UNSTEADY MHD FLOW WITH HEAT TRANSFER IN A DIVERGING CHANNEL

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1 UNSTEADY MHD FLOW WITH HEAT TRANSFER IN A DIVERGING CHANNEL P Y MHONE, O D MAKINDE 1 Applied Mathematics Department, University of Limpopo, Private Bag X116, Sovenga 77, South Africa Received April 1, 6 In this paper, we investigate the combined effects of unsteadiness and a transversely imposed magnetic field on viscous incompressible fluid flow and heat transfer in a slowly varying exponentially diverging symmetrical channel The nonlinear governing equations are obtained and solved analytically using a petubation technique Graphical results are presented and discussed quantitatively for various flow characteristics like axial velocity, temperature, axial pressure gradient, wall shear stress and Nusselt number Key words: Flow unsteadiness, diverging channel, magnetic field, conducting fluid 1 INTRODUCTION MHD flow in diverging channels/ducts has important applications in MHD pumps and generators, liquid metal magnetohydrodynamics and physiological fluid flow In liquid metal magnetohydrodynamics, magnetic fields are used to levitate samples of liquid metal, to control their shape and to induce internal stirring for the purpose of homogenisation of the final product [6] Magnetic fields are also used to control the natural convection of semiconductor melts such as silicon or gallium arsenide to improve crystal quality [7] In physiological fluid flow, Barnothy has reported experiments [4] where the heart rate decreased by exposing biological systems to an external magnetic field In this line of application is magnetic resonance imaging (MRI), a technique for obtaining high resolution images of various organs within the human body in the presence of a magnetic field In 197, Hartmann and Lazarus [11] studied the influence of a transverse uniform magnetic field on the flow of a viscous incompressible electrically conducting fluid between two infinite parallel stationary and insulating plates Since then, this pioneering work in MHD flow has received much attention and has been extended in numerous ways Closed form solutions for the velocity fields under different physical effects have been studied in [8] and [16] A heat 1 Corresponding author: makindeo@ulacza Rom Journ Phys, Vol 51, Nos 9 1, P , Bucharest, 6

2 968 P Y Mhone, O D Makinde transfer problem has been studied in [16] and [] Hall effects, in the case when the magnetic field is strong, have been incorporated in [16] The effect of variable properties (temperature dependent viscosity and thermal conductivity) has been studied in [] An analysis of steady incompressible Hartmann flow through ducts of varying cross-section has been done in [1]; Pure oscillatory MHD flow of blood through channels of varying cross-section has been studied in [15]; In the present work, unsteady Hartmann flow is studied in a diverging channel of slowly varying cross section The unsteadiness is caused by an oscillating flux with constant pulse m The concept of a slowly varying flow forms the basis of a large class of fluid flow problems [6] It is therefore interesting to study the effect of a magnetic field to the flow in this diverging geometry MATHEMATICAL FORMULATION We consider the effect of a uniform transverse magnetic field B on unsteady two dimensional electric conducting fluid flow whose velocities are given by q= ux (, yt, ) i+ vxyt (,, ) j (1) through a symmetric diverging channel { D: < x< +, b( x) < y< b( x)} where (x, y) are Cartesian co-ordinates such that x is the axis of symmetry of the channel and y =± b( x) are the rigid and impermeable walls of the channel The walls of the channel are kept at a constant temperature T w We assume that the fluid is incompressible with uniform properties ie density ρ, kinematic viscocity ν and electrical conductivity σ A volume flux with oscillating frequency δ and pulse m is prescribed as Fig 1 Problem geometry

3 Unsteady MHD flow with heat transfer 969 bx ( ) (1 it udy = Q + me δ ) () A uniform magnetic force is applied in the y-direction A very small magnetic Reynolds number is assumed and therefore the induced magnetic field is neglected Two key physical effects occur when the fluid moves into the magnetic field The first one is that an electric field E is induced in the flow We will assume that there is no excess charge density and therefore that E = Neglecting the induced magnetic field implies that E = and therefore the induced electric field is negligible The second key effect is dynamical ie a Lorentz force ( J B ), where J is the current density acts on the fluid and modifies its motion Therefore there is a transfer of energy ( J E ) from the electromagnetic field to the fluid In this study, relativistic effects are neglected and J is given by Ohm s law: J=σ ( q B ) () Taking into account the Lorentz force and the energy transfer, the equations governing the unsteady flow of a Newtonian fluid are u + v =, x (4) u u u 1 P σb + u + v = +ν u u, t x ρ x ρ v + u v + v v = 1 P +ν v, t x ρ T + u T + v T = t x ( ) k 1 u v v u σb = T + ν u , ρcp cp x x ρc p with conditions: Symmetry : u =, v=, T = on y= (5) (6) (7) (8) Non slip : u+ v db =, T = Tw on y= b( x) (9) dx It is convenient to introduce the stream function Ψ defined by u=/ y, v= / x and after eliminating the pressure term from equations (5) and (6), we obtain

4 97 P Y Mhone, O D Makinde 4 Ψ = ω, (1) ω,ω ( ) σb + =ν ω+ Ψ, t ( y, x) ρ σb T ( T,Ψ) k + = T + ν + 4ν + t ( y, x) ρcp c p x cp x ρcp The corresponding boundary conditions are =, Ψ =, T = on y = db =, T = T Q(1 meit δ ω, Ψ = + ) on y = b( x) x dx (11) (1) (1) (14) The function b(x) is assumed to depend upon a small parameter ε such that ( ) x a,ε = ε (<ε= 1), (15) bx as a L where a is the constant characteristic half width of the channel, L is the constant characteristic length of the channel and S is the function describing the channel wall divergence geometry This assumption helps us to simplify the problem by writting the equations in non-dimensinal form To achieve this, we define T the reference temperature and the following non-dimensional quantities: ω y x εx y Ψ δ t t Q a a Q Tω T ν a ε ε a T T a ω=, =, =,Ψ=, =δ,θ=,α=, Q Re =, P = P ν ρνq (16) In terms of these non-dimensional quantities and after neglecting terms of order ε and higher order as well as the primes symbol for clarity, we obtain y =ω ω,ω ( ) α ω = Re Ω t ( y, x) θ θ t ( y, x) y θ,ψ ( ) α P r = RePr ΩEc E cpr (17) (18) (19)

5 5 Unsteady MHD flow with heat transfer 971 where E c = Q / a ( Tω T ) cp is the Eckert number, P r =ρc p ν/ k the Prandtl number, α the Womersley number, Re the effective flow Reynolds number and Ω=σ BaL /ρ Q is the magnetic field intensity parameter The boundary conditions are =, Ψ =, θ = on y = () =, θ= 1, Ψ = 1 + meit on y = S( x) (1) The problem is now completely specified by the system of equations (17) (1) METHOD OF SOLUTION Due to the nonlinear nature of the equations (17) (19), it is convinient to take a power series expansion in the effective flow Reynolds number (Re) as follows: j it j it Re js me j Re js me j j= j= Re j( it js me j ) j= Ψ= ( Ψ + Ψ ), ω= ( ω + ω ), θ= θ + θ, () where Ψ js, Ψ, j ω js, ω, j θ js and θ j are functions of Sx ( ) and y It is important to note that the real part of the equation () forms the solution of the problem which is physically meaningful Substituting equation () into equations (17) (19) and collecting terms of like order of Re and me it, we obtain Zeroth order: ω =λ1ω () y = ω (4) s λ θ = EP c r θ s ω = (5) (6)

6 97 P Y Mhone, O D Makinde 6 s = ω s (7) s s EP c r θ = y (8) where λ 1 =α i and λ = Pr α i and we require that Pr 1 The boundary conditions are s s s θ θ = =, Ψ =Ψ =, = = on y = = =, Ψ =Ψ s = 1, θ =, θ s = 1 on y= S( x) s First order: 1 (,ωs) ( s,ω) 1 1 ω λ ω = + Ω ( yx, ) ( yx, ) 1 y = ω (,θ ) ( ) 1 s s,ω λ θ 1 = Pr + ΩEc s y ( yx, ) ( yx, ) 1 θ + 1s 1 s EP c r 1s ( s s) s = Ω ω,ω ( yx, ) 1s = ω 1s (9) () (1) () () (4) (5) 1 s ( s s ) s 1 = P s s r Ω EcPr θ,θ ( yx, ) with boundary conditions 1 1s θ1 θ1s 1s = =, Ψ =Ψ =, = = on y = 1 = 1s =, Ψ 1 =Ψ 1s =, θ 1 =θ 1s = on y = S( x) (6) (7) (8)

7 7 Unsteady MHD flow with heat transfer 97 and so on Equations () to (8) are solved for the stream function Ψ, vorticity ω and temperature distribution θ and are presented in the appendix Solutions corresponding to the steady problem and the non-magnetic case are obtained by taking the limits m and Ω The shear stress at the curved wall y = b( x) is given in [15] by db ( dx ) db ( ) 1 4 db ρν y x x y dx τ= 1+ dx (9) Scaling and ignoring terms of order ε and higher order in equation (9), we obtain τ ω = μ = λ 1 sinh( λ1sme ) it 1S x Re 1meit O Re λ1scosh( λ1s) sinh( λ1s) 5 5S = Ω +ω + S ( ) (4) From the axial component of the Navier-Stokes equations, we obtain the dimensionless axial pressure gradient as P = ω Re Ψ + ω Ω α Ψ x y x y x y t On substituting the expression for Ψ, this expands to (41) P λ1 cosh( λ1s) = meit + x S λ1scosh( λ1s) sinh( λ1s) O( Re) (4) The rate of heat transfer at the wall is given by Nu = ah/ k( Tω T) where H = k T/ y is the Fourier law of heat conduction Non-dimensionalising, we obtain Nu = θ/ y Hence EP 6 it c r EP c rλ1 me Sλsinh( λ1s)sinh( λy) Nu = + ( S S λ1scosh( λ1s) sinh( λ1s)) ( λ1 λ)cosh( λs) (4) 4 GRAPHICAL RESULTS AND DISCUSSION The numerical calculations have been performed for the exponentially diverging geometry ie Sx ( ) = e x, where x is slowly varying We choose the

8 974 P Y Mhone, O D Makinde 8 pulse m small so that the ensuing flow is a small oscillatory disturbance about the steady flow We have observed that for every m, the Womersley parameter α can be varied only for a range of values, hence we set α = 5 The effect of varying the Reynolds number Re, the Eckert number E c and the Prandtl number P r to flow structure is well known in the literature and is not investigated here Therefore we set Re = 1, E c = 1 and P r = 71 In the ensuing analysis, we assume that m and Ω can be varied while keeping α, Re, E c and P r fixed This assumption is valid because of the physics of the problem and the range of values of m and Ω that are involved Fig shows that the effect of increasing values of Ω on steady flow is to dampen the velocity profile This is well known for Hartmann flow Moreover for a channel of varying cross section, the dampening is pronounced in the centre of the channel This creates a stagnation point and consequently fluid is pushed to the walls of the channel thereby increasing the velocity in the boundary layer For unsteady flow (m = 1), equation (45) says that axial velocity is oscillatory about the steady flow axial velocity Here, Ω is coupled with the unsteady terms Fig Axial velocity profile ( m =, Ω=, 1, ) Fig Axial velocity profile ( m = 1, Ω=, 1, )

9 9 Unsteady MHD flow with heat transfer 975 in the form f( Ω, )cos( t) + g( Ω, )sin( t), where functions f, g are such that f > g > and both f and g are increasing in Ω This means that increasing values of Ω have the effect of increasing/decreasing axial velocity when axial velocity is increasing/decreasing in time Figs and 4 depict this situation for a smaller range of Ω values This suggests that in this model, increasing magnetic force enhances unsteadiness The pressure gradient, which is trying to accelerate the fluid is counteracted by the magnetic drag Hence we observe a pressure gradient drop in the centre of the channel (y = ) in Fig 5 for t =π/ 4 The drop is increasing for increasing values of Ω There is no marked qualitative difference for figures of other values of t in the oscillation The work being done by the pressure gradient against the magnetic tension is converted to heat Fig 6 shows that Ohmic dissipation at the centre of the channel dominates viscous dissipation at the wall for all times t Increasing values of Ω are associated with larger temperature rises When more and more unsteadiness is introduced into the flow by increasing values of m, the temperature in the center of the channel decreses with increasing m but still rises Fig 4 Axial velocity profile ( m = 1, Ω=, 1, ) Fig 5 Axial pressure gradient ( m = 1, Ω=, 5, 5)

10 976 P Y Mhone, O D Makinde 1 with increasing values of Ω This is depicted in Fig 7 This suggests that unsteadiness has the effect of cooling the fluid Fig 8 shows that heat transfer takes place upstream from the fluid to the wall, consequently the wall is being heated The rate of heat transfer is Fig 6 Temperature profile ( m = 1, Ω=, 1,, R = 1, E = 1, P = 7 1) c r Fig 7 Temperature profile ( m = 1, Ω=, 1,, R = 1, E = 1, P = 7 1) c r Fig 8 Rate of heat transfer at the wall ( m = 1, Ω=, 1,, R = 1, E = 1, P = 71) c r

11 11 Unsteady MHD flow with heat transfer 977 decreasing until the adiabatic condition is attained downstream Increasing values of Ω and m increase the rate of heat transfer for all times This means that in the presence of unsteadiness, the magnetic field causes more heat transfer at the wall This is anticipated from Figs 6 and 7 Shear stress at the wall varies with time upstream Downstream the variations are small as shown in Fig 9 For steady flow, increasing values of Ω increase shear stress and shear stress tends to a positive constant that depends on Ω downstream For unsteady flow Figs 1 and 11 show that increasing values of Ω and m increase shear stress Prandtl s separation criterion fails for time dependent flows [1, 15] However for fixed separation (ie, separation at a constant location), necessary conditions were proposed in [1] Among them is the requirement that π π tdt (44) ( γ,, ) = and ( γ,, tdt xy ) < with non slip boundary conditions at the walls From Figs 1 and 11, we observe that fixed separation is possible only for small values of Ω and m Fig 9 Wall shear stress m = 1, Ω= 1 Fig 1 Wall shear stress ( m = 1, Ω=, 1, )

12 978 P Y Mhone, O D Makinde 1 Fig 11 Wall shear stress ( m = 1, Ω=, 1, ) Acknowledgements The authors would like to thank the Norwegian Universities Fund (NUFU) for their financial support APPENDIX Equations () to (8) are solved to obtain the following expressions for Ψ, ω and θ: y y λ1ycosh( λ1s) sinh( λ1y) Ψ= meit S + 1Scosh( 1S) sinh( 1S) + S λ λ λ ys 7 5 x ysx ysx ysx + Re S7 4S5 8S 56S 5 ysx ysx y +ΩS it + +Ψ 5 1me + O( Re ), 4S S 4S (45) y λ 5 1 sinh( λ1y) it Sx 9y y 99y ω= + me Re 5 1Scosh( 1S) sinh( 1S) + + S S S S 14S + λ λ λ (46) y y +Ω it 1me + ω O( Re ) S 1S +, EP c r 4 4 6EP c rλ θ= 1 ( y S ) + 4 S6 S( λ Scosh( λ S) sinh( λ S)) Ssinh( λ1s)cosh( λy) cosh( λ1s)cosh( λy) + ( λ1 λ)cosh( λs) ( λ1 λ) cosh( λs) ysinh( λ 1y) cosh( λ1y) it PES r c + me Re ( + λ1 λ) ( λ1 λ) 6 16S 4 x y

13 1 Unsteady MHD flow with heat transfer r c x r c x r c x r c x PESy 9PESy PESy 7PESy + + 4S 8S 4S 56S PESy r c x PESy r c x PES r c x 57SPE x r c S8 8S6 11S 56S 7y6 1y4 9y 1 +ΩEP it c r 6 4 1me + + +θ + ORe ( ), 1S 4S 8S 4 (47) where Sx = ds/ dx and Ψ 1, ω 1, θ 1 are the first order solutions REFERENCES 1 R A Alper, Heat transfer in magnetohydrodynamic flow between parallel plates, Int J Heat Mass Transfer, (196), 18 H A Attia, N A Kotb, MHD flow between parallel plates with heat transfer, Acta Mech, 117 (1996), 15 H A Attia, A L Aboul-Hassan, The effect of variable propeties on the unsteady Hartmann flow with heat transfer considering the Hall effect, Appl Math Mod, 7 (), pp M F Barnothy, (ed), Biological effects of magnetic fields, vols 1 and, Plenum Press, New York, (1964,1969) 5 G K Batchelor, An Introduction to Fluid Mechanics, Cambridge University Press, G K Batchelor, H K Moffat, M G Worster, Perspectives in Fluid Mechanics, Cambridge University Press, 7 H Ben Hadid, R Touihri, D Henry, MHD damped convection under uniform magnetic fields, Adv Space Res, vol, no 8, (1998), K R Cramer, S I Pai, Magnetofluid dynamics for engineers and applied physicists, McGraw-Hill, New York, M V Dyke, Slow variations in continuum mechanics, Adv Appl Mech, 5 (1987), pp G Haller, Exact theory of unsteady separation for two dimensional flows, J Fluid Mech 51, 57, (4) 11 J Hartmann, F Lazarus, Kgl Danske Videnskab Selskab Mat-Fys Medd 15 (6,7) J C R Hunt and S Leibovich, Magnetohydrodynamic flow in channels of variable cross-section with strong transverse magnetic fields, J Fluid Mech 8 (1967), pp O D Makinde, Steady flow in a linearly diverging asymmetrical channel, Computer Assisted Mechanics and Engineering Sciences, 4 (1997), pp M J Manton, Low Reynolds number flow in slowly varying axisymmetric tubes, J Fluid Mech, 49 (1971), pp A Rao Ramachandra, K S Deshikachar, MHD oscillatory flow of blood through channels of variable cross section, Int J Engng Sci, vol 4, No 1, (1986), V M Soundalgekar, A G Uplekar, Hall effects in MHD Coutte flow with heat transfer, IEEE Trans Plasma Sci PS-14 (5) (1986)

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