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

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(2)

(3)

The initial condition is

(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

(5)

(6)

(7)

(8)

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

(9)

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

(10)

(11)

(12)

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

(13)

where and

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

(14)

The transient part problem becomes

(15)

(16)

(17)

(18)

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

(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

(20)

The transient part velocity becomes

(21)

where stands for residue and is given by

(22)

We have

(23)

Where

(24)

(25)

The residue at gives steady velocity.

(26)

The transient part velocity from Eq. (21) becomes

(27)

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

(28)

2.1. No Slip Problem

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

(29)

(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 .

(31)

where and .

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

(32)

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

(33)

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

(34)

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

(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[

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for several values of

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Figure 3. Variation of Im[

with

for several values of

and

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Figure 4. Variation of Re[

with

for several values of

and

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Figure 5. Variation of Im[

with

for several values of

and

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Figure 6. Variation of Re[

with

for several values of

and

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Figure 7. Variation of Im[

with

for several values of

and

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Figure 8. Variation of Re[

with

for several values of

and

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Figure 9. Variation of Im[

with

for several values of

and

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Figure 10. Variation of Re[

with

for several values of

and

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Figure 11. Variation of Im[

with

for several values of

and

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Figure 12. Variation of Re[

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for several values of

and

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Figure 13. Variation of Im[

with

for several values of

and

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Figure 14. Variation of Re[

with

for several values of

and

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Figure 15. Variation of Im[

with

for several values of

and

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Figure 16. Variation of Re[

with

for several values of

and

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Figure 17. Variation of Im[

with

for several values of

and

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Figure 18. Variation of Re[

with

for several values of

and

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Figure 19. Variation of Im[

with

for several values of

and

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Figure 20. Variation of Re[

with

for several values of

and

for Couette flow.

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Figure 21. Variation of Im[

with

for several values of

and

for Couette flow.

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Figure 22. Variation of Re[

with

for several values of

and

for Couette flow.

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Figure 23. Variation of Im[

with

for several values of

and

for Couette flow.

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