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Hall Effects on the Unsteady Incompressible MHD Fluid Flow with Slip Conditions and Porous Walls

H. Zaman

Department of Mathematics, Islamia College Chartered University, Peshawar, Pakistan

Abstract: This work is concerned with the influence of Hall current on unsteady incompressible MHD fluid with slip conditions. The effects of Hall current on uniform suction or injection are also seen. As a special case of this problem for no slip condition, the effects of Hall current on Couette flow are discussed. The resulting unsteady problems for velocity are solved by means of Laplace transform, however, the inversion procedure for obtaining the solution is not a trivial matter. The characteristics of the complex transient velocity, complex overall transient velocity, complex steady state velocity are analyzed and discussed for both the cases. Graphical results for the Hall parameter reveal that it has significant influence on the real and imaginary parts of the velocity profiles.

Keywords: hall effects, unsteady, Laplace transform, transient velocity, wall slip, porous walls

Received June 28, 2013; Revised July 20, 2013; Accepted July 29, 2013

Citation: H. Zaman, Hall Effects on the Unsteady Incompressible MHD Fluid Flow with Slip Conditions and Porous Walls. Applied Mathematics and Physics, 1 (2). 31-38. January 2013. doi:10.12691/amp-1-2-3

1. Introduction

The study of Hall current is very important in the presence of a magnetic field [1] A current induced in a direction normal to the electric and magnetic fields is called Hall current. Magnetohydrodynamic (MHD) flows with Hall effect are encountered in power generators, MHD accelerators, refrigeration coils, electric transformers etc. Considerable efforts have been devoted to study the effects of Hall current in various directions. The effect of Hall current on the MHD boundary layer flow is considered by authors [2-13]. Furthermore, the heat transfer and Hall effect in the stretching flow are examined in the Refs. [14-20]. Exact solutions of Navier - Stokes equations are rare due to their inherent nonlinearity. Exact solutions are important because they serve as accuracy checks for numerical solutions. Complete integration of these equations is done by computer techniques, but the accuracy of the results can be established only by comparison with exact solutions. In the literature, there are a large number of Newtonian fluid flows for which exact solutions are possible [21,22,23,24,25] The effects of transverse magnetic field on the flow of an electrically conducting viscous fluid have been studied extensively in view of numerous applications to astrophysical, geophysical and engineering problems [26-31]. If the working fluid contains concentrated suspensions then the wall slip can occur [32]. Khaled and Vafai [23] studied the effect of the slip on Stokes and Couette flows due to an oscillating wall.

However, the literature lacks studies that take into account the possibility of fluid slippage at the walls. Applications of these problems appear in microchannels or nanochannels and in applications where a thin film of light oil is attached to the moving plates or when the surface is coated with special coating such as a thick monolayer of hydrophobic octadecyltrichlorosilane [33]. Yu and Ameel [34] imposed non-linear slip boundary conditions on flow in rectangular microchannels.

In this study, the effects of Hall current on the flow of an incompressible, unsteady, viscous, MHD fluid with slip conditions are considered. Unsteady and steady velocity profiles with mass transfer are presented and solved exactly. There is mass injection from one plate and the same amount of suction on the other plate. When the fluid motion is set up from rest, the velocity field contains transients, determined by the initial conditions which gradually disappeared in time. The effects of Hall current and time on the transient velocity and on overall transient velocity has been seen graphically for injection / suction both. The effect of slip parameter on steady state velocity for injection / suction is also shown graphically in the presence of Hall effects. As a limiting case by taking slip parameter , the effects of Hall parameter on overall transient velocity for Couette flow problem are also discussed in detail.

2. Theoretical Derivation

Consider an MHD incompressible, viscous, unsteady flow problem with Hall effects, in which there is slip between the bottom wall and fluid and also between top wall and fluid. There is mass injection velocity at the bottom wall and mass suction velocity at the top wall, correspond to injection and correspond to suction.

The governing equation for this problem can be obtained as

[picture] | (1)

where /, is the dynamic viscosity, is the kinematic viscosity, is the density of the fluid, is the electric conductivity of the fluid, is the applied magnetic field, is the MHD factor or parameter, is the Hall parameter. For the boundary conditions we consider the existence of slip between the velocity of the fluid at the walls and speed of the walls.

Figure 1. Geometry of the problem.

[picture] | (2)

[picture] | (3)

The initial condition is

[picture] | (4)

where bottom wall is located at , top wall is located at , is the slip parameter, ( gives the usual no slip condition at the wall) and is the velocity at the upper wall. Eqs. (1), (2), (3) and (4) can be non-dimensionalized by defining

Then Eqs. (1), (2), (3) and (4) become

[picture] | (5)

[picture] | (6)

[picture] | (7)

[picture] | (8)

where =, , and (Reynolds number). Decomposing into two parts say transient part and a steady state part,

[picture] | (9)

Then we have two separate problems and the steady state part will be

[picture] | (10)

[picture] | (11)

[picture] | (12)

The solution of Eq. (10) with BCs (11) and (12) can be obtained as

[picture] | (13)

where and

If there is no mass transfer at the walls then , so and , Eq. (13) becomes

[picture] | (14)

The transient part problem becomes

[picture] | (15)

[picture] | (16)

[picture] | (17)

[picture] | (18)

The solution can be derived by using Laplace transformation techniques . The solution can be shown as

[picture] | (19)

where denotes the inverse Laplace transform, and . The inverse Laplace transform of above Eq. (19) shows that we have a simple pole at and infinite number of poles (located on the negative real axis) at , , where = and are given by

[picture] | (20)

The transient part velocity becomes

[picture] | (21)

where stands for residue and is given by

[picture] | (22)

We have

[picture] | (23)

Where

[picture] | (24)

[picture] | (25)

The residue at gives steady velocity.

[picture] | (26)

The transient part velocity from Eq. (21) becomes

[picture] | (27)

Therefore, the overall transient solution from Eq. (9) becomes

[picture] | (28)

2.1. No Slip Problem

When there is no slip between the fluid and the wall than above problem reduces to

[picture] | (29)

[picture] | (30)

This is an unsteady MHD incompressible viscous Couette flow problem with Hall effects, in which the bottom wall is fixed and subjected to a mass injection velocity , and there is mass suction velocity at the top wall. The top plate is stationary when , there is only mass transfer in the transverse direction, say direction. At , the top wall is started impulsively to a constant velocity . The solution of this problem can be derived from above problem by applying no slip condition . Steady state velocity can be deduced from Eq. (13) by putting .

[picture] | (31)

where and .

If there is no mass transfer at the walls then , so and , Eq. (31) becomes

[picture] | (32)

The transient part solution can be obtained from Eq. (27) by putting .

[picture] | (33)

The overall transient solution can be deduced from Eq. (28) by putting .

[picture] | (34)

where , , , . The Eq. (34) can be written as

[picture] | (35)

where =

3. Graphs and Discussion

In this part we discuss the variation of the real and imaginary parts of the complex transient part velocity , complex overall transient velocity , complex steady state velocity with distance from the wall for different values of Reynolds number , Hall parameter , magnetic field parameter , slip parameter and time . Figure 2 and figure 3 hows the variation of real and imaginary parts of the transient part velocity with distance from the wall for several values of Hall parameter by keeping , and fixed. Figure 2 shows that when there is mass suction at the top wall, with increase in Hall parameter, the real part of the transient part velocity increases in magnitude. Figure 3 shows that when there is mass suction at the top wall, with increase in Hall parameter, the imaginary part of the transient part velocity also decreases in magnitude. Figure 4 shows that when there is mass injection at the bottom wall, with increase in Hall parameter, the real part of the transient part velocity increases. Figure 5 shows that when there is mass injection at the bottom wall, with increase in Hall parameter, the imaginary part of the transient part velocity decreases and will become weaker as compared to the case of real. From Figure 6 it is observed that for suction at top wall and for fixed values of , and the real part of the transient part velocity decreases in magnitude with increase in time. From Figure 7 it is observed that for suction at top wall and for fixed values of , and the imaginary part of the transient part velocity also decreases in magnitude with increase in time and will become weaker as compared to the case of real. From Figure 6 and Figure 7 it is seen that the real and imaginary parts of the transient part velocity will decay with time, which is consistent with what we expected.

Figure 8 shows that for injection at bottom wall and for fixed values of , and the real part of the transient part velocity decreases in magnitude with increase in time. Figure 9 shows that for injection at bottom wall and for fixed values of , and the imaginary part of the transient part velocity decreases in magnitude with increase in time. From Figure 8 and Figure 9 it is seen that the real and imaginary parts of the transient part velocity will decay with time. From Figure 6, Figure 7 Figure 8 and Figure 9 it is clear that after a certain time, the complex transient part velocity will die away and velocity will become developed. Figure 10 and Figure 11 indicates variation of the real and imaginary parts of overall transient velocity with for fixed values of and . Figure 10 shows that for , with increase in Hall parameter , real part of the overall transient velocity increases. Figure 11 shows that for , with increase in Hall parameter , imaginary part of the overall transient velocity decreases. Figure 12 shows that for , with increase in Hall parameter , the real part of the overall transient velocity increases. Figure 13 shows that for , with increase in Hall parameter , the imaginary part of the overall transient velocity decreases. The real and imaginary parts of the overall transient velocities for at different times are depicted in Figure 14 and Figure 15. Figure 14 shows that for mass injection at the bottom wall, real part of the overall transient velocity increases with increase in time. Figure 15 shows that for mass injection at the bottom wall, imaginary part of the overall transient velocity decreases with increase in time. Figure 16 and Figure 17 illustrates the variation of real and imaginary parts of the steady state velocity with for several values of slip parameter and for fixed value of and . Figure 16 shows that for suction at the top wall with increase in slip parameter , the real part of the steady state velocity increases. Figure 17 shows that for suction at the top wall with increase in slip parameter , the imaginary part of the steady state velocity increases.

Figure 2. Variation of Re[with for several values of and

Figure 3. Variation of Im[with for several values of and

Figure 4. Variation of Re[with for several values of and

Figure 5. Variation of Im[with for several values of and

Figure 6. Variation of Re[with for several values of and

Figure 7. Variation of Im[with for several values of and

Figure 8. Variation of Re[with for several values of and

Figure 9. Variation of Im[with for several values of and

Figure 10. Variation of Re[with for several values of and

Figure 11. Variation of Im[with for several values of and

Figure 12. Variation of Re[with for several values of and

Figure 13. Variation of Im[with for several values of and

Figure 14. Variation of Re[with for several values of and

Figure 15. Variation of Im[with for several values of and

Figure 16. Variation of Re[with for several values of and

Figure 17. Variation of Im[with for several values of and

Figure 18. Variation of Re[with for several values of and

Figure 19. Variation of Im[with for several values of and

Figure 20. Variation of Re[ with for several values of and for Couette flow.

Figure 21. Variation of Im[ with for several values of and for Couette flow.

Figure 22. Variation of Re[ with for several values of and for Couette flow.

Figure 23. Variation of Im[ with for several values of and for Couette flow.

Figure 18 shows that for injection at the bottom wall with increase in slip parameter , the real part of the steady state velocity decreases. Figure 19 shows that for injection at the bottom wall with increase in slip parameter , the imaginary part of the steady state velocity increases. Figure 20, Figure 21, Figure 22 and Figure 23 are plotted for Hall current on Couette flow with no slip. Figure 20 and Figure 21 indicates variation of the real and imaginary parts of overall transient velocity with for fixed values of and . Figure 20 shows that for , with increase in Hall parameter , real part of the overall transient velocity increases. Fig 21. shows that for , with increase in Hall parameter , imaginary part of the overall transient velocity increases. Figure 22 shows that for , with increase in Hall parameter , the real part of the overall transient velocity increases. Figure 23 shows that for , with increase in Hall parameter , the imaginary part of the overall transient velocity decreases.

4. Final Remarks

In this study exact solutions for the velocity field in the presence of Hall current, magnetic field, porosity and slip parameter are constructed. A uniform magnetic field is applied transversely to the flow. The exact solutions of the velocity field for flow subjected to the slip conditions between the two parallel plates and fluid are obtained by means of Laplace transform. The solutions so obtained, depending on the initial and the boundary conditions, are presented as sum of the steady state transient solutions. The results for Couette flow with Hall effects are also found by taking . Graphical results for mass transfer and Hall current reveal that it has significant influence on the velocity distribution. The current analysis will be useful in dealing with real engineering problems.

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