MHD Flow with Hall Current and Ion-Slip Effects due to a Stretching Porous Disk

A partially ionized fluid is driven by a stretching disk, in the presence of amagnetic field that is strong enough to produce significant hall current and ion-slip effects. 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.The leading order approximations show sinusoidal behavior that is decaying exponentially, as we move away from the disk surface. The twoterm expansions of the radial and azimuthal 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. The effect of mass transfer (suction or injection) through the disk is also considered.


Introduction
The flow due to a stretching 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.When the surface is stretching radially from a point, in a linear manner, the flow is axisymmetric.The velocity components tend to their limits monotonically, in an exponential manner, as we move away from the surface [1].Recently, the problem and variants thereof have received considerable attention by Ariel [2][3][4][5], Ariel et al. [6], and Hayat and coworkers [7,8].Different methods of solution and analysis were applied to cases with or without slip conditions, MHD flow, or second-grade fluid.All these cases exhibited the same monotonic behavior.
One important variant is when the fluid is electrically conducting and a magnetic field is applied normally to the surface.For weak magnetic fields, the flow remains axisymmetric.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.The Hall current is associated with an electromagnetic force which drives an azimuthal flow.The problem loses its axisymmetric nature, maintaining its rotational symmetry, though, with the flow variables being independent of the azimuthal angle.
It is of interest to explore the nature of this MHD flow taking into consideration the 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 asymptotic expansions are, thus, obtained.The presence of the Hall current leads to an exponential tendency to the limits but of sinusoidal nature.This behavior is not altered by including mass transfer through the surface or the electromagnetic effect of ion slip.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 partially ionized fluid is driven by an insulated disk, which is axisymmetrically stretching with speed   that is proportional to the radial distance  from its axis of symmetry .Specifically,   = , where  is a constant of proportionality.The disk is porous, allowing a uniform fluid injection of speed   in the  direction.Otherwise, the fluid would have been quiescent.A uniform magnetic field is applied in the -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 .On the other hand, the magnetic field is strong enough to produce significant curvature in the electrons trajectories, leading to considerable Hall currents.Moreover, the electron motion is dominated by electron-ion collisions, so that the ion-slip effect cannot be overlooked [10].
The fluid is incompressible of density , viscosity , electrical conductivity , Hall coefficient ℎ(= 1/  ), and ion-slip coefficient (= 1/(1 +   /  )  ), all of which are considered constant.Respectively,   and   are the number densities for the electrons and neutral particles,   is the coefficient of friction between ions and neutral particles, and − is electron charge.
The flow is governed by the continuity and Navier-Stokes equations [10] where  is the pressure, V is the velocity vector, B is the magnetic field, and J is the electric current.According to the generalized Ohm's law, J is given by where terms on the right-hand side are due to the effects of Lorentz force, the Hall current, and the ion slip, respectively.Making use of the rotational symmetry, we formulate the problem for a typical meridional plane.The velocity components:  in the -direction, V in the azimuthal direction, and  in the -direction, as well as the pressure , are dependent on  and  only.The governing equations become where subscripts following a comma denote differentiation.The Hall parameter  = ℎ may be positive or negative in accordance with the sign of , that is, depending on whether the magnetic field is directed away from or toward the disk.However, as a simultaneous change of the signs of  and V leaves the problem unaltered, only nonnegative values of  need to be considered.The parameter  = 1 +  2 reduces to unity for zero ion slip.At the surface,  = 0, the adherence conditions  =  and V = 0 apply, together with the injection condition  =   .Far from the disk, as  ∼ ∞, the fluid has pressure  ∞ and velocity components  ∼ 0 and V ∼ 0.
We expand the flow variables in powers of  −1 in the form where  stands for , , , and .The problems for   ,  = 0, 1, 2, 3, . .., are linear.For  0 , we get the solutions [11]  0 =  − cos , (6a) where  and  satisfy  2 −  2 = n and 2 = m.For  1 , the solutions involve secular terms of the form  − sin and  − cos , the removal of which is effected by straining the parameters m and n in the form (5), and the procedure can be continued to higher orders [9].
The following expansions up to O( −2 ) are obtained: where, for conciseness, 1 ,  1 , and V 1 are obtained from and c satisfy , , ã, and b satisfy â and b satisfy and finally  2 ,  2 , and V 2 are obtained from Of interest are the radial and azimuthal components of the shear stress at the surface, as well as the far-field speed.They are represented, respectively, by The expansion for the pressure can be obtained from which is the result of integrating (4d) and the use of (4a) and condition (4j).This is not given here, for brevity.The previously mentioned expansions describe how the flow behaves as  ∼ ∞.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 azimuthal velocity component  (see Appendix).
The numerical results exhibit the attenuating sinusoidal behavior predicted by the asymptotic analysis.This is clearly illustrated in Figure 1 showing the radial and azimuthal velocity profiles () and (), when  = 10,  = 10,  = 1.2, and  = 20, with  ∞ = 350.However, at such low value of , a fixed period of oscillation is not sustained.When  is increased to 100, the  and  profiles, respectively, cross the zero line first at  ≈ 7.98 and 15.67 then at  ≈ 23.37 and 31.10.The two profiles cross the zero line several times later but with much smaller magnitudes, maintaining the same period  ≈ 30.9, all through.

Conclusion
The limiting behavior of the MHD flow due to a porous stretching disk has been studied, as the magnetic field grows indefinitely, taking into consideration the Hall current and ion-slip effects.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 disk.In contrast, when the MHD effect of Hall current is neglected, the exponential decay becomes monotonic.Finite difference solutions have also been calculated.The two-term asymptotic results and the numerical solutions have shown excellent agreement both qualitatively and quantitatively.
This asymptotic relation can be used to determine an expansion for  of form (5), through substituting and equating coefficients of like powers of .The result is It is to be noted that the sinusoidal behavior disappears.Expansions (A.1a) and (A.1b) involve negative exponentials ( − ,  = 1, 2, . ..) only.

Table 1 :
Period of sinusoidal oscillations  for different values of  and .Averaged numerical results each to be compared with corresponding (bold) asymptotic value in the same column. a