Limiting Behavior of MHD Flow with Hall Current, Due to a Porous Stretching Sheet

An electrically conducting fluid is driven by a stretching sheet, in the presence of a magnetic field that is strong enough to produce significant Hall current. The sheet is porous, allowing mass transfer through suction or injection. The limiting behavior of the flow is studied, as the magnetic field strength grows indefinitely. The flow variables are properly scaled, and uniformly valid asymptotic expansions of the velocity components are obtained through parameter straining. The leading order approximations show sinusoidal behavior that is decaying exponentially, as we move away from the surface. The two-term expansions of the surface shear stress components, as well as the far field inflow speed, compare well with the corresponding finite difference solutions; even at moderate magnetic fields.


Introduction
The flow due to a moving surface is an important field of fluid mechanics.It has several applications; for example, in production of glass and paper sheets, drawing of plastic films, and extrusion of metals and polymers.Crane [1] obtained an exact solution for the two-dimensional flow of a Newtonian fluid due to a linearly stretching sheet.The velocity components in this case has simple forms; tending to their limits monotonically in an exponential manner as we move away from the surface.The simplicity of this solution invited several authors to consider different related problems, which exhibit the same behavior; e.g.[2]- [8].
One such problem is when the fluid is electrically conducting and a magnetic field is applied normally to the surface.For weak magnetic fields, the flow remains two-dimensional and a closed-form solution is still possible to attain.The Lorentz force acts to restrain the flow; causing faster exponential tendency to the limits.When the magnetic field is strong enough to produce significant Hall current, the problem changes considerably.It becomes three-dimensional.The Hall current is associated with an electromagnetic force which derives a transverse flow.A closed-form solution is no longer possible.
It is interesting to explore the nature of this MHD flow in the presence of Hall current.To that end, the limiting behavior of the flow as the magnetic field grows indefinitely is studied.The straightforward perturbation analysis leads to secular behavior, which is removed by parameter straining [9].Three-term uniformly valid asy-mptotic expansions are, thus, obtained.The presence of the Hall current leads to an exponential tendency to the limits of sinusoidal nature.The flow involves alternating regions of forward and backward velocity components.
Finite difference solutions are also obtained and show qualitative adherence to the predicted limiting behavior even for moderate magnetic fields.Quantitatively, the two term expansions show excellent agreement with the numerical results.

Formulation of the Problem
A totally ionized fluid is driven by an insulated sheet, which is symmetrically stretching with speed u s that is proportional to the distance x from the axis of symmetry z.Specifically, u s = ωx, where ω is a constant of proportionality.The sheet is porous; allowing a uniform fluid injection of speed w s in the z direction.Otherwise, the fluid would have been quiescent.A uniform magnetic field is applied in the z-direction.The magnetic Reynolds number is small, so that the induced magnetic field can be neglected; and the applied field maintains its uniform magnetic flux density B. On the other hand, the magnetic field is strong enough to produce significant curvature in the electrons trajectories; leading to considerable Hall current [10].
The fluid is incompressible of density ρ, viscosity µ, electrical conductivity σ, Hall coefficient h (=1/en e ); where n e is the number density for the electrons and -e is the electron charge.All these parameters are considered constant.
The velocity components: u in the x-direction, v in the transverse y-direction, and w in the z-direction, as well as the pressure p are dependent on x and z only.They are governed by the following continuity and Navier-Stokes equations, which include components of the Lorentz force corrected for the Hall current, where subscripts denote differentiation, m = σhB is the Hall parameter.Note that m may be positive or negative in accordance with the sign of B; i.e. depending on whether the magnetic field is directed away from or toward the sheet.However, as a simultaneous change of the signs of m and v leaves the problem unaltered, only non-negative values of m need to be considered.
At the surface, z = 0, the adherence conditions u = ωx and v = 0 apply; together with the injection condition w = w s .Far from the sheet, as z~∞, the fluid has pressure p ∞ and velocity components u~0 and v~0.
The problem admits the similarity transformations z

Asymptotic Analysis
We are interested in the limiting behavior of the flow as β~∞ with fixed m; i.e. as β ~∞ with fixed m and n .That F (0) = 1 irrespective of the value of β means that F = O ( β 0 ).The leading term in Equation (2b) is β n F. It can be balanced by the diffusion term F ′′ in a contracting region in which z = O ( β −1/2 ).Then, Equa- tion (2a) gives H = O ( β −1/2 ); consistent with which H s must be O ( β −1/2 ).Consequently, Equation (2d) gives Q = O ( β 0 ).In Equation (2c), the driving force for the transverse flow is the Hall effect expressed by β m F. This requires G′′ ~β m F; leading to G = O ( β 0 ).
New stretched variables where, now, the dashes denote differentiation with respect to η.
We expand the flow variables in powers of β −1 in the form where Z stands for F, G, V, and Q.The problems for Z n , n = 0,1, and n in the form (5); and the procedure can be continued to higher orders.
The following expansions up to O ( β −2 ) are obtained where symbols appearing on the right-hand sides are given in Appendix.
Of interest are the longitudinal and transverse components of the shear stress at the surface, as well as the farfield speed.They are represented, respectively, by The expansion for the pressure can be obtained from which is the result of integrating Equation (4d) and the use of Equation (4a) and Condition (4j).This is not given here for brevity.
The above expansions describe how the flow behaves as β~∞ with fixed m.They reveal a sinusoidal behavior that dies out exponentially as we move away from the surface.This behavior is solely due to the Hall effect, which is also responsible for the presence of the transverse velocity component G.When the Hall current is neglected; i.e. m = 0, Expansion (6b) leads to γ = 0, Expansion (6d) gives G = 0, Expansions (6c) and (6e) reduce , respectively, while Expansion (6a) translates to α satisfying . These results coincide with the exact solutions of the degenerate cases of Gupta and Gupta [2] and Andersson [3].

Results and Discussion
The asymptotic expansions obtained above are tested against corresponding numerical results.The problem described by Equation ( 4) is solved numerically in double precision, using Keller's two-point, second-order-accurate, finite-difference scheme [11].A uniform step size ∆η = 0.01 is used on a finite domain 0≤ η ≤ η ∞ .The value of η ∞ is chosen sufficiently large in order to insure the asymptotic satisfaction of the farfield conditions (4h,j).(As pointed out by Pantokratoras [12], a small value of η ∞ can lead to erroneous results.)The non-linear terms are quasi-linearized, and an iterative procedure is implemented; terminating when the maximum errors in F ′ (0), G′ (0), and H(∞) become less than 10 −10 .
The numerical results exhibit the attenuating sinusoidal behavior predicted by the asymptotic analysis.This is clearly illustrated in Fig. 1 showing the longitudinal and transverse velocity profiles F(η) and G(η), when β = 20, m = 10, V s = 20, and η ∞ = 500.However, at such low value of β, a fixed period of oscillation is not sustained.When β is increased to100 the F and G profiles, respectively, cross the zero line first at η ≈ 7.79 and 15.44 then at η ≈ 23.15 and 30.87; thus sustaining the same period τ ≈ 30.8.The two profiles cross the zero line several times later, but with much smaller magnitudes; maintaining the same period, though.How quantitatively useful the asymptotic expansions of the previous section can be, is next investigated.To generate numerical values, for given β, m and V s , we need to determine the coefficientsα, γ, λ, v 0 , f 1 , g 1 , v 1 …, etc.
To that end, we use Expansions (6a) and (6b) and the pertinent equations of Appendix A to obtain expansions of these coefficients in the form (5). In particular, the coefficients of the expansions which are involved to O ( β −1 ) are found to be Results of the two-term asymptotic expansions (i.e., up to ( β -1 ) )are proved to be in close agreement with the numerical results, even at values of β as low as 20.This is illustrated in Figures 2 and 3 for F ′ (0) and G′ (0) with different values of m, and in Figure 4 for V(∞) with different values of V s .
The period of the sinusoidal oscillations τ = 2π/γ takes the form

Conclusion
The limiting behavior of the MHD flow due to a porous stretching sheet has been studied, as the magnetic field grows indefinitely, taking into consideration the Hall current.Three-term uniformly-valid asymptotic expansions have been derived using parameter straining.The velocity components show sinusoidal behavior that attenuates exponentially, as we move away from the surface.(When the Hall current is neglected, the exponential decay becomes monotonic.)Finite difference solutions have also been calculated.The two-term asymptotic and the numerical solutions have shown excellent agreement both qualitatively and quantitatively.

=
being the magnetic interaction number; and a dash denotes differrentiation with respect to ζ.

FQ
•••, are lin.ear.For Z 0 , we get the basic flow solutions 0 = −EC, where α and γ satisfy α 2 − γ 2 = n ˆ and 2αγ = m ˆ; with S = sinγη, C = cosγη and E = exp(−αη), for short.For Z 1 , the solutions in- volve secular terms of the form ηES and ηEC, the removal of which is effected by straining the parameters m