Numerical solution of free convection MHD micropolar fluid flow between two parallel porous vertical...

Numerical solution of free convection MHD micropolarfluid flow between two parallel porous vertical plates

R. Bhargava a, L. Kumar a, H.S. Takhar b,*

a Department of Mathematics, Indian Institute of Technology, Roorkee 247667, Indiab Department of Engineering, Manchester Metropolitan University, Manchester M1 5GD, UK

Received 7 August 2002

(Communicated by E.S. S�UHUB_II)

Abstract

The fully developed electrically conducting micropolar fluid flow between two vertical porous parallel

plates is studied in the presence of temperature dependent heat sources including the effect of frictional

heating and in the presence of a magnetic field. Profiles for velocity, microrotation and temperature are

presented for a wide range of Hartmann numbers and the micropolar parameter. The skin friction, couple

stress and Nusselt numbers at the plates are shown in the tables.

� 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction

With the enormous expansion of engineering industries based on non-Newtonian fluids, such aspseudoplastic, dilatant, rheopetic, suspension flows, a lot of research attention has been given inthis direction. The theory of microfluids as developed by Eringen [1], which includes the effect oflocal rotary inertia, the couple stresses and inertial spin satisfactorily provides a model for thenon-Newtonian behavior observed in the polymers, paints, lubricants, suspended fluids and bloodetc.Eringen [2] also developed the theory of micropolar fluids for the case where only microrota-

tional effects and microrotational inertia exist. He [3] extended the theory of thermomicropolarfluids and derived the constitutive laws for fluids with microstructure. Simple problems on the

*Corresponding author. Tel.: +44-161-247-3668.

E-mail address: [email protected] (H.S. Takhar).

0020-7225/03/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0020-7225(02)00157-X

International Journal of Engineering Science 41 (2003) 123–136www.elsevier.com/locate/ijengsci

mail to: [email protected]

flow of such fluids were studied by a number of researchers and a review of this work was given byAriman et al. [4].Balaram and Sastry studied the fully developed free convection flow in a micropolar flow [5].

Later Jena and Mathur [6] examined the free convective heat transfer to a micropolar fluid along anon-isothermal vertical plate. Agarwal and Dhanapal [7] have analysed the effect of temperaturedependent heat sources on the fully developed free convection micropolar fluid flow when aconstant suction (or injection) is applied on the plates and the fluid.The extension of above type of flows to include magnetohydrodynamic effects has become

important due to several engineering applications such as in MHD generators, designing coolingsystem for nuclear reactors, flow meters, etc. where the microconcentration provides an impor-tant parameter for deciding the rate of heat flow. By simulating it, one can obtain the desiredtemperature in such equipments. Several investigators have made theoretical and experimentalstudies of micropolar flow in the presence of a transverse magnetic field during the last threedecades.Kasivishwanathan and Gandhi [8] have studied a class of exact solutions for the MHD flow of

a micropolar fluid confined between two infinite, insulated, parallel, non-coaxially rotating disks.Mohammeadein and Gorla [9] investigated the transverse magnetic field on mixed convection ina micropolar fluid flowing on a horizontal plate with vectored mass transfer. Later Gorla andTakhar [10] examined the simultaneous occurrence of buoyancy and magnetic forces in the flowof an electrically conducting micropolar fluid along a hot vertical plate in the presence of astrong cross magnetic field. The results indicate that the micropolar fluids reduce drag andsurface heat transfer rate. The general theory of magneto-micropolar fluids can be found inEringen [11].In the present paper, we have analysed the effect of temperature dependent heat sources on the

fully developed free convection electrically conducting micropolar fluid between two parallelporous vertical plates in a strong cross magnetic field. The governing differential equations havebeen solved by using the Quasi-linearization method. The velocity, microrotation and temperaturefunctions are shown graphically and the effects of magnetic field and micropolar parameter arestudied.

2. Mathematical formulation

Consider the fully developed steady, laminar free convection flow of an incompressible mi-cropolar fluid flowing between two infinite parallel porous flat plates distance h apart and orientedin the direction of the body force. The plates are maintained at constant temperature T1 and T2, inthe presence of strong magnetic field H0 normal to the plate. The x axis is taken along one of theplates and y axis normal to it. Since the boundaries in the x direction are of infinite dimensionstherefore without any loss of generality we assume that the physical quantities, for example ve-locity, microrotation and temperature depends on y only. The velocity field is taken to be ðu; t; 0Þand microrotation as ð0; 0;NÞ.It is assumed that the fluid possesses constant properties except the density variation due to

temperature difference is used only to express the body force term as the buoyancy term. Thus thegoverning equations of such type of flow can be written as:

124 R. Bhargava et al. / International Journal of Engineering Science 41 (2003) 123–136

otoy

¼ 0; giving t ¼ t0 ðconstantÞ ð1Þ

qt0ouoy

¼ ðl þ KÞ o2uoy2

þ KoNoy

þ qf � opox

þ rH 20 u ð2Þ

qjt0oNoy

¼ co2Noy2

� K 2N�

þ ouoy

�ð3Þ

qcpt0oToy

¼ Kfo2Toy2

þ ðl þ KÞ ouoy

� �2

þ coNoy

� �2

þ 2K N 2

�þ N

ouoy

�þ c0t0ðT � T0Þ ð4Þ

where c0 is the constant of proportionality, c0t0ðT � T0Þ the amount of heat generated per unitvolume in unit time is assumed to be a linear function of temperature, p the pressure, q thedensity, f the body force, cp the specific heat, l the coefficient of viscosity, K the gyroviscosity,c the material constant, j the microinertia and Kf the coefficient of thermal conductivity andT0 is the temperature in hydrostatic state.The boundary conditions are given by

at y ¼ 0 : u ¼ 0; t ¼ t0; T ¼ T1; N ¼ 0

at y ¼ h : u ¼ 0; t ¼ t0; T ¼ T2; N ¼ 0ð5Þ

where t0 < 0 implies injection at y ¼ h and suction at y ¼ 0 while t0 > 0 represents injection aty ¼ 0 and suction at y ¼ h.The body force term is expressed as a buoyancy term,

qf � opox

¼ qbðT � T0Þfx ð6Þ

where fx ¼ �f , pD ¼ ðp � p0Þ is the pressure difference, p0 being the hydrostatic pressure and b isthe coefficient of volumetric expansion.As the plates are infinitely long, the pressure p can be taken equal to the hydrostatic pressure p0,

i.e. pD ¼ 0:Using Eq. (6), Eqs. (2)–(4) can be written as

d2udy2

� qt0ðl þ KÞ

dudy

þ Kðl þ KÞ

dNdy

þ qbfxðT � T0Þðl þ KÞ � rH 2

0 uðl þ KÞ ¼ 0 ð7Þ

d2Ndy2

� qjt0c

dNdy

� Kc

2N�

þ dudy

�¼ 0 ð8Þ

R. Bhargava et al. / International Journal of Engineering Science 41 (2003) 123–136 125

d2ðT � T0Þdy2

� qcpt0Kf

dðT � T0Þdy

þ ðl þ KÞKf

dudy

� �2

þ cKf

dNdy

� �2

þ 2KKf

N 2

�þ N

dudy

�

þ c0t0ðT � T0ÞKf

¼ 0 ð9Þ

Introducing the following dimensionless variables

y ¼ gh; u ¼ KfUqfxbh2

; ðT � T0Þ ¼Kfl:T �

q2f 2x b2h4

; N ¼ KfN �

qfxbh3ð10Þ

and substituting the above transformations into the Eqs. (7)–(9), we obtain

d2Udg2

� Reð1þ RÞ

dUdg

þ T �

ð1þ RÞ þR

ð1þ RÞdN �

dg� H 2

a CU ¼ 0 ð11Þ

d2N �

dg2� Reb

AdN �

dg� RA

2N ��

þ dUdg

�¼ 0 ð12Þ

d2T �

dg2� RePr

dT �

dgþ ð1þ RÞ dU

dg

� �2

þ AdN �

dg

� �2

þ 2R N �2�

þ N � dUdg

�þ c1RePrT

� ¼ 0

ð13Þ

where R ¼ ðK=lÞ is the micropolar parameter, Re ¼ ðqt0h=lÞ the cross flow Reynolds number,A ¼ ðc=lh2Þ, B ¼ ðj=h2Þ and C ¼ ðh2=ð1þ RÞÞ the micropolar material constants, Pr ¼ ðlcp=Kf Þthe Prandtl number, c1 ¼ ðc0h=qcpÞ the dimensionless vertical distance, Ha ¼ ððrH 2

0 =lÞÞ1=2

is thelocal Hartmann number.The corresponding boundary conditions given in Eq. (5) then reduces to

at g ¼ 0; U ¼ 0; N � ¼ 0; T � ¼ Q

at g ¼ 1; U ¼ 0; N � ¼ 0; T � ¼ eQð14Þ

where Q ¼ ðPrGrbfxh=cpÞ is a dimensionless group, Gr ¼ ðq2bfxh3ðT � T0Þ=l2Þ is the Grasshofnumber and e ¼ ððT2 � T0Þ=ðT1 � T0ÞÞ is the non-dimensional heating parameter.The dimensionless shear stress, couple stress and the Nusselt number at the plates are given by

s� ¼ ð1�

þ RÞ dUdg

�at g¼0;1

M� ¼ dN �

dg

� �at g¼0;1

ð15Þ


Nu ¼ � 1

QdT �

dg

� �at g¼0;1

; for e ¼ 1

¼ 1

ðe � 1ÞQdT �

dg

� �at g¼0;1

; for e 6¼ 1

ð16Þ

3. Method of solution: Quasi-linearization method

The set of Eqs. (11)–(13) are highly non-linear therefore this system of equations cannot besolved analytically. The Quasi-linearization method has been used to solve it, which is illustratedas follows:The Quasi-linearization method, which is also known as the generalized Newton–Raphson

method, was given by Bellman and Kalaba, details of which can be seen in [12]. However anexplanation of the method is given in Appendix A. This method is used due to following ad-vantages.

(1) The method is quadratically convergent, starting from the initial guess value.(2) The solution is valid for a large range of parameters. Even when the required number of initial

conditions are not given, this method converges at a fast speed.

In order to implement the quasi-linearization technique, the system of Eqs. (11)–(13) is con-verted to a system of first order differential equations as follows:

U ;U 0;N �;N �0 ; T �; T �0� �

¼ ðx1; x2; x3; x4; x5; x6Þ ð17Þ

The Eqs. (11)–(13) are replaced by the following equivalent system of six first order ordinarydifferential equations,

dx1dg

¼ x2

dx2dg

¼ Reð1þ RÞ x2 �

Rð1þ RÞ x4 �

x5ð1þ RÞ þ H 2

a Cx1

dx3dg

¼ x4

dx4dg

¼ ReBA

x4 þRAð2x3 þ x2Þ

dx5dg

¼ x6

dx6dg

¼ RePr x6 � ð1þ RÞx22 � Ax24 � 2R x23�

þ x3x2�� c1RePr x5

ð18Þ


Applying the quasi-linearization algorithm as given in Appendix A on the set of Eq. (18) we get:

dxðkþ1Þ1

dg¼ xðkþ1Þ2

dxðkþ1Þ2

dg¼ Re

ð1þ RÞ xðkþ1Þ2 � R

ð1þ RÞ xðkþ1Þ4 � xðkþ1Þ5

ð1þ RÞ þ H 2aCx

ðkþ1Þ1

dxðkþ1Þ3

dg¼ xðkþ1Þ4

dxðkþ1Þ4

dg¼ ReB

Axðkþ1Þ4 þ R

A2xðkþ1Þ3

�þ xðkþ1Þ2

�

dxðkþ1Þ5

dg¼ xðkþ1Þ6

dxðkþ1Þ6

dg¼ RePr xðkþ1Þ6 � ðRþ 1Þ 2xðkþ1Þ2 xk2

h� xk2x

k2

i� A 2xðkþ1Þ4 xk4

h� xk4x

k4

i

� 2R 2xðkþ1Þ3 xk3h

� xk3xk3 þ xk2x

ðkþ1Þ3 þ xk3x

ðkþ1Þ2 � xk2x

k3

i� c1RePr x

ðkþ1Þ5

ð19Þ

Boundary conditions given by (14) now reduces to:

xkþ11 ð0Þ ¼ 0; xkþ11 ð1Þ ¼ 0

xkþ13 ð0Þ ¼ 0; xkþ13 ð1Þ ¼ 0

xkþ15 ð0Þ ¼ Q; xkþ15 ð1Þ ¼ eQ

ð20Þ

The system given by Eq. (19) is linear xðkþ1Þi and its general solution can be obtained by using theprinciple of superposition.The initial values for xðkþ1Þi are chosen as follows:

(i) For the particular solution

P ðkþ1Þi ð0Þ ¼ ð0; 0; 0; 0;Q; 0Þ ð21Þ

(ii) For the homogeneous solution

Hi2ð0Þ ¼ ð0; 1; 0; 0; 0; 0ÞHi4ð0Þ ¼ ð0; 0; 0; 1; 0; 0ÞHi6ð0Þ ¼ ð0; 0; 0; 0; 0; 1Þ

ð22Þ

The general solution of system of Eq. (19) is given by

xðkþ1Þi ðgÞ ¼ P ðkþ1Þi ðgÞ þ C2Hi2ðgÞ þ C4Hi4ðgÞ þ C6Hi6ðgÞ ð23Þ


where C2;C4 and C6 are the unknown constants and are determined by considering the boundaryconditions at g ¼ 1:The following expressions obtained for C2;C4 and C6 arise, viz

C2 ¼D1

D; C4 ¼

D2

D; C6 ¼

D3

D

where

D ¼ H12ð1Þ H34ð1ÞH56ð1Þ½ � H36ð1ÞH54ð1Þ � H14ð1Þ H32ð1ÞH56ð1Þ½ � H36ð1ÞH52ð1Þþ H16ð1Þ H32ð1ÞH54ð1Þ½ � H52ð1ÞH34ð1Þ

D1 ¼ �P ðkþ1Þ1 ð1Þ H34ð1ÞH56ð1Þ½ � H36ð1ÞH54ð1Þ � H14ð1Þ

h� P ðkþ1Þ

3 ð1ÞH56ð1Þ

� eQ�

� P ðkþ1Þ5 ð1Þ

�H36ð1Þ

iþ H16ð1Þ

h� P ðkþ1Þ

3 ð1ÞH54ð1Þ � eQ�


�H34ð1Þ

i

D2 ¼ H12ð1Þh� P ðkþ1Þ

3 ð1ÞH56ð1Þ � eQ�


�H36ð1Þ

iþ P ðkþ1Þ

1 ð1Þ H32ð1ÞH56ð1Þ½

� H36ð1ÞH52ð1Þ þ H16ð1Þ eQ�h


�H32ð1Þ þ P ðkþ1Þ

3 ð1ÞH52ð1Þi

D3 ¼ H12ð1Þ eQ�h


�H34ð1Þ þ P ðkþ1Þ

3 ð1ÞH54ð1Þi� H14ð1Þ eQ

�h� P ðkþ1Þ

5 ð1Þ�H32ð1Þ

þ P ðkþ1Þ3 ð1ÞH52ð1Þ

i� P ðkþ1Þ

1 ð1Þ H32ð1ÞH54ð1Þ½ � H34ð1ÞH52ð1Þ

The set of differential equations given in (19), have been solved by employing the fourth orderRunge Kutta method for the boundary condition given in (20) and solution is obtained by in-voking the principle of superposition. The whole range is divided into 40 equal parts with equalsub intervals of width 0.025; and accuracy has been taken as 0.0005.

4. Results and discussion

On solving the system of equations, given by (11)–(13), with boundary conditions (14), thevelocity, microrotation and temperature distribution are obtained. The Prandtl number Pr, ma-terial constant c1 and the micropolar material constants A, B and C are taken to be constant, at0.733, 1.0, 1.0, 0.001 and 1.0 respectively whereas the effect of other important parameters, namelymicropolar parameter R, the cross flow Reynolds number Re and local Hartmann number Ha hasbeen studied for these functions and the corresponding profiles are shown in the Figs. 1–6.Fig. 1 depicts the variation of velocity function U with Hartmann number Ha for a prescribed

value of micropolar parameter R ¼ 3: Profiles are shown both for positive Re ð¼ 2Þ (i.e. for in-jection at g ¼ 0 and suction at g ¼ 1) and negative Re ð¼ �2Þ (i.e. for suction at g ¼ 0 and in-jection at g ¼ 1). The velocity profiles are parabolic in nature and attain their maximum near themiddle. However for negative cross flow Reynolds number, the maximum is shifted towards


g ¼ 0. It is observed that the velocities are higher for injection as compared to suction at g ¼ 0.The velocity continuously decreases with an increase in Hartmann number. The same pattern isobserved for both values of cross flow Reynolds number Re. This indicates that the fluid velocitycan be reduced by an increase in the magnetic field, a similar phenomenon is observed in [10].The variation in the velocity with reference to the variation in micropolar parameter R for

constant magnetic field is presented in Fig. 2. The results are given for both positive and negativecross flow Reynolds number Re. It is observed that the velocity values for injection are more thanthat for suction at g ¼ 0. As the micropolar parameter R increases, the velocity continuouslydecreases, the maximum being shifted towards g ¼ 1. But for suction the maximum is closer tog ¼ 0. This is however different as compared to viscous flow which correspond to R ¼ 0. It isreally interesting to observe this behavior of micropolar flow. The boundary layer thickness, asexpected, continuously decreases with increase in microconcentration.

Fig. 1. Velocity distributions for various H 2a .

Fig. 2. Velocity distributions for various R.


The effect of the Hartmann number Ha on microrotation when the micropolar parameter R

kept fixed at 3, is shown in Fig. 3. The values of microrotation are negative in the first halfwhereas in the second half, these are positive, thus showing a reverse rotation near the twoboundaries. Moreover for injection these are higher numerically as compared to suction at g ¼ 0.An increase in the magnetic field leads a to decrease in microrotation. The intensity of themagnetic field thus can be used for decreasing the angular rotation especially in suspension flows,commonly arising in lubrication problems.Fig. 4 illustrates that the microrotation continuously increases with an increase in the micro-

polar parameter R for a fixed value of the magnetic field. As expected the increase in micro-concentration enhances the angular rotation. It is clear from this figure that the microrotation is

Fig. 3. Microrotation for various H 2a .

Fig. 4. Microrotation for various R.


negative in the first half while it is positive in the second half. The values for injection are againhigher numerically as compared to suction at g ¼ 0.The temperature variation is shown in Figs. 5 and 6 both for the injection and suction cases. It

is observed that the temperature distribution depends upon e, however for Re > 0; it increases anda attains maximum and then decreases. While for Re < 0; it continuously decreases. This happensfor e > 1 as well as for e < 1:The skin friction and Nusselt number has been given for Re ¼ 2 and �2 in the Tables 1 and 2

respectively which show their variation with respect to the micropolar parameter R and Hartmannnumber Ha. From the numerical results obtained as given in Table 1, it is noticed that the skin

Fig. 5. Temperature distribution for various H 2a .

Fig. 6. Temperature distribution for various R.


friction for injection increases with an increase in the Hartman number Ha while keeping themicropolar parameter R fixed, thus causing more resistance. However it decreases with an in-crease in the micropolar parameter R, for a fixed Ha. But for the suction case, the skin frictiondecreases with an increase in Hartmann number Ha, for a fixed R. A similar pattern is observedfor injection with an increase in the micropolar parameter R with a fixed Hartmann number Ha.From Table 2, it is observed that the rate of cooling increases with an increase in the Hartmann

number Ha and also with the micropolar parameter R. Similar pattern is observed for suction.This implies that the magnetic field as well as microconcentration are the important parameters inreducing heating effects and can be used for controlling the rate of heat transfer, which is desiredin many MHD applications.

5. Conclusions

The velocity decreases with an increase in Hartmann number. Values for positive cross flowReynolds number are higher as compared to those for )ve Reynolds number.The microrotationincreases with an increase in microconcentration while it decreases with an increase in the mag-netic field. The micropolar effects behave as a coolant and are thus effective in reducing the coolingrate and help in producing the desired temperature. The magnetic field can be used effectively forcontrolling the rate of heat transfer as required in MHD applications like MHD generators,nuclear reactors, where it is used to control enormous temperatures.

Table 1

Values of f�f 00ð0Þg for different values of the Hartmann number (Ha) and micropolar parameter (R) with Re þveand )ve

Re ¼ 2 Re ¼ �2H 2

a R ¼ 3 R H 2a ¼ 5 H 2

a R ¼ 3 R H 2a ¼ 5

0 0.146179 0 0.418457 0 0.270853 0 1.744551

1 0.153360 1 0.313665 1 0.268153 1 0.648696

5 0.172302 3 0.172302 5 0.260610 3 0.260610

10 0.184977 5 0.108876 10 0.254872 5 0.152236

e ¼ 1, Q ¼ 1, B ¼ 0:001, Pr ¼ 0:733, A ¼ 1:0, c ¼ 1:0, C ¼ 1:0.

Table 2

Values of f�T 0ð0Þg for different values of the Hartmann number (Ha) and micropolar parameter (R) with Re þveand )ve

Re ¼ 2 Re ¼ �2H 2

a R ¼ 3 R H 2a ¼ 5 H 2

a R ¼ 3 R H 2a ¼ 5

0 )0.623848 0 )0.628725 0 0.777345 0 0.768676

1 )0.621962 1 )0.622106 1 0.778835 1 0.777926

5 )0.617790 3 )0.617790 5 0.782115 3 0.782115

10 )0.615644 5 )0.616200 10 0.783783 5 0.783436

e ¼ 1, Q ¼ 1, B ¼ 0:001, Pr ¼ 0:733, A ¼ 1:0, c ¼ 1:0, C ¼ 1:0.


Appendix A

Let us consider the vector equation

dxidt

¼ gi xj; t� �

; ði; j ¼ 1; 2; . . . . . . . . . ; nÞ ðA:1Þ

where �t�, is the independent variable, defined in the closed domain [a,b], subject to the boundaryconditions:

xpðaÞ ¼ ap ðp ¼ 1; 2; . . . . . . . . . ; nÞxqðbÞ ¼ bq ðq ¼ 1; 2; . . . . . . . . . ; nÞ

ðA:2Þ

here xi, gi are taken as vectors in the nth dimensional space. The ðk þ 1Þth approximation xðkþ1Þi tothe solution of (A.1) is obtained by expanding the functional gi about the kth approximation xki ,and retaining only the linear and constant terms. The kth approximation xki is assumed to be a setof known approximate solutions.The general form of the quasi-linearization algorithm is,

dxðkþ1Þi

dt¼ gi xkj ; t

� �þXn

j¼1

ogioxj

� �k

xðkþ1Þj

�� xkj

�ðA:3Þ

where ðogi=oxjÞk is the Jacobian matrix evaluated at the kth approximation and is defined as:

ogioxj

� �k

¼

og1oxðkÞ1

og1oxðkÞ2

. . .og1oxðkÞn

. . . . . . . . . . . .ognoxðkÞ1

ognoxðkÞ2

. . .ognoxðkÞn

266664

377775 ðA:4Þ

The superscripts in the parenthesis denote the iteration number. The boundary conditions for theðk þ 1Þth approximation xðkþ1Þi are given by,

xðkþ1Þp ðtÞjt¼a ¼ ap ðA:5Þ

xðkþ1Þq ðtÞjt¼b ¼ bq ðA:6Þ

The system of differential Eq. (A.3) is linear in xðkþ1Þi and is non-homogeneous. Its solution can beobtained by invoking the principle of superposition, i.e. by adding the �n� homogeneous solutionsof the corresponding �n� homogeneous equations (obtained by making the non-homogeneous termequal to zero), to the particular solution. A particular solution can be obtained by integrating(A.3) with approximate initial conditions. For simplicity, the initial values may be assumed as,

PiðtÞjt¼a ¼ xiðtÞjt¼a ¼ hi ðA:7Þ

where hi is the ith component of null vector, and Pi is the corresponding particular integral.


Besides the particular solution, there exist �n� linearly independent solutions of the homoge-neous equations,

dxðkþ1Þj

dt¼

Xn

j¼1xðkþ1Þj

ogioxj

� �k

ðA:8Þ

The initial conditions for the jth solution of this equation may be assumed as

H ðkþ1Þij ðtÞjt¼a ¼ xðkþ1Þij ðtÞjt¼a ¼ dij; i; j ¼ 1; 2; . . . . . . . . . ; n ðA:9Þ

where

dij ¼ 1 for i ¼ j¼ 0 for i 6¼ j

Hence the general solution may be written as,

xðkþ1Þi ¼ Pi þXn

j¼1AjHij ðA:10Þ

The coefficients Aj may be obtained by satisfying the conditions at the boundaries (A.5) and (A.6).Hence

Pnj¼1

AjHmjðtiÞ ¼ Bi � PmðtiÞ; i ¼ 1; 2; . . . . . . . . . ; n

m ¼ Mi

ðA:11Þ

The approximation xðkþ1Þi is next feed back into the algorithm to obtain the approximation xðkþ2Þi

and the process is repeated until satisfactory results are obtained.

References

[1] A.C. Eringen, Simple micro fluids, Int. J. Eng. Sci. 2 (1964) 205–217.

[2] A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966) 1–16.

[3] A.C. Eringen, Theory of thermomicrofluids, J. Math. Anal. Appl. 38 (1972) 480–496.

[4] T. Ariman, M.A. Turk, N.D. Sylvester, Micro continuum fluid mechanics––A review, Int. J. Eng. Sci. 11 (1973)

905–930.

[5] M. Balaram, V.U.K. Sastry, Micropolar free convection flow, Int. J. Heat Mass Transfer 16 (1973) 437–440.

[6] S.K. Jena, M.N. Mathur, Similarity solution for laminar free convection flow of a thermomicropolar fluid past a

non-isothermal vertical flat plate, Int. J. Eng. Sci. 19 (1981) 1431–1439.

[7] R.S. Agarwal, C. Dhanapal, Numerical solution of free convection micropolar fluid flow between two parallel

porous vertical plates, Int. J. Eng. Sci. 26 (1988) 1247–1255.

[8] S.R. Kasivishwanathan, M.V. Gandhi, A class of exact solutions for the magnetohydrodynamic flow of a

micropolar fluid, Int. J. Eng. Sci. 30 (1992) 409–417.

[9] A.A. Mohammeadein, R.S.R. Gorla, Effects of transverse magnetic field on a mixed convection in a micropolar

fluid on a horizontal plate with vectored mass transfer, Acta. Mech. 118 (1966) 1–12.


[10] R.S.R. Gorla, H.S. Takhar, A. Slaouti, Magnetohydrodynamic free convection boundary layer flow of a

thermomicropolar fluid over a vertical plate, Int. J. Eng. Sci. 36 (1998) 315–327.

[11] A.C. Eringen, Microcontinuum Field Theories. II: Fluent media, Springer-verlag, New York, 2001.

[12] R.E. Bellman, R.E. Kalaba, Quasilinearization and Nonlinear Boundary-value problems, Elsevier Publishing Co.

Inc., New York, 1965.


Numerical solution of free convection MHD micropolar fluid flow between two parallel porous vertical...

Documents

Transcript of Numerical solution of free convection MHD micropolar fluid flow between two parallel porous vertical...