 17 Temperature profile h for different values of y at m = 0.

Pr 0.71

Figure 18 Temperature profile h for different values of y at m = 1.

studied effects of physical parameters R, b and Pr on the velocity as well as micro-rotation profiles under the asymmetric/ symmetric heating thermal condition on the vertical walls. In this study, we have focused on the physical parameters M and presented their influences through Figs. 1-20 on the velocity, micro-rotation temperature and velocity vector profiles for asymmetric and symmetric heating of the vertical walls. Odd numbered figures are for m = 0 and rest are for m = 1.

Figs. 1 and 2 display the effect of magnetic parameter M on the velocity profiles. It is observed that the amplitude of the velocity as well as the boundary layer thickness decreases when M is increased. Physically, it may also be expected due to the fact that the application of a transverse magnetic field results in a resistive type force (called Lorentz force) similar to the

drag force, and upon increasing the values of M, the drag force increases which leads to the deceleration of the flow.

The influence of material parameter b on velocity profiles is shown in Figs. 3 and 4. In both thermal cases, we can observe that the velocity of fluid has attained steady state by increasing with the time.

The graphical results for vortex viscosity R are shown in Figs. 5 and 6. For both thermal cases we observed that the vortex viscosity parameter has decreasing tendency on the velocity profiles as well as on the steady state time.

Figs. 7 and 8 exhibit the velocity profiles for different values of Prandtl number Pr, when the other parameters are fixed. It is observed that velocity of the fluid decreases with increasing Prandtl number.

Velocity Vector u (y,t)

Figure 19 velocity vector u(y, t) for different values of y & t at m = 0, R = 0.5, b = 0.1, M =5 and Pr = 1.

Velocity Vector u (y,t)

Figure 20 velocity vector u(y, t) for different values of y & t at m = 1, R = 0.5, b = 0.1, M =5 and Pr = 1.

Figs. 9 and 10 are plotted to show the effects of magnetic parameter M on the micro-rotation profiles. In both thermal cases, the steady state time and the magnitude of the microrotation have increasing tendency with the material parameter.

Figs. 11 and 12 display the effect of material parameter b on the micro-rotation profiles. It is observed that the magnitude of the micro-rotation increases with increase in b.

The influence of vortex viscosity R on micro rotation profiles is shown in Figs. 13 and 14. The magnitudes of the micro-rotation profiles show the increasing tendency with R.

Figs. 15 and 16 reveal that magnitude of micro rotation decreases with increasing Prandtl number.

It is depicted from Figs. 17 and 18 that the temperature decreases as the Prandtl number Pr increases. It is justified due to the fact that thermal conductivity of the fluid decreases

with increasing Prandtl number Pr and hence decreases the thermal boundary layer thickness.

Figs. 19 and 20 display the velocity vectors for different values of t & y at M = 0.1, b = 0.1, R = 0.5 and Pr = 1 for asymmetric and symmetric cases.

A close study of these figures reveals that the steady state time in case of asymmetric heating is more than the symmetric heating.

6. Conclusion

We investigated the effect of magnetic, material and viscosity parameters on natural convective flow along vertical walls in case of both asymmetric and symmetric heating (and cooling) of walls. An exact analysis is performed to investigate the

steady boundary layer momentum, angular momentum and energy equations of flow of an incompressible micropolar fluid under uniform magnetic field between vertical walls. The governing unsteady boundary layer problem is solved numerically.

The main conclusions of this study are as follows:

1. Velocity of the fluid decreases with increasing Prandtl number Pr.

2. The amplitude of the velocity as well as the boundary layer thickness decreases when magnetic parameter M is increased.

3. Velocity is an increasing function of time t.

4. Temperature decreases as the Prandtl number Pr increases.

5. Temperature increases with increasing time t .

6. Magnitude of the micro-rotation has increasing tendency with the material parameter M, material parameter b and vortex viscosity R while decreases with increase in Prandtl number Pr.

7. The steady state time of fluid velocity as well as microrotation is more for symmetric cases compared to asymmetric cases.

8. The velocity and micro-rotation profiles of fluid decrease at any point of fluid regime with magnetic parameter.

9. The velocity decreases and micro rotation profile of fluid increases at any point of fluid regime with vortex viscosity parameter.

10. The steady state time of velocity profile and micro-rotation have decreasing tendency with material parameter.

Appendix A.

A4 = A7 =

A = a « =

b2 = b5 =

. (1+R)(1+0.5R) ' RbM2 . -1 ' M2

2MlRb (1+R)(1+0.5R)

b11 A3 : A10 A13

= P-A4

b1 + b3

= b5(-A2 - 1) = A1C3b5 + A2C4b5 - A3b5

_ A(ep1 -1) = e P1 -eP1

= -2b5A3 - Abs + A7 = b9ep1 - A3b5e-p1 + A' A1C3 + C4A2 + A3

= b5(-B1 - 1) = C7b5B1 + Cgb5B2 + b5B3

_ Aep1 -A

C6 = bu

B12 = B15 B18

C11 H1

b20 H5 H8

' e P1 -eP1 ■ -Ab5 + A7

^ b14epi - b5B3e-pi + A6 + A7 B1C7 + C8B2 + B3 = (-C11 - C12 - A)b17

1-e p2 sin Pj

= -B12b17 - b6

= (-sinP1 - A12cosP1 )b 17 - b6e-

_ B19B15 -B16B18

_ Cospi-Cospi

sin P1

= (-C15 - A)b17 = H2b17 - b12

= H-b17 cos P1 - bn cos Pi

C15 = h =

_ H9HS-H6H8 " H4H8-H7H5

e«1 -e «1 e«1

A' = A6 + A7

— (1-m)Rb a3 = (1+R)(1+0.5R) P _ a1+P P1 = 2

q2 = V1"

b3 = P1 A5

b6 = b2 + b4

b9 = b5(-A3 - A)

- eP1 -e?2 A1 = e P1 -eP1

A8 = -2b5A1 - b5 + b6

A11 = b7eP1 - A1b5e-Pi +

C4 _ A13A8-A10A11 4 A11A9-A8A12 C1 = -C2 - C3 - C4 - A b 14 = b5(B3 - A)

b _ eP1 CosP

B1 e P1 -eP1

B4 = -2b5B1 - b5

B7 = b13eP1 - b5B1e-P1 + b12sinP2

C _ B4B9 B6B7 C8 = B5B7 B8 B4

C5 = -C6 - C7 - A b 19 = b 17( C11B11 - C12B12 - B13)

B _ A(cosP]-1) 13 sin P]

B16 = -B13b17 + A7

B19 = (-A sinp1 - A13 cosP1)b17 + A'

C10 = B11C11 + C12B12 + B13

H — -s'"Pi

2 sin P! b21 = b 17( C15H1 - H3) H6 = H3b5 + A7

H9 = -Ab17 sinp1 - b17H3 cosp1 + A'

C14 = H1C15 - C16H2 + H3

T _ e q1

12 — Tür

(1+R)(1+0.5R) + (1+0.5R) + (T+RJ

k = p2 = a- - 4a2

P — a1-P p2 = 2

b1 = P31 A4

b4 = A5P2

b7 = b5(-A1 - 1)

b10 = C3b7 + C4b8 + b9

A _ -e p2 A2 = e P1 -eP1

A9 = -2b5A2 - b5 - b6

A12 = b8eP1 - A2b5e-pi - b6e-Pl

3 A11A9-A8A12 b12 = b2 - b4

b15 = C7b13 + C8B-b5 + b14

B _ s'npi

B2 e P1 -eP1

B5 = -bi + b4

B8 = b5eP1 B2 - b5B2e-P1 - b12cosp2

C B5B9 B6B8

7 B5B7 B8 B4

b17 = b1 - b3

B = 1-epl

11 sin P1 B14 = —B11 b17 + b6

B17 = (-sinp1 - B11 cosp1 )b 17 + b6e

_ B19B15-B16B18 C12 B14B18-B17B15

C9 = -C11 - C12 - A

r> _AC0SP1 -A

3 sin Pi

H4 = — H1b 17 H7 = (-sin P1 -C _ H9H4-H6H7 C16 = H5H7-H8H4

C13 = -C15 - A

, _ A(eq1 -1) '3 = e«1

- H1 cosp1 )b17 + b12 sinp2

¿22 - q\A4 ¿23 - q2A4 ¿24 - q1A5

¿25 - b22 + ¿24 ¿26 - 3b23 + A5 ¿27 - ¿25 + ¿26

¿28 - ¿25 - ¿26 ¿29 - C19/1 — C20/2 + /3 ¿29 - ¿25 ( C19 — A) + b26b29

b30 - ¿25 ¿29 ¿29 - —¿25 C19 + ¿26 ¿30 - — C20b25

u-~ = -2b25 + /^26 /5 - — №6 + ¿26 /6 - — Ab22 + /3 ¿26 — Ab24 + A7

I7-- = —¿25 (eql + e—ql ) + Iieqi ¿27 /8 - —/2eql ¿27 — e—ql ¿28 /9 - —Aeql ¿25 + /3eql ¿27 + A6 + A7

C20 _ I914 I617 /5/7-/8/4 C19 /4/8 —/7/5 Cl8 - /1C19 — C20/2 + /3

Cl7 = — C19 — A J1 - — tanq! J2 - — tanql

J3 -- A(cos qi — 1) cos qi - C23 Jl + C24J2 + J3 ¿31 - 3b23 — A5 ¿32 - ¿22 — ¿24

¿33 ¿34 - —Abз2 ¿35 - ¿32¿33

¿36 - — ¿3^33 — C2зbз2 ¿37 - —c24b32 J4 - —J^¿31 — ¿32

J 5 -- - —J2bзl J6 - — Jзbзl + A7 J - J^31 — ¿32, J7 - J2bз2 — ¿31

4 = = J2bзl — ¿32 J9 - —Abз2 + Jзbзl J7 - sin q1 J1 ¿32 — cos q1 J6

J8 = - sin q1J7 — cos qJ J9 - sin q1J9 — J3 cos q1b31 + A' C _ J9J4—J6J7 C24 J5J7—J8J4

C23 _ J9J5—J6J8 J4J8—J7J5 C22 - J1C23 — C24J2 + J3 C21 - —A

M1 - — tan v1 M2 - — tan v1 ,, A(cos Vi ev2 — 1) M3 - cos v1ev2

¿38 - v2A4 ¿39 - v22v1 A4 ¿40 - V2A4

¿41 - v3 A4 ¿42 - v1 v2A4 ¿43 - v1 v32A4

¿44 - V2V1A4 ¿45 - v1A5 ¿46 - v2A5

¿46 - v2A4 ¿47 - ¿38 — ¿40 ¿48 - 3bз9 — ¿41

¿49 = 3b44 + ¿46 ¿50 - A(¿46 + ¿49) ¿51 - ¿41 — ¿45

¿52 - 3b47 — ¿50 + ¿46 + A5 ¿53 - —¿48 — ^¿42 — ¿45 ¿54 - —¿51 + 3bз9

¿55 - ¿46 + ¿50 ¿56 - Mlb52 + ¿54 ¿57 - Ml ¿53 + ¿55

¿58 = ¿54 + ¿42 ¿59 - ¿55 + 3bз8 — 2b40 + A5 ¿60 - ^^¿52 + ¿58

¿61 = M2b53 + ¿59 ¿62 - ¿55 + (¿46 + A5) ¿63 - Ab50 — Ab46

¿64 - Mзb62 + ¿63 ¿65 - Mзb53 + Ab54 ¿66 - 3b47 + A5

¿67 - 3bз8 — 2b40 + A5 ¿68 - ¿48 + ¿45 ¿69 - M1C27 + M2C28 + M3

¿70 - ¿54 + ¿42 — Ab55 ¿71 - ¿55 + ¿67 + Ab68 ¿72 - b66b69 + ¿70

¿73 = b55b69 + ¿68 ¿74 - —¿42¿69 + ¿71 ¿75 - —b68b69 + ¿55

M4 - ^47 + A5)Mi + ¿48 + ¿45 M5 - (3b47 + A5)M2 + 6b42 M6 - (3b47 + A5)M3 — ¿50 + A7

M7 - (cos v1b56 + sin v^57)ev2 M8 - ev2 (cos v^60 + sin v1 ¿61) M9 - ev2 (cos v1b64 + sin v1 ¿65)

C28 M9M4—M6M7 M5M7—M8M4 C M9M5—M6M8 C27 - M4M8—M7M5 C26 - Ml C27 + C28M2 + M3

C25 = —A L1 - — tan q2 L A (cos q2eq2 — 1) L2 - cos q2eq2

¿77 = q2A4 ¿78 - q2A5 ¿79 - —2b77 + ¿78

¿80 - 2b77 + ¿78 ¿81 - L^79 + L1A5 ¿8l - ¿81 + ¿80

¿82 - ¿82 + ¿79 ¿83 - ¿81 + ^¿76 ¿84 - ¿82 + A5

¿85 - ¿81 — Ab79 ¿86 - ¿82 + Ab80 ¿87 - C31 + C32

¿87 - A(A5 — ¿79) ¿88 - —6b87Ll — 6A ¿82 - — L^80 — 6L^76

¿83 = C31 ¿80 + LlA5b87 + 6cз2b76 + ¿87 ¿84 - —¿80¿81 + ¿79 ¿85 - C31 ¿79 — b76b88 + C32A75L1

L4 = 2b77 + A5L1 + ¿79 L5 - 6b76 + A5L1 L6 - 2Ab77 — A^9 + A5L2 + A7

L7 = (cos q2b8l +sin q2b82)eq2 L8 - (cosq2b83 + sinq^^e® L9 - (cosq2b85 + sinq2b86)eq2 + A'

C32 L9L4 —L6L7 L5L7—LgL4 C L9L5—LL C31 - L4L8—¿7i5 C30 - L1C31 + C32L1 + L2

C29 - —A z'1 cos r—e' G1 - e'—e ' / ' sin r G2 - e—e~

G3 (03/2^1) e'—e ' ¿86 - r3 A4 ¿87 - A5r

¿88 - ¿86 + ¿87 ¿89 - ¿86 — ¿87 ¿90 - —G1C35 — G2C36 + G3

¿91 = ¿88 ¿90 ¿92 - C35 ¿89 ¿93 - c36b89

¿94 — A - a. -A6 a G4 - ( —2Gl — 1^88 G5 - —2G2b88 — ¿89

G6 - 2Gзb88 + A7 G7 - ¿88 (Gl(—er — e—') — er) + sin '¿89 G8 - ¿88 G2 ( er — e—r) + cos '¿89

G9 - ¿88 G3 ( er — e—') — f + A' „ G9G4 G6G7 C36 - G5G7—G8G4 „ G9G5 G6G8 35 G4G8 -G7G5

C34 - G1C35 + C36G2 — G3 C33 - —C34 — C35 ¿95 -1

¿96 - 3C40 ¿97 - 2C39 + A6 + A4Û3 ¿98 - C39 + C40 + 24

¿99 - 6C40A4 — ¿98A5 + A7 Dl - — A5, D2- —6A4

D3 —25a3 , a - "24 + A7 D4 - D1 + 2 D5 - D2 + Dl + 3

D6 - A4C39 — Af + ¿95 + A6 + A7 C _ D6Dl —D3D4 C40 - D2D4 —D5D1 C _ D6D2—D3 D5 C39 - fl1fl5—d4D2

C38 - —C39 — C40 — § C37 - 0

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Dr. Hari R. Kataria is a Professor, and Head, in the Department of Mathematics, Faculty of Science, The Maharaja Sayajirao University of Baroda, Vadodara, Gujarat, India. He has completed his Ph.D. in Fluid Dynamics from the NIT, Surat, in 2003. He has authored and co-authored over 12 papers in reputed National/ International journals.

Mr. Harshad R. Patel has completed his M. Phil. in Applied Mathematics from VNSGU, Surat, in 2012, and M.Sc. degree in Applied Mathematics from The Maharaja Sayajirao University of Baroda; He is now in the position of Assistant Professor of Mathematics, Department of Applied Science and Humanities, SVIT, VASAD, Gujarat, India. He has authored and co-authored over 1 paper in reputed international journals.

Dr. Rajiv Kumar Singh has completed his Ph. D. degree in Fluid Mechanics from the B.H. U., Varanasi, in 2010. He is now in the position of Assistant Professor of Mathematics, Department of Mathematics, Faculty of Science, The Maharaja Sayajirao University of Baroda, Vadodara, Gujarat, India. He has authored and co-authored over 12 papers in reputed National/International journals.