UNSTEADY MHD FLOW WITH HEAT TRANSFERIN A DIVERGING CHANNEL

P. Y. MHONE, O. D. MAKINDE1

Applied Mathematics Department, University of Limpopo, Private Bag X1106,Sovenga 0727, South Africa

Received April 12, 2006

In this paper, we investigate the combined effects of unsteadiness and atransversely imposed magnetic field on viscous incompressible fluid flow and heattransfer in a slowly varying exponentially diverging symmetrical channel. The nonlineargoverning equations are obtained and solved analytically using a petubation technique.Graphical results are presented and discussed quantitatively for various flow characte-ristics like axial velocity, temperature, axial pressure gradient, wall shear stress andNusselt number.

Key words: Flow unsteadiness, diverging channel, magnetic field, conductingfluid.

1. INTRODUCTION

MHD flow in diverging channels/ducts has important applications in MHDpumps and generators, liquid metal magnetohydrodynamics and physiologicalfluid flow. In liquid metal magnetohydrodynamics, magnetic fields are used tolevitate samples of liquid metal, to control their shape and to induce internal stirringfor the purpose of homogenisation of the final product [6]. Magnetic fields arealso used to control the natural convection of semiconductor melts such as siliconor gallium arsenide to improve crystal quality [7]. In physiological fluid flow,Barnothy has reported experiments [4] where the heart rate decreased by exposingbiological systems to an external magnetic field. In this line of application ismagnetic resonance imaging (MRI), a technique for obtaining high resolutionimages of various organs within the human body in the presence of a magnetic field.

In 1937, Hartmann and Lazarus [11] studied the influence of a transverseuniform magnetic field on the flow of a viscous incompressible electricallyconducting fluid between two infinite parallel stationary and insulating plates.Since then, this pioneering work in MHD flow has received much attention andhas been extended in numerous ways. Closed form solutions for the velocityfields under different physical effects have been studied in [8] and [16]. A heat

1 Corresponding author: makindeo@ul.ac.za

Rom. Journ. Phys., Vol. 51, Nos. 9–10, P. 967–979, Bucharest, 2006

968 P. Y. Mhone, O. D. Makinde 2

transfer problem has been studied in [16] and [2]. Hall effects, in the casewhen the magnetic field is strong, have been incorporated in [16]. The effectof variable properties (temperature dependent viscosity and thermal conductivity)has been studied in [3]. An analysis of steady incompressible Hartmann flowthrough ducts of varying cross-section has been done in [12]; Pure oscillatoryMHD flow of blood through channels of varying cross-section has beenstudied in [15];

In the present work, unsteady Hartmann flow is studied in a divergingchannel of slowly varying cross section. The unsteadiness is caused by anoscillating flux with constant pulse m. The concept of a slowly varying flowforms the basis of a large class of fluid flow problems [6]. It is thereforeinteresting to study the effect of a magnetic field to the flow in this diverginggeometry.

2. MATHEMATICAL FORMULATION

We consider the effect of a uniform transverse magnetic field B onunsteady two dimensional electric conducting fluid flow whose velocities aregiven by

( ) ( )u x y t v x y t= , , + , ,q i j (1)

through a symmetric diverging channel { ( ) ( )}D x b x y b x:− ∞ < < +∞, − < <where (x, y) are Cartesian co-ordinates such that 0x is the axis of symmetry ofthe 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 Tw. We assume thatthe fluid is incompressible with uniform properties i.e. density ρ, kinematicviscocity 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

( )

0(1 )

b xi tudy Q me δ= + .∫ (2)

A uniform magnetic force is applied in the y-direction. A very smallmagnetic Reynolds number is assumed and therefore the induced magnetic fieldis neglected. Two key physical effects occur when the fluid moves into themagnetic field. The first one is that an electric field E is induced in the flow. Wewill assume that there is no excess charge density and therefore that 0.∇ ⋅ =ENeglecting the induced magnetic field implies that 0∇× =E and therefore theinduced electric field is negligible. The second key effect is dynamical i.e. aLorentz force ( ),×J B where J is the current density acts on the fluid andmodifies its motion. Therefore there is a transfer of energy ( )⋅J E from theelectromagnetic field to the fluid. In this study, relativistic effects are neglectedand J is given by Ohm’s law:

( )= σ × .J q B (3)

Taking into account the Lorentz force and the energy transfer, theequations governing the unsteady flow of a Newtonian fluid are

0u vx y∂ ∂+ = ,∂ ∂

(4)

2021 Bu u u Pu v u u

t x y xσ∂ ∂ ∂ ∂+ + = − + ν∇ − ,

∂ ∂ ∂ ρ ∂ ρ(5)

21v v v Pu v vt x y y∂ ∂ ∂ ∂+ + = − + ν∇ ,∂ ∂ ∂ ρ ∂

(6)

( )2 2 22

02 212 ,2p p p

T T Tu vt x y

Bk u v v uT uc c x y x y c

∂ ∂ ∂+ + =∂ ∂ ∂

⎛ ⎞ σ⎛ ⎞ ⎛ ⎞ν ∂ ∂ ∂ ∂= ∇ + + + + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ρ ∂ ∂ ∂ ∂ ρ⎝ ⎠ ⎝ ⎠⎝ ⎠

(7)

with conditions:

Symmetry 0 0 0 on 0u Tv yy y∂ ∂: = , = , = = .∂ ∂

(8)

Non slip 0 on ( )wdbu v T T y b xdx

− : + = , = = (9)

It is convenient to introduce the stream function Ψ defined by u y= ∂Ψ/∂ ,v x= −∂Ψ/∂ and after eliminating the pressure term from equations (5) and (6),we obtain

970 P. Y. Mhone, O. D. Makinde 4

2−∇ Ψ = ω, (10)

2 2022

( )( )

Bt y x y

σ∂ Ψ,ω∂ω ∂ Ψ+ = ν∇ ω+ ,∂ ∂ , ρ ∂

(11)

2 2 222 2 2 022 2

( ) 4( ) p p p p

BTT k Tt y x c c c x y c yy x

σ⎛ ⎞∂ ,Ψ ⎛ ⎞ ⎛ ⎞∂ ν ∂ Ψ ∂ Ψ ν ∂ Ψ ∂Ψ+ = ∇ + − + + .⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ , ρ ∂ ∂ ρ ∂∂ ∂ ⎝ ⎠ ⎝ ⎠⎝ ⎠(12)

The corresponding boundary conditions are

2

2 0 0 0 on 0T yyy

∂ Ψ ∂= , Ψ = , = =∂∂

(13)

0 (1 ) on ( )i tdb T T Q me y b xy x dx

δω

∂Ψ ∂Ψ− = , = , Ψ = + = .∂ ∂

(14)

The function b(x) is assumed to depend upon a small parameter ε such that

00

0( ) (0 1)

axb x a Sa L

⎛ ⎞ε, ε = < ε = ,⎜ ⎟⎝ ⎠

(15)

where a0 is the constant characteristic half width of the channel, L is the constantcharacteristic length of the channel and S is the function describing the channelwall divergence geometry. This assumption helps us to simplify the problem bywritting the equations in non-dimensinal form. To achieve this, we define T0 thereference temperature and the following non-dimensional quantities:

2 20 0 0

0 0 020

a T T ayxx y t tQ a a Q T T

aQRe P P

Q

ω

ω − δε Ψ′ ′ ′ ′ ′ω = , = , = , Ψ = , = δ , θ = , α = ,− ν

εε ′= , = .ν ρν

(16)

In terms of these non-dimensional quantities and after neglecting terms oforder ε2 and higher order as well as the primes symbol for clarity, we obtain

2

2y∂ Ψ− = ω∂

(17)

2 2

2 2( )Re( )t y xy y

∂ Ψ,ω⎛ ⎞∂ ω ∂ω ∂ Ψ−α = −Ω .⎜ ⎟∂ ∂ ,∂ ∂⎝ ⎠(18)

2 22 2 2

2 2 2( )( )r r c c rP Re P E E P

t y xy y y

⎛ ⎞⎛ ⎞ ⎛ ⎞∂ θ,Ψ∂ θ ∂θ ∂ Ψ ∂ Ψ⎜ ⎟− α = −Ω −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ,∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠(19)

5 Unsteady MHD flow with heat transfer 971

where 2 20 0( )c pE Q a T T cω= / − is the Eckert number, r pP c k= ρ ν/ the Prandtl

number, α the Womersley number, Re the effective flow Reynolds number and20 0B a L QΩ = σ /ρ is the magnetic field intensity parameter. The boundary

conditions are

2

2 0 0 0 on 0yyy

∂ Ψ ∂θ= , Ψ = , = =∂∂

(20)

0 1 1 on ( )itme y S xy

∂Ψ = , θ = , Ψ = + = .∂

(21)

The problem is now completely specified by the system of equations (17)–(21).

3. METHOD OF SOLUTION

Due to the nonlinear nature of the equations (17)–(19), it is convinient totake a power series expansion in the effective flow Reynolds number (Re) asfollows:

0 0

0

( ) ( )

( )

j it j itjs j js j

j j

j itjs j

j

Re me Re me

Re me

∞ ∞

= =

∞

=

Ψ = Ψ + Ψ , ω = ω + ω ,

θ = θ + θ ,

∑ ∑

∑(22)

where jsΨ , jΨ , jsω , jω , jsθ and jθ are functions of ( )S x and y. It is

important to note that the real part of the equation (22) forms the solution of theproblem which is physically meaningful. Substituting equation (22) intoequations (17)–(19) and collecting terms of like order of Re and ,itme we obtainZeroth order:

20 2

1 02y∂ ω

= λ ω∂

(23)

20

02y∂ Ψ

= −ω∂

(24)

2 2 20 0 02

2 02 2 22 sc rE P

y y y∂ θ ∂ Ψ ∂ Ψ

−λ θ = −∂ ∂ ∂

(25)

202 0s

y∂ ω

=∂

(26)

972 P. Y. Mhone, O. D. Makinde 6

20

02s

sy∂ Ψ

= −ω∂

(27)

220 02

s sc rE P

yy∂ θ ∂Ψ⎛ ⎞= ⎜ ⎟∂∂ ⎝ ⎠

(28)

where 21 iλ = α and 2

2 rP iλ = α and we require that 1.rP ≠ The boundaryconditions are

2 20 0 0 0

0 02 2 0 0 0 on 0s ss y

y yy y∂ Ψ ∂ Ψ ∂θ ∂θ

= = , Ψ = Ψ = , = = = .∂ ∂∂ ∂

(29)

0 00 0 0 00 1 0 1 on ( )s

s s y S xy y

∂Ψ ∂Ψ= = , Ψ = Ψ = , θ = , θ = = .

∂ ∂(30)

First order:22

0 0 0 0 01 21 12 2

( ) ( )( ) ( )

s sy x y xy y

∂ Ψ ,ω ∂ Ψ ,ω ∂ Ψ∂ ω− λ ω = + −Ω

∂ , ∂ ,∂ ∂(31)

21

12y∂ Ψ

= −ω∂

(32)

20 0 0 0 0 01 2

2 12( ) ( )

2( ) ( )

s s sr cP E

y x y x y yy∂ Ψ ,θ ∂ Ψ ,ω ∂Ψ ∂Ψ∂ θ ⎛ ⎞− λ θ = + − Ω⎜ ⎟∂ , ∂ , ∂ ∂∂ ⎝ ⎠

–

2 2 220 1 01

2 2 2 22 s sc rE P

y y y y

⎛ ⎞∂ Ψ ∂ Ψ ∂ Ψ∂ Ψ− +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

(33)

2 21 0 0 02 2

( )( )

s s s sy xy y

∂ ω ∂ Ψ ,ω ∂ Ψ= −Ω

∂ ,∂ ∂(34)

21

12s

sy∂ Ψ

= −ω∂

(35)

22 2 21 0 0 0 102 2 2

( )2

( )s s s s ss

r c rP E Py x yy y y

∂ θ ∂ Ψ ,θ ∂ Ψ ∂ Ψ∂Ψ⎛ ⎞= −Ω −⎜ ⎟∂ , ∂∂ ∂ ∂⎝ ⎠(36)

with boundary conditions

221 11 1

0 12 2 0 0 0 on 0s ss y

y yy y∂ Ψ ∂θ∂ Ψ ∂θ

= = , Ψ = Ψ = , = = = .∂ ∂∂ ∂

(37)

111 1 1 10 0 0 on ( )s

s s y S xy y

∂Ψ∂Ψ= = , Ψ = Ψ = , θ = θ = = .

∂ ∂(38)

7 Unsteady MHD flow with heat transfer 973

and so on. Equations (23) to (38) are solved for the stream function Ψ, vorticityω and temperature distribution θ and are presented in the appendix. Solutionscorresponding to the steady problem and the non-magnetic case are obtained bytaking the limits 0m → and 0.Ω→

The shear stress at the curved wall ( )y b x= is given in [15] by

( )( )

22 2 2

2 2

2

1 4

1

db dbdx x y dxy x

dbdx

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

⎛ ⎞⎛ ⎞⎛ ⎞∂ Ψ ∂ Ψ ∂ Ψρν − − −⎜ ⎟⎜ ⎟⎜ ⎟ ∂ ∂⎝ ⎠∂ ∂⎝ ⎠⎝ ⎠τ = − .+

(39)

Scaling and ignoring terms of order ε2 and higher order in equation (39),we obtain

2

2

21 1 2

12 21 1 1

12sinh( )3 1 ( )cosh( ) sinh( ) 5 35

itx it

y

SS meRe me O Re

S S SS S

ω

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

∂ Ψτ = −μ =∂

λ λ= + + Ω − + ω +

λ λ − λ

(40)

From the axial component of the Navier-Stokes equations, we obtain thedimensionless axial pressure gradient as

2 2P Rex y y x y x y t y

⎛ ⎞∂ ∂ω ∂Ψ ∂ Ψ ∂Ψ ∂Ψ ∂ Ψ= − − + ω−Ω −α⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠(41)

On substituting the expression for Ψ, this expands to

311

31 1 1

cosh( )3 ( )cosh( ) sinh( )

itSP me O Rex S S SS

⎛ ⎞λ λ∂ = − − +⎜ ⎟∂ λ λ − λ⎝ ⎠(42)

The rate of heat transfer at the wall is given by 0 0( )Nu a H k T Tω= / − whereH k T y= − ∂ /∂ is the Fourier law of heat conduction. Non-dimensionalising, weobtain .Nu y= −∂θ/∂ Hence

21 2 1 2

3 3 2 21 1 1 1 2 2

3 6 sinh( )sinh( )( cosh( ) sinh( )) ( )cosh( )

itc r c rE P E P me S S y

Nu …S S S S S S

⎛ ⎞λ λ λ λ= − +⎜ ⎟λ λ − λ λ − λ λ⎝ ⎠

(43)

4. GRAPHICAL RESULTS AND DISCUSSION

The numerical calculations have been performed for the “exponentially”diverging geometry i.e. ( ) ,xS x e= where x is slowly varying. We choose the

974 P. Y. Mhone, O. D. Makinde 8

pulse m small so that the ensuing flow is a small oscillatory disturbance aboutthe steady flow. We have observed that for every m, the Womersley parameter αcan be varied only for a range of values, hence we set α = 0.5. The effect ofvarying the Reynolds number Re, the Eckert number Ec and the Prandtl numberPr to flow structure is well known in the literature and is not investigated here.Therefore we set Re = 0.1, Ec = 1 and Pr = 7.1. In the ensuing analysis, weassume that m and Ω can be varied while keeping α, Re, Ec and Pr fixed. Thisassumption is valid because of the physics of the problem and the range ofvalues of m and Ω that are involved.

Fig. 2 shows that the effect of increasing values of Ω on steady flow is todampen the velocity profile. This is well known for Hartmann flow. Moreoverfor a channel of varying cross section, the dampening is pronounced in the centreof the channel. This creates a stagnation point and consequently fluid is pushedto the walls of the channel thereby increasing the velocity in the boundary layer.For unsteady flow (m = 0.01), equation (45) says that axial velocity is oscillatoryabout the steady flow axial velocity. Here, Ω is coupled with the unsteady terms

Fig. 2. – Axial velocity profile ( 0 0 100 200)m = , Ω = , , .

Fig. 3. – Axial velocity profile ( 0 01 0 10 20)m = . , Ω = , , .

9 Unsteady MHD flow with heat transfer 975

in the form ( )cos( ) ( )sin( ),f t g tΩ,… + Ω,… where functions f, g are such that

f > g > 0 and both f and g are increasing in Ω. This means that increasing valuesof Ω have the effect of increasing/decreasing axial velocity when axial velocityis increasing/decreasing in time. Figs. 3 and 4 depict this situation for a smallerrange of Ω values. This suggests that in this model, increasing magnetic forceenhances unsteadiness.

The pressure gradient, which is trying to accelerate the fluid is counteractedby the magnetic drag. Hence we observe a pressure gradient drop in the centre ofthe channel (y = 0) in Fig. 5 for 4.t = π/ The drop is increasing for increasingvalues of Ω. There is no marked qualitative difference for figures of other valuesof t in the oscillation. The work being done by the pressure gradient against themagnetic tension is converted to heat. Fig. 6 shows that Ohmic dissipation at thecentre of the channel dominates viscous dissipation at the wall for all times t.Increasing values of Ω are associated with larger temperature rises. When moreand more unsteadiness is introduced into the flow by increasing values of m, thetemperature in the center of the channel decreses with increasing m but still rises

Fig. 4. – Axial velocity profile ( 0 01 0 10 20)m = . , Ω = , , .

Fig. 5. – Axial pressure gradient ( 0 01 0 2 5 5)m = . , Ω = , . , .

976 P. Y. Mhone, O. D. Makinde 10

with increasing values of Ω. This is depicted in Fig. 7. This suggests thatunsteadiness has the effect of cooling the fluid.

Fig. 8 shows that heat transfer takes place upstream from the fluid to thewall, consequently the wall is being heated. The rate of heat transfer is

Fig. 6. – Temperature profile ( 0 01 0 10 20 0 1m R= . , Ω = , , , = . ,1 7 1)c rE P= , = . .

Fig. 7. – Temperature profile ( 0 1 0 10 20 0 1m R= . , Ω = , , , = . ,1 7 1)c rE P= , = . .

Fig. 8. – Rate of heat transfer at the wall ( 0 01 0 10 20m = . , Ω = , , ,0 1 1 7 1)c rR E P= . , = , = . .

11 Unsteady MHD flow with heat transfer 977

decreasing until the adiabatic condition is attained downstream. Increasingvalues of Ω and m increase the rate of heat transfer for all times. This means thatin the presence of unsteadiness, the magnetic field causes more heat transfer atthe wall. This is anticipated from Figs. 6 and 7.

Shear stress at the wall varies with time upstream. Downstream thevariations are small as shown in Fig. 9. For steady flow, increasing values of Ωincrease shear stress and shear stress tends to a positive constant that depends onΩ downstream. For unsteady flow Figs. 10 and 11 show that increasing values ofΩ and m increase shear stress. Prandtl’s separation criterion fails for timedependent flows [10, 15]. However for fixed separation (i.e., separation at aconstant location), necessary conditions were proposed in [10]. Among them isthe requirement that

2 22 3

2 20 0( 0 ) 0 and ( 0 ) 0t dt t dt

y xy

π π∂ Ψ ∂ Ψγ, , = γ, , <∂ ∂∫ ∫ (44)

with non slip boundary conditions at the walls. From Figs. 10 and 11, weobserve that fixed separation is possible only for small values of Ω and m.

Fig. 9. – Wall shear stress 0 1 10.m = . , Ω =

Fig. 10. – Wall shear stress ( 0 01 0 100 200).m = . , Ω = , ,

978 P. Y. Mhone, O. D. Makinde 12

Fig. 11. – Wall shear stress ( 0 1 0 100 200).m = . , Ω = , ,

Acknowledgements. The authors would like to thank the Norwegian Universities Fund(NUFU) for their financial support.

APPENDIX

Equations (23) to (38) are solved to obtain the following expressions for Ψ,ω and θ:

31 1 1

31 1 1

7 5 3

7 5 3

5 32 2

15 3

cosh( ) sinh( )32 cosh( ) sinh( )2

3 3 33 356280 40 280

( )4040 20

it

x x x x

x x it

y S yy yme

S S S SS

y S y S y S ySRe

SS S S

y S y S yS me O Re

SS S

λ λ − λ⎛ ⎞Ψ = − + +⎜ ⎟λ λ − λ⎝ ⎠

⎛ −+ + − + +⎜

⎝⎞⎛ ⎞

+Ω − + − + Ψ + ,⎟⎜ ⎟⎝ ⎠ ⎠

(45)

2 5 31 1

3 2 5 31 1 1

32

13

sinh( )3 9 3 99cosh( ) sinh( ) 14020 2

3( )

102

xit

it

Syy y y yme Re

S S S SS S S S

y yme O Re

SS

⎞⎟⎟⎠

⎛⎛ ⎞λ λ ⎛ ⎞ω = + + − + +⎜ ⎟⎜⎜ ⎟λ λ − λ ⎝ ⎠⎝⎝ ⎠

⎛ ⎞+Ω − + ω + ,⎜ ⎟

⎝ ⎠

(46)

214 4

6 31 1 1

1 2 1 22 2 2 2 21 2 2 1 2 2

21 1

2 2 2 2 21 2 1 2

3 61 ( )

4 ( cosh( ) sinh( ))

sinh( )cosh( ) 2cosh( )cosh( )( )cosh( ) ( ) cosh( )

3sinh( ) 2 cosh( )( ) ( )

c r c r

r cit

E P E Py S

S S S S S

S S y S yS S

P E Sy y yme Re

λθ = − − + ⋅

λ λ − λ

⎛ λ λ λ λ⋅ + −⎜ λ − λ λ λ − λ λ⎝

⎞⎛ ⎞λ λ− + +⎟⎜ ⎟λ − λ λ − λ⎝ ⎠⎠

4

616x y

S

⎛−⎜

⎝

13 Unsteady MHD flow with heat transfer 979

2 8 2 2 2 6 8

10 4 8 1

6 4 2

8 6 2 2

6 4 22

16 4 2

9 9 3 27224 8 40 560 0

3 99 1011 5710 280 1120 560

7 21 9 31 ( )4010 40 8

r c x r c x r c x r c x

r c x r c x r c x x r c

itc r

P E S y P E S y P E S y P E S yS S S S

P E S y P E S y P E S S P ES S S S

y y yE P me O Re

S S S

⎞⎟⎟⎟⎠

− − + − +

+ − + + +

⎛ ⎞+Ω − + − + + θ + ,⎜ ⎟

⎝ ⎠

(47)

where xS dS dx= / and Ψ1, ω1, θ1 are the first order solutions.

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