Journal of Naval Architecture & Marine Engineering 2008

ISSN 1813-8235 (Print), 2070-8998 (Online)

Delft University of Teciinoiogy j ^ n e 2008

Ship Hydromechanics laboratory 5 j ^ q ^

L i b r a r y " '

Mekelweg 2 26282 CD Delft

Phone: +31 (0)15 2786873 E-mail: p.w.deheer@tudelft.nl

JOURNAL OF NAVAL ARCHITECTURE AND

MARINE ENGINEERING

An International Research Publication

HYDROMAGNETIC T H R E E DIMENSIONAL C O U E T T E F L O W AND HEAT T R A N S F E R

S. S. Das, M. Mohan ty, J. P. Panda and S. K. Sahoo 1-10 SEPARATION POINTS O F AAAGNETOHYDRODYNAMIC

BOUNDARY LAYER F L O W ALONG A V E R T I C A L P L A T E WITH EXPONENTIALLY DECREASING F R E E STREAM V E L O C I T Y

M.A. A lim, M. M. Rahman, M. M. Karim 11-18

SYNTHESIS, CHARACTERIZATION AND E F F E C T O F MICROSTRUCTURE ON S L U R R Y EROSION RESISTANCE O F CAST F E - T I C COMPOSITES

M K Manoj, R K Galgali, S. K Nath, S. Ray 19-26 NUMERICAL SOLUTIONS O F HEAT AND AAASS T R A N S F E R

E F F E C T S O F AN UNSTEADY MHD F R E E C O N V E C T I V E F L O W PAST AN INFINITE V E R T I C A L P L A T E WITH CONSTANT SUCTION

V.Ambethkar 27-36

http://Jname. 8m. net/

and

DOI: 10.3329/jname.v5il.l785

Journal Of Naval Architecture and Marine Engineerins June, 2008 /i/tp://jname.8in.nel

NUMERICAL SOLUTIONS O F HEAT AND AAASS T R A N S F E R

E F F E C T S O F AN UNSTEADY MHD F R E E C O N V E C T I V E F L O W

PAST AN INFINITE V E R T I C A L P L A T E WITH

CONSTANT SUCTION

V.Ambethkar

Department of Matliematics, University of Delhi, Delhi, India. Email: vambethl<ar@!maths.du.ac.in. Tel: 91-11-27666658.

Abstract

The objective of Ihis worli is lo study lieat and mass transfer in an unsteady MHD free convective flow past an infinite vertical plate with constant suction numerically. Dimensionless governing equations of Ihe problem have been solved by using finite difference technique. Numerical solulions for temperature, velocity, concentration have been obtained for suitable parameters like Graslwff number, mass Grashoff number, Prandtl number and Schmidt number Rate of heal transfer and mass transfer are studied The resuils obtained are discussed wilh Ihe help of graphs and tables lo observe effeel of various parameters concerned in the problem imder investigation. Slabilit}' and convergence of the finite difference scheme is established.

Key words: MHD, unsteady, constant suction, finite difference technique, Heat and mass transfer.

N O M E N C L A T U R E

Cp concentration near plate (3 Bo uniform magnetic field a C„ concentration o f the ambient fluid p

g acceleration due to gravity v

G,. Grashof number | i

ii thermal conductivity o f the fluid co N,i Nusselt number

P, Prandtl number p Sc Schmidt number

t dimensionless time oo

T temperature o f the fluid in the p

boundary layer

temperature o f the ambient fluid

T temperature at the plate ' u, V The dimensionless x and

y-component o f velocity.

L Introduction

Combined heat and mass transfer problems are important in many processes and have therefore received a considerable amount o f attention. In many mass transfer processes, heat transfer considerations arise due to chemical reaction and often due to the very nature o f the process. In processes, such as drying, evaporation at the surface o f a water body, energy transfer in a wet cooling tower and the flow in a desert cooler, heat and mass transfer occur simultaneously. Unsteady free convection flow past a vertical porous plate was investigated by

volumetric coefficient o f thermal expansion the scalar electrical conductivity

density o f the fluid

reference kinematic viscosity Viscosity o f the fluid frequency o f oscillation wall conditions Subscript ambient temperature wall conditions Superscript

differentiation with respect toy

V.Ambethkar Journal of Naval Aichileclwe unci Marme Engineering 3(200KJ 28-36

Helmy (1998). Achaiya et al. (2000) have studied free convection and mass transfer flow through a porous medmm bounded by vertical infinite surface with constant suction and heat flux. But in those studies they considered the flow to be steady. Coming back to unsteady case, K i m (2000) investigated unsteady M H D convection heat transfer past a semi- infinite vertical porous moving plate with variable suction. Little extension to those problems has been done by Chamkha (2004). In this study author extended the problem for the case o f mass transfer but restricted to the case o f semi-infinite moving plate.

Unsteady oscillatory free convection flow plays an important role in chemical engineering, turbo machines, and aerospace technology. Such flows arise due to unusual motion o f boundary or boundary temperature. Recently Singh et al. (2003) have investigated the effect o f oscillatory suction velocity on free convection and mass transfer flow o f a viscous fluid past an infinite vertical porous plate. Sahoo et al. (2003) have analyzed M H D unsteady free convective fiow past an infinite vertical plate with constant suction and heat sink. Extension to this problem has been done by Muthucumaraswamy and Kumar (2004). In this study thermal radiation effect on moving infinite vertical plate in presence o f variable temperature and mass diffusion is considered.

Because o f the importance o f suction in the fields o f aerodynamics and space science our present study is motivated towards this direction. Our main purpose is to investigate numerically the problem o f combined heat and mass transfer o f an unsteady M H D flow past an infinite plate with suction. None o f the above stated studies discusses completely about this. This is our motivation to the present study. The results o f this study discussed for various numerical values o f t h e parameters.

T=0 U=0

Heated Plate _{Cooled Plate}

Fig. 1: Geometry o f the problem

V.Ambethkar Jownal of Naval ArchilecUire and Marine Bngmceiing 5(2008) 28-36

2. Mathematical Model

Consider an unsteady two dimensional free convective flow o f an electrically conducting viscous and incompressible fluid past an infinite, porous and vertical plate with constant suction. A magnetic field Bg is apphed perpendicular to the plate. A system o f rectangular coordinate axes ox, y.z, is taken such that yi=0 on the plate and z, is along its leading edge. A l l the fluid properties like density, velocity, pressure, temperature, concentration, and viscosity etc. are considered.

The influence o f t h e density variation with temperature is considered only in the body force term. Its influence in other terms o f the momentum and the energy equations is assumed to be negligible. The variation o f expansion coefficient with temperature is considered to be negligible. This is the well-known Boussinesq approximation. Thus, under these assumptions, the physical variables are functions o f y, and t, only and the problem is governed by the following system o f equations

dv,

Continuity equation: = 0 , ₍₁₎

^ " i , .. 5«i _^or^ ^ . . vd\_aBlu^

.2 Momentum equations: — - -f- v, = gp{T, -T^,) + (2) Energy equation: h v

^ dT, _ dT, _ kd%

dt, dy, dy

1

P

(3)

^. . p • dC, D'd'C,

Mass transfer equation: - + v, IA\

dt, ' dy, dy]

The initial and boundary conditions o f the problem are ti.<0,u,(y,,t,) = 0, Ti.(yi,t,) = T„ (5) C r ( y „ t , ) = C„ ; t, > 0 , u,(0,t, ) = Vo, T , ( 0 , t | ) =Tp+E(Tp-T„)e'"'|t,, a t y , = 0 . (6) C , ( 0 , t | ) =Cp+E(Cp-C„)e'",t,. tl > 0 , U , ( o o , t , ) ^ 0 , T . K t,) ^ T„, (7) C|(a), t|) C,,, as y, ^ cx).

Since the plate is assumed to be porous and through it suction with uniform velocity occurs, Equation (1) integrates to v, = - v , , is the constant suction velocity. Here, U B | B is the velocity o f the fluid, TBps the temperature o f the fluid near the plate, T B ^ B the temperature o f the fluid far away from the plate, Cp the concentration near the plate, C„ the concentration far away from the plate, g the acceleration due to gravity, p the coefficient o f volume expansion for heat transfer, P' the coefficient o f volume expansion for concentration,

V the kinematic viscosity, a the scalar electrical conductivity, co the frequency o f oscillation, B B Q B the applied uniform magnetic field, p B . B the density o f t h e fluid, k the thermal conductivity and t B | B is the time.

From Equation (1) we observe that V, is independent o f space co-ordinates and may be taken as constant. We define the following non-dimensional variables and parameters.

V.Ambethkar ' .Iinimal of Naval ArchilecUire and Marine Engiiweriiig 5(2008) 28-36

4v

4v

' k '

D'

(8)

M

' (I , CO CO,

J

Now taking into account Equations (5), (6), (7) and (8), Equations (2), (3) and (4) reduce to the following non-dimensional form With

du

,

du

. d^u „

4 4 +

4(^

J _

4 ^ „

+

r

dt dy

dy^

dT

_^dT _

4

d^T

dt dy~P^

dy' '

dC

^dC _ 4 d'T

dt dy

~ dy'

(9) (10) ( H ) t. < 0, u(y, t) = 0, T ( y , t ) = 0; t > 0 , u (0, t) = 0, T(0,t) = 1 + C (0,t) = 1 + se'"'. y, 0 t > 0 , u (oo, t) = 0, T (oo,t) = 0 C (oo,t) = 0. as yi ^ CO. (12) (13) (14)

The Grashof number G,. > 0 represents external cooling o f t h e plate and G,. < 0 denotes external heating o f t h e plate. Gm the modified Grashof number , S^the Schmidt number, and Pr the Prandtl number.

3. Method of solution

Here we sought a solution by finite difference technique o f implicit type namely Crank- Nicolson implicit finite difference method which is always convergent and stable. This method has been used to solve Equations (9) (10) (11) subject to the conditions given by (12) (13) and (14). To obtain the difference equations, the region o f the flow is divided into a gird or mesh of lines parallel to y and t axes. Solutions o f difference equations are obtained at the intersection o f these mesh lines called nodes. The values o f t h e dependent variables T , u and C at the nodal points along the plane y = 0 are given by T(0,t) and u(0,t) and C(0,t) hence are known from the boundary conditions.

V.Ambethkar Jownal of Naval ArchilecUire and Marine Engineering 5(2008) 28-36

In the above Fig. 2, Ay , At are constant mesh sizes along y and t directions respectively. We need a scheme to fmd single values at next time level in terms o f known values at an earlier time level. A forward difference approximation for the first order partial derivatives o f u, T and C w.r.t. t and y and a central difference approximation for the second order paitial derivative of u , T and C w.r.t. y are used. On introducing finite difference approximations for;

/ j+2 j+1 t j At j - 1 j-2 \ / j+2 j+1 t j At j - 1 j-2 j+1) _{(iJ+1) (i+l J+1)} / j+2 j+1 t j At j - 1 j-2 ' i / j+2 j+1 t j At j - 1 j-2 (i-lJ) (io) f ( ' + l j ) / j+2 j+1 t j At j - 1 j-2 ( i - l j - l ) ( i j - l ) ( i + l j - 1 ) / j+2 j+1 t j At j - 1 j-2 ' •

i-2 i-1 i 1+1 i+2 > y <

Fig. 2: Finite difference grid.

fdT^

f - 1

.dt

Kr T —T 4-T —T 4 ( A j ' ) C ~C -t-C -C A{Ay) ' < / . l j - Z < , - l . j + " w j + | - M , - - u ^ l 4(A3.) •',y+i At { dl J,^ At dl - 2T Al _ iii,,j+ii,_,j-2ii^ j +v,,, j^, +",-i,,>i - 2 » , , y , i 2{Ayf (15)

J

V.Ambethkar .Joumal of Nam/ ArchilecUire ami Marine Engineering, 5(2008) 28-36

The finite difference approximation o f Equations (9), (10) and (11) are obtained with substituting Equation (15) into Equations (9), (10) and (11) then

^MJ-^i-^J+^^XJ.l-^-^J^\ _{% U +%U+i +H-Urt -2"v+i}

+ 4 G , 7 ; . ^ + 4 G , „ C , , - 4 M » . ^ . (16) A/ A/ •T. -S. - 2 ? ; . - 2 ? ; , , . , (Ay)^

Multiplying both sides o f Equations (16) (17) and (18) by (A?)and after simplifying, we obtain

At At (Ayr Ay At (17) (18) A?

k.l., + " , - , , 7 _{-2w,-,J+— -z/,„,_J + 4G,.A/7;., +4G„,AfC,, - 4MAtu,}

V^/+l,7+l lAt 2 -(,7+1 Ay _Pi-m •(•^•+1,7+1 ^-l./+l ) 2At P^^y fe:,7+?;-i,7-2^J+7;,,+^ (?;.,,,-?;_,,,) (19) (20)

c

+ - - ^ c

—ic ~c )

2At Sc { ^ y y Ay '(^/+l,7+l + Q-1,7+1) 2A^ SAy ( ^ ' - ^ ^ - . - 2 C J + C „ , + ^ ( C , „ , - C , _ J (21)

Now for Crank- Nicolson implicit method, let

{ ^ y f

this condition the above equations can be written as

= ;• - 1 (method is always stable and convergent), under

2^AyJ !«/+l,7+l + _{v 4 v 2j} '-1,7+1

^ l ^ A t }

y2^Ayj l"'+i.7 +

^l_At_^ 2 Ay

+ 4G,A^r.^. + 4G„,ArC,. . - 4MA^M.^. ₍₂₂₎

V.Ambethkar Jowmil of Naval ArchilechiK and Mmim Engineering 5(2008) 28-36 V PrJ P,. Z i-\,j+\

+

P,. T. f ^ y P . . Pi^x.i + Prj Pr ^y. z '-Uj A ? _

2

A ; ^ ~ 5 ; c

2^

j-ij+i ^ s . _'cJ ^ _ 2 _ _ A / ^ •S-. A ; ; (23) 1 — (24)

4. Numerical solutions and their accuracy

To get the numerical solutions o f the temperature T, velocity u and concentration C, Necessary code is developed m code in MathematicaS.O. The logic o f t h e program is divided into 4 modules as follows:

Module 1: Main, initially it creates three tables to hold the Numerical Solutions o f Temperature Velocity and concentration whose coefficients are allotted in the Module 2. After this, it calculates the numerical values at the next time step level. In order to do this, it uses another sub module named, Tridiagonal, which solves the t r i -diagonal matrix by using Gauss-Elimination method. Further it moves to the Module 3, for comparison o f t h e numerical solutions with analytical solutions.

Module 2: Coeff Mat, we know that all the terms and their coefficients on RHS o f Equations (22), (23) and (24) are known values from initial and boundary conditions. At every time step, for different values o f ' i ' the finite difference approximation o f Equation (24) gives a linear system o f equations. Then, for j = 0 and i = 1 2 n - l Equation (24) gives a linear system o f ( n - l ) equations for the ( n - l ) unknown values o f ' C ' in the first time row m terms o f known initial and boundary values. This module maintains coefficients o f this linear system o f equauons. Similarly the above process repeats for the remaining Equations (23) and (22) to obtain the values o f u and T.

Module 3: Comparison, It compai-es the numerical solution with the analytical solution at every time step level By making use o f T and C into Equation (22), the numerical solutions for ' u ' are obtained.

To ensure the validity o f our numerical solutions, we have compared this numerical solution for temperature velocity and concentration for the case o f suction for different Prandtl numbers with the available exact solutions in the literature. Table 1 and Table 2 show comparisons between the numerical values o f temperature and velocity for P, =6.75 and 0.733 respectively obtained from the present study. It is clearly seen from these tables that results are in excellent agreement. The comparison tables Table 1 and Table 2 have been plotted and shown in Fig. 3 and Fig. 4. As the accuracy o f the numerical solutions is very good, the curves corresponding to exact and numerical solutions are laying very close to the other. To ensure the efficiency o f our code for velocity, we have given a table o f numerical solution for velocity for water (P,=6.75) for the cases of suction These values have been plotted under Fig. 4.

5. Results and Discussion

For the purpose o f discussing the results some numerical solutions are obtained for non-dimensional temperature T, velocity u concentration C. By using temperature the rate o f heat transfer and by using concentration rate o f mass transfer is obtained. The numerical solutions for the case o f suction for temperature have been shown in Table 1. It can be seen from the table that the transient temperature decreases for the increase o f y. Similarly temperature field due to variation in P, for air, water, mercury etc has been found and observed that mercury has a stationary temperature. The numerical solutions for the case of suction for concentration have been shown in Table 5. It can be seen from the table that the transient concentration profiles decreases for the increase o f y. The concentration profiles due to variation o f S^ for gases like hydrogen, oxygen and water vapor has been found but not giving here due to almost similar calculations. It can be found that hydrogen can be used for maintaining effective concentration field. The numerical solutions for the case o f

V.Ambethkar .lourmil of Naval ArchilecUire unci Marine Engmeering 5(200S) 28-36

suction for velocity have been shown in Table 2. It can be seen from the table that the transient velocity profiles decrease for the increase o f y. While finding velocity profiles numerical values for G„ G„„ M have been chosen suitably.

Table 1: Comparison o f Temperatare profiles for ?,=6,15, t=0.1 for the case o f suction

y Analytical Solution Numerical Solution 0 1 1 0.05 0.408622 0.409811 0.1 0.168723 0.167945 0.15 0.689642 0.068825 0.2 0.021486 0.028205 0.25 0.011598 0.011558 0.3 0.004986 0.004736 0.35 0.001875 0.001940 0.4 0.000985 0.000793 0.45 0.000456 0.000321 0.5 0.000369 0.000030 0.55 0.000892 0.000032 0.6 0.000098 0.000021 0.65 0,000069 0.000011 0.7 0.000086 0.000001 0.75 0.0000045 0.00000001 0.8 0.0000001 0.1E-07 0.85 0.1E-05 0.1E-08 0.9 0.1E-06 0.1E-09 0.95 0.1E-06 0.1E-11 1 0 0

Table 2: Comparison o f velocity for Pr =0.733, M=2, Gr=4, G„,=2, T=0.0975

y Analytical Numerical Solution Solution 0 1 1 0.05 0.952677 0.909804 0.1 0.896203 0.820757 0.15 0.832823 0.733967 0.2 0.764716 0.650451 0.25 0.693946 0.571107 0.3 0.62241 0.49668 0.35 0.551797 0.427745 0.4 0.483555 0.364691 0.45 0.418869 0.307724 0.5 0.358648 0.256872 0.55 0.303532 0.211999 0.6 0.253906 0.172823 0.65 0.20992 0.138938 0.7 0.171526 0.109839 0.75 0.138509 0.084943 0.8 0.11053 0.063609 0.85 0.08716 0,04516 0.9 0.067916 0,02889 0.95 0.05229 0,014079 1 0.039778 0

V.Ambethkar Jownal of Naval ArchilecUire and Marine Engmeermg 5(200H) 28-36

Table 3: Numerical values o f rate o f heat transfer _{Table 4:.Numerical solutions o f Mass Transfer}

t Numerical values of N , 0.0025 18.69578 0.005 16.4067 0.0075 14.66101 0.01 13.29947 0.0125 12.21477 0.015 11.33348 0.0175 10.60449 0.02 9,991646 0.0225 9.468969 0.025 9,017448 0.0275 8,622961 0.03 8,274846 0.0325 7,964935 0.035 7,686876 0.0375 7,435664 0.04 7,207309 0.0425 6,998585 0.045 6,806863 0.0475 6,629974 0.05 6,466115 S,No _{So Sh} 1 0,22 0,74986 2 0,60 0,59846 3 0,78 0,77986

Table 5: Comparison o f concentration profiles for

water vapor 8^=0,60, t=0,0025 Analytical Numerical y Solution Solution 0 1 1 0,05 1.14693 1,14682 0,1 0.307983 0,30729 0,15 0.08635 0,0823382 0,2 0.022062 0,0220624 0,25 0.005986 0,00591161 0,3 0,001258 0,00158401 0,35 0,0001148 0,00011372 0,4 0,000985 0,00003044 0,45 0,000456 8,06984E-6 0,5 0,000369 1,83187E-6 0,55 0,000892 7,4E-7 0,6 0,000098 4,8E-6 0,65 0,000069 0,6E-4 0,7 0,000086 0,2E-3 0,75 0,0000045 0,1 E-4 0,8 0,0000001 0,1E-07 0,85 0,1E-05 0,1E-08 0,9 0,1E-06 0,1E-09 0,95 0,1E-06 0,1E-11 1 0 0

From the technological point o f view, it is important to know the rate o f heat transfer between the plate and the fluid. This can be found by using the non-dimensional quantity, the Nusselt number, N„, The Nusselt number is defined as -ve gradient o f t h e temperature. The numerical values o f the Nusselt number against time t are shown in Table 3. As t increases, the rate o f heat transfer at the plate decreases gradually. Finally for mass transfer we need the -ve gradient o f concentration. This is denoted and defined as Schmidt number Sc. The numerical values of rate o f mass transfer Sh in terms o f Sherwood number are obtained and have been shown in Table 4. From this table it can be observed that rate o f mass transfer first increases gradually and then decreases as per gradual increase o f the Schmidt number.

6. Conclusions

The heat and mass transfer in an unsteady M H D free convective flow past an infinite vertical plate with constant sucdon is studied numerically. From the above study, following conclusions can be drawn:

i) The transient temperature decreases for the increase o f y.

ii) The transient concentration profiles decreases for the increase o f y. iii) The transient velocity profiles decreases for the increase o f y. iv) The rate o f heat transfer at the plate decreases gradually.

v) The rate o f mass transfer first increases gradually and then decreases.

V.Ambethkar Jommil of Naval Architecliire and Mariiw Engineering 5(2008) 28-36

References

Carnahan, P., Luther, H.A. and Wilkes, J. O. (1969): Applied Numerical Methods, John Wiley & Sons, USA. Helmy, K.A., (1998): M H D Unsteady Free Convection Flow Past a Vertical Porous Plate, Z A M M Vol 78(4) pp. 225-270. doi:10.1002/(SICI)1521-4001(199804)78:4<255::AlD-ZAMM255>3.0.CO:2-V

Acharya, M . , Dash, G.C. and Singh, L . P. (2000): Magnetic Field Effects on the Free Convection and Mass Transfer Flow Through Porous Medium with Constant Suction and Constant Heat Flux, Indian J. Pure Appl Math. 31(1), pp.I-18.

Young, J. K. (2000): Unsteady M H D Convective Heat Transfer Past A Semi-Infinite Vertical Porous Moving Plate with Variable Suction, Int. J. Engg. Sci, Vol 38, pp. 833-845. doi: 10.1016/50020-7225(99)00063-4 Chamkha, A. J. (2004): Unsteady M H D Convective Heat and Mass Transfer Past A Semi-Infinite Vertical Permeable Moving Plate with Heat Absorption, Int. J. Engg. Sci, Vol 42, pp. 217-230. doi:10.1016/S0020-7225(03)00285-4

Singh, A. K., Singh, A. K. and Singh, N . P. (2003), Heat And Mass Transfer In M H D Flow of a Viscous Fluid Past a Vertical Plate Under Oscillatoty Suction Velocity, Indian. J. Pure Appl. Math., 34(3), pp. 429-442. Sahoo, P.K., Datta, N . and Biswal, S. (2003): Magnetohydrodynamic Unsteady Free Convection Flow Past an Infinite Vertical Plate With Constant Suction and Heat Sink, Indian J. Pure Appl. Math., 34(l),pp .145-155. Muthucumaraswamy, R. and Kumar, G. S. (2004): Heat and Mass Transfer Effects on Moving Vertical Plate in The Presence of Thermal Radiation, Theoret. Appl. Mech., Vol 31, No.1, pp. 35-46.

doi:10.2298/TAM0401035M

D O I : 10.3329/jname.v5il.I784

Journal of Naval Architecture and Marine Engineering

June, 2008

littp://jname. 8m. mt

HYDROAAAGNETIC T H R E E DIAAENSIONAL C O U E T T E F L O W

AND H E A T T R A N S F E R

S. S. Das', M . Mohanty^ J . P. Panda^ and S. K . Sahoo''

^Department of Physics, KBDAV College, Nirakarpur, Khurda-752 019 (Orissa), India, Email: drssd2@yahoo.com ^Department of Physics, Christ College, Mission Road, Cuttack-753 001 (Orissa), India

^Department ofMathematics, Synergy Institute of Engg. & Tech., Dhenkanal-759 001 (Orissa), India Department of Computer Science & Engg., KIIT University, Bhubaneswar-751 024 (Orissa), India

Abstract

This paper theoretically analyzes three dimensional couette flow ofa viscous incompressible electrically conducting fluid behveen hvo infinite horizontal parallel porous flat plates in presence ofa transverse magnetic field The stationary plate and the plate in uniform motion are, respectively, subjected to a transverse sinusoidal injection and uniform suction ofthe fluid The governing equations ofthe flowfield are solved by using series expansion method and the expressions for the velocity field the temperature field skin friction and heatflicx in terms of Nusseh number are obtained The effecis of the flow parameters on the velocity temperature, skin friction and heatflux have been studied and analyzed with the help of figures and tables It is observed lhat the magnetic parameter (M) has a retarding effect on the main velocity (it) and an accelerating effect on the cross velocity (w,) ofthe fiow field The suction parameter (RJ has a retarding effect on the main velocity as well as on the temperature field. The Prandtl number (P,) reduces the temperature of the flow fleld and increases the rale of heat transfer at lhe wah (NJ. The effect of suction parameter is to reduce the x-component of skin friction and to enhance the magnitude of z-componem ofthe skin friction at Ihe wall. The problem is very much significam m view ofits several engineering, geophysical and industrial applications.

Keywords: Hydromagnetic, couette f l o w , heat transfer, three dimensions

N O M E N C L A T U R E

Bo uniform magnetic field / distance between the plates

M magnetic parameter N,i Nusselt number Pr Prandtl number p* pressure Re Reynold's Number

r * temperature

T dimensionless temperature To temperature at the lower plate

r„, temperature at the upper plate

U uniform velocity o f the upper plate u,v,w dimensionless velocity components

1. Introduction

;/*,v* ic* velocity components a l o n g x * y * , z * direction respectively

K constant suction velocity v*(z*) sinusoidal injection velocity

x,y,z dimensionless Cartesian coordinates x*,y*,z* Cartesian coordinates

Greek Symbol

a thermal diffusivity

E a small positive constant ( 0 < E « 1 )

p density

V kinemafic viscosity

CJ electrical conductivity

The problem o f hydromagnetic couette flow w i t h heat transfer has been a subject o f interest o f many researchers because o f its possible applications in many branches o f science and technology. Channel flows have several engineering and geophysical applications, such as, in the field o f chemical engineering f o r filtration and purification processes; in the field o f agricultural engineering to study the underground water resources; in petroleum industry to study the movement o f natural gas, o i l and water through the oil channels and reservoirs.

S. S. Das. M. Mohanly, J. P. Panda and S. K. Sahoo/ Journal of Naval Arclüteclure and Marine Engineermg 5(2008) 1-10

In V i e w o f these applications a series o f investigations have been made by different scholars where the medium is either bounded by horizontal or vertical surfaces. Gersten and Gross (1974) studied the flow and heat transfer along a plane wall with periodic suction. Gulab and Mishra (1977) analyzed the unsteady M H D flow o f a conducting fluid through a porous medium. Kaviany (1985) explained the laminar flow through a porous channel bounded by isothermal parallel plates. Vajravelu and Hadjinicolaou (1993) have investigated the heat transfer in a viscous fluid over a stretching sheet with viscous dissipation and intemal heat generation. Attia and Kotb (1996) explained the M H D flow between two parallel plates with heat transfer.

The unsteady hydromagnetic natural convection in a fluid saturated porous channel was studied by Chamkha (1996). Attia (1997) analyzed the transient M H D flow and heat transfer between two parallel plates with temperature dependent viscosity. Krishna et al. (2004) presented the hydromagnetic oscillatory flow o f a second order R i v l i n -Erickson fluid in a channel. Sharma and Yadav (2005) analyzed the heat transfer through three dimensional couette flow between a stationary porous plate bounded by porous medium and a moving porous plate. Sharma et al (2005) explained the steady laminar flow and heat transfer o f a non-Newtonian fluid through a straight horizontal porous channel in the presence o f heat source. Recently, Jain et al. (2006) discussed the three dimensional couette flow with transpiration cooling through a porous medium in the slip flow regime.

The proposed study considers the three dimensional couette flow o f a viscous incompressible electrically conducting fluid between two inflnite horizontal parallel porous flat plates in presence o f a transverse magnetic fleld. The stationary plate and the plate in uniform motion are, respectively, subjected to a transverse sinusoidal injection and uniform suction o f t h e fluid. The governing equations o f t h e flow field are solved by using series expansion method and the expressions f o r the velocity field, the temperature field, skin friction and heat flux in terms o f Nusselt number are obtained. The effect o f the flow parameters on the velocity field, temperature fleld, skin friction and Nusselt number have been studied and analyzed with the help o f figures and tables.

i i \ i 3o i i V T i i O y To O y /N A A v*(z*) 4\ A /fy .X*

Fig. A : Physical sketch and geometry o f the problem

2.

Formulation ofthe Problem

Consider the three dimensional couette flow o f a viscous incompressible electtically conducting fluid bounded between two infinite horizontal parallel porous plates in presence o f a uniform transverse magnedc field 5„. The physical model and geometry o f the problem is shown in Fig. A . A coordinate system is chosen with its origin at the

•S. S. Das. M. Mohanly, J. P. Panda and S. K. Salwo/ Joinml of Naval ArchitecHire and Marine Engmeering 5(2008) I- 10

lower stationary plate lying horizontally in x'-z' plane and the upper plate at a distance / f r o m it is subjected to a uniform velocity (J.

T h e / - a x i s is taken normal to the planes o f the plates. The lower and the upper plates are assumed to be at constant temperatures T„ and T„, respectively, with T,, > T„. The upper plate is subjected to a constant suction velocity V whereas the lower plate to a transverse sinusoidal injection velocity o f t h e form:

v\z')= y{l+ecos7iz'/ IJ,

where E ( « 1 ) is a very small positive constant quantity, / is taken equal to the wavelength o f t h e injection velocity. Due to this kind o f injection velocity the f l o w remains three dimensional. A l l the physical quantities involved are independent o f x for this f u l l y developed laminar f l o w . Denoting the velocity components u, v, w' in x\ y , z* directions, respectively and the temperature by T*, the problem is governed by the f o l l o w i n g equations:

dv*

BM* - + - ^ = 0, (2)

dy' dz'

. du , du

^d'u d'u

+ w

. dv'

V —- + w

dy dz

= V + • t

dw*

t

dw'

V T + W

dz* \dy*' ' dz"

t

dv* I dp*

1

dp*

-u ,

• + v +

-dy*' dz*

dy'

. dT'

' dy'

+ v

dz' pdz' [dy" dz'

5

2 • -,2 *

M' d w

+ -

oB'

_{-w ,}

•+ w

. dT'

dz'

= a

dy" dz'

+

(3) (4) (5) (6) where p is the density, a is the electrical conductivity,/j'is the pressure, v is coefficient o f t h e kinematic viscosity and a is the thermal diffusivity.

The boundary conditions o f the problem are

ll' = 0, V* = V(\+£cosm/IJ, w' = 0,T*=T* at / = 0,

ii*=U, v=V, u'*=0,

T*=T*.

a t / = / .

Introducing the following non-dimensional quantities,

* * • * * *

y z u V w p

Z U V

w

y — r ' = — ' " = — . 1' =

—

,w = —

,p=-I ,p=-I U V V -'''

_ 1 _

R

f csl

d'u d'u

Equations (2) - (6) reduce to the f o l l o w i n g forms

dv dw ^

dy dz

du du

V

- h w —

dy dz

dv dv

V \-w —

dy dz

dw dw

V

h w

pV'

T*-T*

-'o

T*-T* '

dp

dy'

dz'

_J

Ml

R

1

rcsi

dy

dz

dz R

d'v d'v

+

-dy-'

dz'

_J

^d'w d'w '

+

-dy' dz'

J

M'

w ,

R

(7)

(8)

(9) (10) (11) (12)

S. S. Das, M. Mohanty. J. P. Panda and S. K. Sahoo/Jownal of Naval Archileclwe and Marine Engmeermg 5(2008) I-W

dT dT

V + VI'

^ d^T d'T

dy

dz

_R.P

r

V

d f

+

dz-

(13)

VI

where R^^—, Reynolds number, V 2 ;2 , Magnetic parameter. (14) (15) 0 < e « 1 o f P, = — , Prandtl number. a

The corresponding boundary conditions become // = 0, v=\+scosnz, w = Q, T=QnX y = 0,

u=\, v = 1, w = 0 , r = l a t ; ' = l .

3. Method of Solution

In order to solve the problem, we assume the solutions o f t h e following form because the amplitude the permeability variation is very small:

ll (y, z) = iio(y) + E ll, (y z) + 112 (y z) +

1' (y, z) = Vo(yJ + E V| (y z) +E^V2(yz)+

It' (y, z) = wofy) + e U'l (y, z) + E' W2 (y z) +

P (y z) =Po(y) + Ep, (y z) + E^ p2(yz)+

T(y z) = To(y) + E 7 , (y, z) + E' T2 (y z) + (16)

When E =0, the problem reduces to the two dimensional free convective M H D f l o w which is governed by the following equations:

^ = 0,

d y d^Ur,

^ dun

9

dy

dy

dT.

dy

dy

= 0

The corresponding boundary conditions become

2/0=0, V o = l , ro = O a t ; ' = 0,

!/o= 1, Vo=I, ro = l a t > ' = l .

The solutions o f these equations for this two dimensional problem are

U o ( y ) =

To(y) =

,RePry _{- 1} RePr

with Vo = 1, vfo =0, Po =constant, where

1

(17) (18) (19) (20) (21) (22) (23) X, =•

_R,-^R:+4M'

When Ef^O, substituting (16) into Equations (9) - (13) and comparing the coefficients o f like powers o f e, neglecting those o f we get the f o l l o w i n g first order equations with the help o f Equation (23):

S. S. Das, M. Mohanty, J. P. Panda and S. K. Sahoo/Jowiml of Naval Archileclwe and Marine Engineering 5(2008) 1-10

dv, dw.

dy dz

= 0,

diiQ ^ dui

• +

-cy dy

du, du,

—h+-dy'

dz'

_J

M^m

R.

5vi 'df 9M', dy Re dy'

dz-dy

dz R.

dy'

dz'

:

(d^Ti d^T^

dy dy R,P,

dy'

dz'

(24) (25) (26) (27) (28)

The corresponding boundary conditions are z / i = 0 , v^=cos7TZ, u ' | = 0 , T,=OsXy=Q,

z/i=0, V|=0, u/|=0, r | = O a t ; ' = l . ₍₂₉₎

Equations (24)-(28) are the linear partial differential equations which describe the M H D three-dimensional f l o w through a porous medium. For solution we shall first consider three Equations (24), (26) and (27) being independent o f t h e main f l o w component w, and the temperature field T,. We assume v,, u', and /?, o f t h e f o l l o w i n g f o r m :

v^(y,z) = v^^(y)cosTiz, ^3^^

w,(y,z) = -v'Jy)simiz, p i )

Pi(y,z) = Pn(y)cos'KZ.,

₍₃₂₎

where the prime in V,', { y ) denotes the differentiation with respect to y. Expressions for v,(y, z) and w^(y. z) have been chosen so that the equation o f continuity (24) is satisfied.

Substituting these expressions (30)-(32) into (26) and (27) and solving under corresponding transformed boundary conditions, we get the solutions o f v,, u ' l a n d p , as:

^Xy'^) = - ^ W + A, e + A, e + A, e ]cos nz,

A.

Pv

where

R

m, =-^ +

1

iy> 0 = — [a (^K +M'y>'yA, [TIR^ - M ^ y - ' '

TlR^Jx

cos nz

(33)

(34)

(35)

A = {n-m,ln + m^)[e"'^-''

+ e " ' i + ' ^ J + ^

+nlm^-n)y^"'^ +e"''-^\-2n{m^

- « 7 , ) [ e " ' ' + " ' 2

+1

A, = -2nm^ + n{m^ + n)e''^"'' - n{n - ,

A2 =

2 7 t w , - 7t(7t + >"'i-'^ - 7r(wi -

ny"^"-^ ,

Hydromagnelic ihree dimensional coiielle flow and heal transfer

S. S. Das, M. Mohanly. J. P. Panda and S. K. Sahoo/Journal of Naval Archileclwe and Marine Engmeermg 5(2008) 1-10

|g'"l+"'2

To solve Equations (25) and (28) for ii, and T,, we assume

u / y ) = u,,(y)cosnz,

T,(y,z) = TJy)cosnz.

Substituting the values o f w, and T, from Equations (36) and (37) into Equations (25) and (28), we get

where the primes denote the differentiation with respect to y. The corresponding boundary conditions are

y=0: «11=0, 7^1,=0,

y=\: «11=0, rn=0.

Solving Equations (38) and (39) under the boundary conditions (40) and using Equations (36) and (37), we get

M, = [ 5,e"""^' -

Bf"'' + B,e""'' + B

^e^"'^*"'^^y -

Bf"''' +

5^e("''+"'-''

- _ B,e^"''^^^>' + Bf""-""^'' ]cosnz

(36) (37) (38) (39) (40)

cos nz

where (41) (42) I7P

+ n ' ,

+ n' , A, =B,+B^-B,+B,-B., ~B,+B,

2

V

4

A, =Bf"' + Bf"'^'"' - B f " ' +nB,e""^^ - B f - ' ' + B f ' ^ ^ - B f " - \

A,=D,+D,+D,+D,, A,= Df^'-''^ + Df^'-'' + Df'-''' + D f - ' '

A

-Af

1- e Wj +«14

-A.

B,=

J 1 _g"'-l+"'4

R..A,

R.A2

RA

B = ^

7L4(e''

-e^'Xm, +n + m,)' ' nAf -e^'^m, -n + m,)

R.A.

B. =•

J

^ 1

ReA,

-

_ C H

A[e^^-e'''lm2+n + m,){m^yn-mf ' A f - e'^ -n + m, \m^-n-mf

R f

A =

A =

A^ -A,e

_AJ -

A^e

o"'b-"h 2 D2 A2R'Pr

A f ' - i X m , +R^P^-mjm, + R^.P^-m,)'^' -\lm, + R^-m,lm,+RA.-m,)'

A f P , .

Af"'-\lm,yR^P^-m,\m, y R^-mJ'

A f P ^

S. S. Das. M. Mohanly, J. P. Panda and S. K. Sahoo/Jownal of Naval Architecmre and Marine Engineering 5(2008) 1-10

A.RlP'

Substituting tlie values o f Z/Q, W,. and T, from Equations (21), (41), (22) and (42) in Equation (16), the solutions for velocity and temperature are given by

ll =•

_{+ e [ B f " ' ' - B f ' -V B}

_f

_{B f"'^'"'^' - B f " ' ' + B,e^"'^^^^'}

-B,e

(»i|-;r)>

' -B.e^"'^'''^^' +B,e^"'^-^^^' ] c o s ; r z +

- + z [ D f + D,e""'' + Df"'^"'''^'

T =

,KPry

+ D f " ' ^ ' ' ' ' ^ ' + D f " " ^ ' ' ' ' ' ^ y + D f ^ ' - ' ' ^ ' ] c o s n z + o [ s ' )

Skin Friction

The X- and z-components o f skin friction at the wall are given by

du, ^ T.. = 1 I + S ^ and T = S dy yv=o dy Jv=0 (43) (44) (45)

Rate of Heat Transfer

The rate o f heat transfer i.e. heat f l u x at the wall in terms o f Nusselt number ( V J is given by -hS f f dy f f dy y= (46)

4. Results and Discussion

The hydromagnetic three dimensional couette flow o f a viscous incompressible electrically conducting fluid between two infinite horizontal parallel porous flat plates with heat transfer has been analyzed. The governing equations o f the flow field are solved by using series expansion method and the expressions for the velocity fleld, temperature field, skin f r i c f i o n and heat flux in terms o f Nusselt number are obtained. The effect o f the flow parameters on the velocity field and temperature field have been studied and discussed w i t h the help o f velocity profiles shown in Figs. 1-3 and temperature profiles shown in Figs. 4-5 and the effects o f t h e flow parameters on the skin friction and heat flux have been discussed with the help o f Tables 1 and 2 respecdvely.

4.1. Main velocity field

The major change in the main velocity {u) o f the flow fleld is due to the variadon o f magnetic parameter (M) and sucdon / injecdon parameter (R^). The magnetic parameter affects the main velocity o f the flow field to a greater extent than the suction / injecdon parameter. The effects o f these parameters have been presented in Figs. 1 and 2 respectively.

4.1.1. Effect of suction / injection parameter (/?e)

In Fig. 1, we present the variation in the main velocity o f the flow field due to the change o f t h e suction / injection parameter keeping other parameters o f the flow field constant. It is observed that the suction / injection parameter retards the velocity o f t h e flow fleld at all points. As the suction / injection o f t h e fluid through the plate increases, the plate is cooled down and i n consequence o f which the viscosity o f t h e flowing fluid increases. Therefore, there is a gradual decrease in velocity o f the fluid as Re increases. Further, the velocity increases slowly from zero to its

S. S. Das, M. Mohanly, J. P. PandamidS. K. Sahoo/ Jourml of Naval Architeclw-e and Marme Engmeermg 5(2008) 1-lQ

maximum value as we proceed f r o m the inlet section. But in absence o f suction / injection (^e=0), there is a rapid mcrease m velocity and the velocity is proportional to the distance f r o m the inlet section.

Fig. 1: Velocity Profile against y for different values o f Fig. 2: Velocity Profiie against y for different values o f

Re w i t h z=0, e=0.02, M = l M with z=0, 8=0.02, R,=0.2

4.1.2. Effect of magnetic parameter (M)

Fig. 2 depicts the effect o f the magnetic parameter on the main velocity o f the f l o w field. The curve with M=0 corresponds to the f l o w in absence o f magnetic field. The main velocity is observed to increase slowly f r o m zero to its maximum value as we proceed from the inlet section. But in absence o f magnetic field (A/=0), there is a uniform variation in the velocity o f t h e f l o w field. Comparing the curves o f Fig. 2, it is observed that the magnetic parameter has a retarding effect on the main velocity o f the f l o w field due to the action o f Lorentz force on the f l o w field Further, comparing the curves o f Figs. 1 and 2 it is observed that the magnetic parameter has a very dominant effect on the main velocity field over the sucdon / injection parameter.

4.2. Cross flow velocity field

The variation in the magnitude o f the cross flow velocity (U'l) o f the flow fleld is shown in Fig. 3 for three different values o f the magnetic parameter {M = 3, 5, 10). The magnetic parameter has an accelerating effect on the cross velocity o f the flow fleld near the lower plate. It is further observed that the cross velocity at flrst increases sharply to a peak value and then decreases to zero.

4.3. Temperature field

The temperature o f the flow field is affected by the variation o f Prandtl number and the suction / injection parameter. These variations are shown in Figs. 4 and 5 respecdvely. The suction / injection parameter affects the temperature field to a greater extent than the Prandtl number.

0.14

Fig. 3: Cross flow velocity profile against y for

different values o f M with z=0.5, 8 = 0 . 0 2 , Re=0.2

Hydromagnelic three dimensional couette flow and heat transfer

Das, M. Mohanty, J. P. Panda and S. K. Sahoo/ Jowval of Naval AixMleclure wid Marine Engmeeriiig 5(2008) 1-10

4.3.1. Effect of Prandtl number ( F , )

In Fig. 4, we discuss tiie effect o f Prandti number (P,) on the temperature o f the f l o w field. Fig. 4 is a plot o f temperature against the non-dimensional distance for three different values o f / ' r (=0.71, 1, 2). A comparison o f t h e curves o f t h e said figure shows that the Prandtl number decreases the temperature at all points o f t h e f l o w field. With the increase o f Prandtl number, the thermal conduction in the f l o w field is lowered and the viscosity o f the flowing fluid becomes higher. Consequenfly, the molecular motion o f the fluid elements is lowered down and therefore, the flow fleld suffers a decrease in temperature at all points as we increase P^.

4.3.2. Effect of suction / injection parameter (R^)

The effect o f suction / injection parameter on the temperature o f t h e flow field is shown in Fig. 5. The temperature o f t h e flow field is found to decrease in presence o f growing suction / injection. The temperature profile becomes very much linear in absence o f suction / injection (R=Q). In presence o f higher suction / injection more amount o f fluid IS pushed mto the flow field through the plate due to which the flow field suffers a decrease in temperature o f the flow fleld at all points.

0.8 -I 0.6 T 0.4 0.2 - ^ E e = l --*-R«=1.5 -•-E«=2.5 0 0.2 0.4 0.6 0.8 1 S.

Fig. 4: Temperature proflle against y for different Fig. 5: Temperature profile against y f o r different

values o f Pr with z = 0 , e=0.02, M = l , Re=0.5 values o f with z = 0 , 8=0.02, M = l , P^ = 0 . 7 1

4.4. Skin friction

The skin friction at the wall for different values o f suction / injection parameter (R^ has been entered in Table 1. The sucfion / injection parameter reduces the skin friction at the wall in x-direction while it enhances the magnitude o f z-component o f the skin friction at the w a l l .

Table 1: Values o f skin friction at the wall for different

values o f suction / injection parameter (R^

Re _Tz

0 0.8509 -0.0773

0.01 0.8467 -0.0817

0.2 0.7616 -0.7228

0.5 0.6785 1.3491

Table 2: Values o f rate o f heat transfer at the wall

for different values o f Prandtl number (P^)

0.71 -1.5030

1 -0.7494

2 0.0936

7 0.1437

4.5. Rate of heat transfer

The rate o f heat transfer in terms o f Nusselt number ( V J for different values o f t h e Prandtl number (P,) is presented in Table 2. The Prandtl number (P,) is found to enhance the rate o f heat transfer at the wall. It is interesting to

I S. Dns, M. Mohanty, J. P. Panda and S. K. Sahoo/Journa/ of Naval Archileclwe and Marine Engmeering 5(2008) l-IO

observe that for lower value o f Pr(<\), the rate o f heat transfer assumes negative values while f o r higher values (P, >1), it takes positive values.

5. Conclusion

The present analysis brings out the following interesting results o f physical interest on the velocity and temperature o f the flow field:

i) The magnetic parameter (M) retards the main velocity (M) at all points o f t h e flow field due to the magnedc pull o f t h e Lorentz force acting on the flow field and accelerates the cross velocity (w,) o f t h e flow field. ii) The suction / injecdon parameter (R^) decelerates the main velocity o f the flow field while no appreciable

effect is observed for cross velocity o f the flow fleld.

iii) The Prandtl number (P,) reduces the temperature o f t h e flow field at all points.

iv) The sucdon / injection parameter (R^) diminishes the temperature o f t h e flow fleld at all points.

v) The suction / injecdon parameter reduces the x-component o f skin friction and enhances the magnitude o f z-component o f the skin friction at the wall.

vi) The rate o f heat transfer at the wall (N^) increases with the increase o f the Prandtl number (P,) o f the flow field.

References

Atda, H . A . (1997): Transient M H D Flow and Heat Transfer between Two Parallel Plates with Temperature Dependent Viscosity, Mech. Res. Commun. 26, 115-121. doi: 10.1016/80093-6413(98)00108-6

Attia, H . A . and Kotb, N . A . (1996): M H D Flow Between Two Parallel Plates W i t h Heat Transfer, Acta Mech 117 215-220. doi:lQ.1007/BF01181049

Chamkha, A . J. (1996): Unsteady Hydromagnetic Natural Convection in a Fluid Saturated Porous Medium Channel Advances Filtra. Sep. Tech. 10, 369-375.

Gersten, K . and Gross, J. F. (1974): Flow and Heat Transfer Along a Plane Wall W i t h Periodic Suction, Z. Angew Math. Phys. 25 (3), 399-408. doi:10.1007/BF01594956

Gulab, R. and Mishra, R. 8. (1977): Unsteady Flow Through Magnetohydrodynamic Porous Media, Ind. J Pure Appl. Math. 8, 637-642.

Kaviany, M . (1985): Laminar Flow Through a Porous Channel Bounded by Isothermal Parallel Plates Int J Heat Mass Transfer 28, 851-858. doi: 10.1016/0017-9310(85)90234-0

Krishna, D . V . , Rao, P. N . and Sulochana, (2004): P. Hydro Magnetic Oscillatory Flow o f a Second Order R i v l i n -Erickson Fluid in a Channel, B u l l . Pure A p p l . Sci. E 23(2), 291-303.

Jain, N . C , Gupta, P. and Sharma, B . (2006): Three Dimensional Couette Flow W i t h Transpiradon Cooling Through Porous Medium in Slip Flow Regime, A M S E J. M o d . Meas. Cont. B 75 (5), 33-52.

Sharma, P. R., Gaur, Y . N . and Sharma, R. P. (2005): Steady Laminar Flow A n d Heat Transfer o f a Non-Newtonian Fluid Through a Straight Horizontal Porous Channel in The Presence o f Heat Source, Ind. J. Theo. Phys. 53(1),

37-Sharma, P. R. and Yadav, G. R. (2005): Heat Transfer Through Three Dimensional Couette Flow Between a Stationary Porous Plate Bounded by Porous Medium and a Moving Porous Plate, Ultra Sci. Phys Sci 17(3M) 351¬

360.

J • y J,

Vajravelu, K . and Hadjinicolaou, A . (1993): Heat Transfer in a Viscous Fluid over a Stretching Sheet With Viscous Dissipation and Internal Heat Generation, Int. Commun. Heat Mass Transfer 20, 417-430 doi: 10.1016/0735-1933(93)90026-R

'purnaTof Naval Architecture and Marine E^^^

June, 2008

http://jname.8m.net

DOI: 10.3329/jname,v5il,1868

SEPARATION POINTS O F AAAGNETO-HYDRODYNAMIC

BOUNDARY LAYER F L O W ALONG A V E R T I C A L PLATE WITH

EXPONENTIALLY DECREASING F R E E STREAM V E L O C I T Y

M. A. Alim', M. M. Rahman^ and M. M. Karim^

'Department of Mathematics, Bangladesh University of Engineering and Teclmology Dhaka-1000, Bangladesh e-maih maalitn@math.buet.ac.bd

Postgraduate Student, Department of Naval Architecture and Ocean Engineering, Graduate School of Engineering Osaka University, Osaka, Japan, e-mail: rahman@naoe.eng.osaka-u.ac.jp

^Department of Naval Architecture and Marine Engineering, Bangladesh University of Engineering and Technology Dhaka-1000, Bangladesh, e-mail: minkarim@name.buet.ac.bd

Abstract:

The points of separation of iiiagneto-hydrodynamic mixed convection boundary layer flow along a vertical plate have been investigated The fi-ee stream velocity is considered decreasing exponentially in the stream wise direction. The governing boundaiy layer equations are transformed into a non-dimensional form and Ihe resulting nonlinear system of partial differential equations are reduced to local non-similar boimdaty layer equations which are solved numerically by implicit finite difference method known as Keller box scheme. Here we have focused our attention lo flnd Ihe effecis of suction, magnetic field and olher relevant physical parameters on Ihe position of boimdaty layer separation. The numerical results are expressed in terms of local shear stress showing the effects of suction, buoyancy Prandh number and magnetic field on the shear stress as well as on Ihe points of separation.

Keywords: Separation points, inagneto-hydrodynamic, mixed convection, boundaiy layer, suction, finite

difference method, Keller box scheme.

N O M E N C L A T U R E Greek symbols

Bo magnetic field strength a suction parameter

g acceleration due to gravity _{P volumetric coefficient o f thermal expansion} k coefficient o f thermal diffusivity _{yo transpiration (suction) velocity}

h'l Magneto-hydrodynamic parameter _{>i similarity variable}

Pr Prandtl number e _{dimensionless temperature function}

Re Reynolds number V coefficient of viscosity

T temperature o f the fluid _{a scaled stream wise coordinate}

T„ temperature o f the heated surface _{p density ofthe fluid}

To. temperature o f the ambient fluid a electric conductivity

u,v velocity along the x & y direction _{stream function}

X _{stream wise coordinate measuring distance}

along the surface

y stream wise coordinate measuring distance normal to the surface

1. Introduction

M. A. Alim. M. M. Rahman and M. M. Karim/ Joiimu! of NmmI Arclülcchn-e ami Marine Engmeermg 5(2008) II-IS

near the point at which the sicin friction vanishes. The determination of the separation point in boundary layer flow has been the subject of many investigations over the past few decades. The usual procedure is to apply numerical methods to the governing partial differential equation, compute the full-field solution, and thereby obtain the stream wise station at which the wall shear stress becomes zero. Brown & Stewartson (1969) and Laura et al. (1990) investigated the points of separation in their works. Curie (1981) investigated lhe development of a steady two-dimensional incompressible laminar boundary layer when the external flow velocity is given by = U^l-se^ ) , Q<s <\. When s is very small and ^ is not too large then se^ is also

small and ii, is approximately constant and hence the flow is just a perturbation o f t h e Blasius flow. However, as £ increases the effect of se"- is felt more and more. As £ approaches log(£•-'), u, falls rapidly causing the boundary layer to separate. This type of problem was analyzed by Curie (1981) and he used the equation ^ ' m +-^^'7'; '^'^^{p^P'nn " ' ^ / ; - ^ ^ ' , ) = - Ê K ^ ) . This is obtained from two-dimensional boundary layer equation after introducing the variable ^ _

2v X

1/2

y and the stream function

Curie solved the above equation by writing F{£,ri) = F„{n) + ee-F,(£,ij) + e^e'^F,(£,n) + ••• where Fo(77) satisfies the Blasius equation. For small £, F,(£;;) is further expanded as

,n) = fAn) + ^ f f n ) + £' f i n ) +

-and the results were used to calculate the skin friction -and displacement thickness. For large £ inner -and outer asymptotic expansions were determined and matched. The skin friction is expressed in a power series as:

Chiam (1998) numerically solved quite similar type of problem for no suction and for uniform suction at the plate.

A study of the flow of electrically conducting fluid in presence of magnetic field is also important from the technical point of view and such types of problems have received much attention by many researchers. The specific problem selected for study is the flow and heat transfer in an electrically conducting fluid adjacent to the surface. The interaction of the magnetic field and the moving electric charge carried by the flowing fluid induces a force, which tends to oppose the fluid motion. And near the leading edge the velocity is very small so that the magnetic force, which is proportional to the magnitude of the longitudinal velocity and acts in the opposite direction, is also very small. Consequently, the influence ofthe magnetic field on the boundaiy layer is exerted only through induced forces within the boundary layer itself, with no additional effects arising from the free stream pressure gradient. Magneto-hydrodynamic was originally applied to astrophysical and geophysical problems, where it is still very important, but more recently to the problem of fusion power, where the application is the creation and containment of hot plasmas by electromagnetic forces, since material walls would be destroyed. Astrophysical problems include solar structure, especially in the outer layers, the solar wind bathing the earth and other planets, and interstellar magnetic fields. The primaiy geophysical problem is planetaiy magnetism, produced by currents deep in the planet, a problem that has not been solved to any degree of satisfaction.

The hydrodynamic behavior of boundary layers along a flat plate in the presence of a constant transverse magnetic field was first analyzed by Rossow (1958), who assumed that magnetic Reynolds number was so small that the induced magnetic field could be ignored. M H D free convection flow of visco-elastic fluid past an infinite porous plate was investigated by Chowdhuiy and Islam (2000). Raptis and Kafoussias (1982) investigated the problem of magneto-hydrodynamic free convection flow and mass transfer through a porous medium bounded by an infinite vertical porous plate with constant heat fiux. Moreover, Hossain et al. (1997) discussed both forced and free convection boundary layer flow of an electrically conducting fluid in presence of magnetic field. The magneto-hydrodynamic boundary layer flow and heat transfer on a continuous moving wavy surface was investigated by Hossain & Pop (1996).

The present work considered the magneto-hydrodynamic boundaiy layer flow along a vertical plate, with exponentially decreasing free stream velocity and has solved the problem numerically using the implicit finite

^- Rfl'mun and M. M. Karim .loumul vf Naval ArchilecUire and Marine Engmeermg 5(2008) 11-18

difference metiiod togetlier with the Keller-Box scheme (1978). This method is described in details by Cebeci & Bradshaw (1984) and used by Hossain et al. (1997, 1998 and 1999). The purpose of this paper is to study the MHD boundaiy layer flow and to show the effect of MHD on the points of separation. The numerical results are expressed in terms of local shear stress showing the effects of suction, buoyancy, Prandlt number and magneüc field on the shear stress as well as on the points of separation and some results are compared with those of Chiam (1998).

2. Formulation of the problem

A steady two-dimensional incompressible boundary layer flow along a vertical plate with external velocity is considered. It is assumed that the surface temperature ofthe plate is and the temperature ofthe ambient fluid

IS r^, where r„. >T^^ The physical coordinates (x. y) are chosen such thatx is measured from the leading edge in

the stream wise direction and y is measured normal to the surface of the plate. The flow configuration and the coordinate system are shown in Fig.1.

X Ba Thermal boundary layer Momentum boundary layer ^ y Ue(x)

Fig. 1: The co-ordinate system and the physical model

du d\>

= 0 1 = 0 dx,

dy

= 0 du

d

u u -t- V : = U

d

X

dy

e

dT

dr

d'T

u — -I-v — - a

dy'

dx

_dy

dy'

(1) dx d y ' " " " ^ ^ p (2) (3) where, u and v are velocity components in the x and y directions respectively, v is the coefficient of viscosity, p is the density of ambient fluid, g is the acceleration due to gravity, a is the electric conductivity, 5„ is the magnetic field strength, fi is the volumetric coefficient of thermal expansion, K is the coefficient of thermal conductivity and Tis the temperature of the fluid. The boundary conditions for the present problem are'

y = ^: u = 0,v = Vo, T = T,„ ,

y ^ ^ - . u - ^ u X x ) , T^T^ " . = " o ( l - ^ e - ) , 0 < . < l (4)

A7. A. Alim. h'l. hi Rahman and M. M. Karim .hninml nf Naval Archileclwe ami Murine Engmeermg 5(2008) 11-18

In equation (4), v„ represents tlie suction velocity of fluid through the surface of the plate. Here, for suction or withdrawal of fluid the transpiration velocity v„ is negative whereas for blowing or injection óf fluid v„ > 0. Near the leading edge, the boundary layer is very much like that of the free convection boundary layer in the absence of suction. Therefore the following group of transformations may be introduced:

T] = y

r

\i/2

_T(x,y)-T^ T -T

where the stream function ij/(x, y) satisfies the mass conservation equation with

du/ dw

U= ,v = —

d y dx

Equations (2) and (3) can be transformed into

1 0 + i ! L t l . ö = / / - ^ - ^ ^

Gr

Re^ •Jn Jnn Pr (5) (6) (7) (8) (9) (10) where m is a dimensionless pressure gradient parameter and M is magnetic parameter defined respectively by

m =

^ and

M ^ ^ ^ ^

dx

pu (11)

And Pr - i/A:is the Prandtl number. Equations (9)-(l 1) are the local non-similarity equations governing the flow under consideration. The boundary condition (4) takes the form

= 0, 6» = 1 at = 0 = 1 , (9 = 0 as 7 - » 0 0 (12) which leads to ( 1 - g l + + (13) where the suction parameter « = -2v„

(14) Chiam (1998) assumed

r _ { \ - s e f ^ ^

\ + {\+^)se^

and obtained the solution numerically.

The solutions of Equations (9)-(l 1) enable us to compute the local shear stress / „ ( < f , 0 ) and the local rate o f heat transfer 9,, 0) from the wall values of

f . f , r ] ) and 9 f , r i ) ^,5^ In this paper only the effect of Fon shear stress as well as separation point is considered.

I. A. Alim. M. M. Rahman and M. U Karim Journal of Naval Archileclwe ami Marine Engmeermg 5(2008) I l-IH

3. Method of solution

The numerical method used here is finite difference method known as Keller box Scheme (1978) which has been described in details by Cebeci and Bradshow (1984). The method has been used successfully by Hossain & Ahm (1997) and Hossain et al. (1998, 1999) and it has also been used by many authors in a wide variety of boundary layer problems. To employ the finite difference method, the system of partial differemial equations (9)-(10) are first converted into a system of first order differential equations. The discretization of momentum and energy equations carried out with respect to non-dimensional coordinates £ and rj to express the equations m fmUe difference form by approximating the functions and their derivatives in terms o f t h e central differences in both coordinate directions. Denoting the mesh points in the ^,;7-plane by £i and rij where ; = 1, 2, . . . , M mé

j=\,2,...,N, central difference approximations are made, such that those equations involving £ explicidy are

centered at , /;,.;,) and the remainder at (£ , qj-ri), where ^ , = L(,^^ + ) etc. The above central difference approximations reduces the system of first order differential equations to a set of non-linear difference equations for the unknown at £, in terms of their values at 4 ; . The resulting set of nonlinear difference equations are solved by using the Newton's quasi-linearization method. The Jacobian matrix has a block-tridiagonal structure and the difference equations are solved using a block-matrix version of the Thomas algorithm; the details of the computational procedure have been discussed further in the book by Cebecci and Bradshow (1984).

4. Results and Discussion

Equations ( 9 - 1 1 ) subject to the boundary conditions (12) are solved numerically using implicit finite difference method of Keller (1978), which is described by Cebeci and Bradshow (1984). The numerical results oif,„{£,r]) for ri = 0 denoted by / , , ( ^ , 0 ) are obtained for representative values of the suction parameter

a = -Iv^yjL/u^y between 0.0 and 1.0 for different values of Prandtl number Pr, the buoyancy parameter

Gl•/Re^ magneto-hydrodynamic parameter M and for several values of s. Fig.2 shows the results of separation points for Pr=1.0, M= 0.0, e=0.1 and Gr/Re' = 0.0 considering the effect of r ( E q " 14) and compares with the results of Chiam (1998). The effect of r i s clearly demonstrated in the figure, AX a = 0.0 the effect of r i s insignificant thus the present soludon gives the identical result as Chiam (1998), As the value of a increases, the effect of r becomes prominent as shown in the figure. It is well known that sucdon tends to delay separation. This can readily be seen from Table 1 where £, is shown for three representadve values ofthe suction parameter

a for each s. The percentage shift of to higher values is more pronounced for larger values of s.

Fig. 2: Plot o f t h e wall values of ij) Versus £ for Fig. 3: Plot o f t h e wall values o f / , / ^ , / / ; Versus

different values of the sucdon parameter a when ,f for different values of the sucdon parameter a Pi-=1,0, Gr/Re^=0.0, M=0.0 and £-=0,1 when Pi= 1,0, Gr/Re^ =0,0, A/=0,5, and s=0.] Figs, 3-5 show plots o f t h e wall values o{f„^(^.7j) against ^, the stream wise coordinate. The behavior o f t h e curves can be understood on the basis of the interplay of two effects: the adverse pressure gradient tending to bring about separation and suction tending to delay it. Fig, 3 show the effect of the suction parameter a for Pr=I,0, M=0.5, Gr/Re' =0,0 when g = 0,1, The effect of a is also shown for s = 0,05 and 0.01 for the same