A partially ionized fluid is driven by a stretching disk, in the presence of a magnetic 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 two-term 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.
1. 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 [2–5], 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.
2. Formulation of the Problem
A partially ionized fluid is driven by an insulated disk, which is axisymmetrically stretching with speed us that is proportional to the radial distance r from its axis of symmetry z. Specifically, us=ωr, where ω is a constant of proportionality. The disk is porous, allowing a uniform fluid injection of speed ws 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 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 ρ, viscosity μ, electrical conductivity σ, Hall coefficient h(=1/ene), and ion-slip coefficient k(=1/(1+ne/na)κai), all of which are considered constant. Respectively, ne and na are the number densities for the electrons and neutral particles, κai is the coefficient of friction between ions and neutral particles, and -e is electron charge.
The flow is governed by the continuity and Navier-Stokes equations [10](1a)∇·V=0,(1b)ρV·∇V=μ∇2V-∇p+J×B,
where p 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
(1c)J=σ[V×B-hJ×B-kB×(J×B)],
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: u in the r-direction, v in the azimuthal direction, and w in the z-direction, as well as the pressure p, are dependent on r and z only. The governing equations become (2a)u,r+ur+w,z=0,(2b)ρ(uu,r-v2r+wu,z)=μ[u,rr+(ur),r+u,zz]-p,r-σB2n2+m2(nu-mv),(2c)ρ(uv,r+uvr+wv,z)=μ[v,rr+(vr),r+v,zz]-σB2n2+m2(nv+mu),(2d)ρ(uw,r+ww,z)=μ[w,rr+1rw,r+w,zz]-p,z,
where subscripts following a comma denote differentiation. The Hall parameter m=σhB may be positive or negative in accordance with the sign of B, 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 m and v leaves the problem unaltered, only nonnegative values of m need to be considered. The parameter n=1+σkB2 reduces to unity for zero ion slip.
At the surface, z=0, the adherence conditions u=ωr and v=0 apply, together with the injection condition w=ws. Far from the disk, as z~∞, the fluid has pressure p∞ and velocity components u~0 and v~0.
The problem admits the similarity transformations z=(μ/ρω)1/2ζ, u=ωrF(ζ), v=ωrG(ζ), w=(ωμ/ρ)1/2H(ζ), and p=p∞+ωμQ(ζ), leading to the following problem:(3a)2F+H′=0,(3b)F′′-HF′-F2+G2-β(n^F-m^G)=0,(3c)G′′-HG′-2FG-β(n^G+m^F)=0,(3d)Q′=H′′-HH′,(3e)F(0)=1,(3f)G(0)=0,(3g)H(0)=Hs,(3h)F(∞)~0,(3i)G(∞)~0,(3j)Q(∞)~0,
where a dash denotes differentiation with respect to ζ, β=σB2/ρω is the magnetic interaction number, and (m^,n^)=(m,n)/(m2+n2).
3. Asymptotic Analysis
We are interested in the limiting behavior of the flow as β~∞ with fixed m^ and n^. That F(0)=1 irrespective of the value of β means that F=𝒪(β0). The leading term in (3b) is βn^F. It can be balanced by the diffusion term F′′ in a contracting region in which ζ=𝒪(β-1/2). Then, (3a) gives H=𝒪(β-1/2), consistent with which Hs must be 𝒪(β-1/2). Equation (3d), then, gives Q=𝒪(β0). In (3c), the driving force for the azimuthal flow is the Hall effect expressed by βm^F. This requires G′′~βm^F, leading to G=𝒪(β0).
New scaled variables η=β1/2ζ and V=β1/2H are introduced, transforming Problem (3a), (3b), (3c), (3d), (3e), (3f), (3g), (3h), (3i), and (3j) to the form(4a)2F+V′=0,(4b)F′′-n^F+m^G=β-1(VF′+F2-G2),(4c)G′′-n^G-m^F=β-1(VG′+2FG),(4d)Q′=V′′-β-1VV′,(4e)F(0)=1,(4f)G(0)=0,(4g)V(0)=Vs,(4h)F(∞)~0,(4i)G(∞)~0,(4j)Q(∞)~0,
where, now, the dashes denote differentiation with respect to η.
We expand the flow variables in powers of β-1 in the form
(5)Z~Z0+β-1Z1+β-2Z2+⋯,
where Z stands for F, G, V, and Q. The problems for Zn, n=0,1,2,3,…, are linear. For Z0, we get the solutions [11](6a)F0=e-αηcosγη,(6b)G0=-e-αηsinγη,(6c)V0=Vs-2[α+e-αη(γsinγη-αcosγη)]α2+γ2,(6d)Q0=-2e-αηcosγη,
where α and γ satisfy α2-γ2=n^ and 2αγ=m^. For Z1, the solutions involve secular terms of the form ηe-αηsinγη and ηe-αηcosγη, 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(β-2) are obtained:(7a)n^~(α2-γ2)+β-1αv0+β-2αv1+⋯,(7b)m^~2αγ+β-1γv0+β-2γv1+⋯,(7c)F~EC+β-1[E(Cf1+Sg1)-E2f1]+β-2[E(Cf2+Sg2)+E2c-+E3(Ca-+Sb-)]+⋯,(7d)G~-ES+β-1[E(Cg1-Sf1)-E2g1]+β-2[E(Cg2-Sf2)+E2c~+E3(Ca~+Sb~)]+⋯,(7e)V~[v0+2λE(-γS+αC)]+β-1×[v1+2λE{C(γg1+αf1)+S(αg1-γf1)}-E2f1α]+β-2[E2c-αv2+2λE{C(γg2+αf2)+S(αg2-γf2)}hhhhhh+E2c-α+E3(Ca^+Sb^)]+⋯,
where, for conciseness, (8a)E=e-αη,(8b)C=cosγη,(8c)S=sinγη,(8d)λ=(α2+γ2)-1,(8e)v0=Vs-2λα,f1, g1, and v1 are obtained from
(8f)(3α2+γ2)f1+2αγg1=λ(α2-γ2),(8g)-2αγf1+(3α2+γ2)g1=2λαγ,(8h)v1=-2λ(γg1+αf1)+f1α,c- and c~ satisfy
(8i)(3α2+γ2)c-+2αγc~=v0(γg1+αf1)-2λ(α2-γ2)f1,(8j)-2αγc-+(3α2+γ2)c~=v0(αg1-γf1)-4λαγf1,a-, b-, a~, and b~ satisfy
(8k)8α2a--6αγb-+2αγa~=-(1-4λα2)f1,(8l)6αγa-+8α2b-+2αγb~=(1-4λα2)γf1α-2g1,(8m)-2αγa-+8α2a~-6αγb~=-2(1-2λα2)g1+γf1α,(8n)-2αγb-+6αγa~+8α2b~=-4λαγg1+f1,a^ and b^ satisfy
(8o)3αa^-γb^=2a-,(8p)γa^+3αb^=2b-,
and finally f2, g2, and v2 are obtained from
(8q)f2=-c--a-,(8r)g2=-c~-a~,(8s)v2=-2λ(γg2+αf2)-c-α-a^.
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(9a)F′(0)~-α+β-1(αf1+γg1)+β-2(-αf2+γg2-2αc--3αa-+γb-)+⋯,(9b)G′(0)~-γ+β-1(αg1-γf1)+β-2(-αg2-γf2-2αc~-3αa~+γb~)+⋯,(9c)V(∞)~v0+β-1v1+β-2v2+⋯=V∞.
The expansion for the pressure can be obtained from
(9d)Q=-2F+12β-1(V∞2-V2)
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 β~∞. 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 G (see Appendix).
4. Results and Discussion
The asymptotic expansions obtained previously are tested against corresponding numerical results. The problem described by (4a), (4b), (4c), (4d), (4e), (4f), (4g), (4h), (4i), and (4j) is solved numerically in double precision, using Keller’s two-point, second-order accurate, finite-difference scheme [12]. A uniform step size Δη=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 far-field conditions (4h)–(4j). (As pointed out by Pantokratoras [13], a small value of η∞ can lead to erroneous results.) The nonlinear terms are quasi-linearized, and an iterative procedure is implemented, terminating when the maximum errors in F′(0), G′(0), and V(∞) become less than 10-10.
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 F(η) and G(η), when β=10, m=10, n=1.2, andVs=20, with η∞=350. However, at such low value of β, a fixed period of oscillation is not sustained. When β is increased to 100, the F and G profiles, respectively, cross the zero line first at η≈7.98 and 15.67 then at η≈23.37 and 31.10. The two profiles cross the zero line several times later but with much smaller magnitudes, maintaining the same period τ≈30.9, all through.
Velocity profiles, β=20, m=10, n=1.2, and Vs=20.
How quantitatively useful the asymptotic expansions of the previous section can be is next investigated. To generate numerical values, for given β, m, n, and Vs, we need to determine the coefficients α, γ, λ, v0, f1, g1, v1,…, and so forth. To that end, we use expansions (7a) and (7b) as well as (8a), (8b), (8c), (8d), (8e), (8f), (8g), (8h), (8i), (8j), (8k), (8l), (8m), (8n), (8o), (8p), (8q), (8r), and (8s) to obtain expansions of these coefficients in form (5). In particular, the coefficients of the expansions which are involved to 𝒪(β-1) are found to be α0=[(1/2){n^+(m^2+n^2)1/2}]1/2, γ0=m^/2α0, λ0=(α02+γ02)-1, v00=Vs-2λ0α0, α1=-(1/2)v00, γ1=0, λ1=-2λ02(α0α1+γ0γ1), v01=-2(λ0α1+λ1α0), f10=λ0(n^θ1-m^θ2)/θ3, g10=λ0(n^θ2+m^θ1)/θ3, and v10=-λ0(n^f10+m^g10)/α0, where θ1=3α02+γ02, θ2=m^, and θ3=θ12+θ22. The following expansions are also obtained:(10a)F′(0)=-α0+β-1(-α1+α0f10+γ0g10)+𝒪(β-2),(10b)G′(0)=-γ0+β-1(-γ1-γ0f10+α0g10)+𝒪(β-2),(10c)V(∞)=v00+β-1(v01+v10)+𝒪(β-2).
Results of the two-term asymptotic expansions (i.e., up to 𝒪(β-1)) proved to be in close agreement with the numerical results (with η∞=40), even at values of β as low as 10. This is illustrated in Figure 2 for F′(0) with different values of m, in Figure 3 for -G′(0) with different values of n, and in Figure 4 for V(∞) with different values of Vs.
Radial surface shear F′(0) versus β, with different values of n; m=1, Vs=0.
Azimuthal surface traction -G′(0) versus β, with different values of m; n=1, Vs=0.
Far-field normal velocity V(∞) versus β, with different values of Vs; m=1, n=1.
The period of the sinusoidal oscillations τ=2π/γ takes the form
(11)τ=4π[(1/2){n^+(m^2+n^2)1/2}]1/2m^+𝒪(β-2)
which shows that, to 𝒪(β-2), τ is independent of Vs. Partial differentiation of the leading term expression for τ with respect to m and n reveals that τ monotonically increases with n, while it has a local minimum when m=n3. Typical results are presented in the first six rows of Table 1. They are in excellent agreement with the corresponding averaged numerical results, examples of which, calculated with β=100, Vs=0, η∞=100, are given in the last row of Table 1, each to be compared with the corresponding (bold) asymptotic value in the same column.
Period of sinusoidal oscillations τ for different values of m and n.
m
n=1.0
1.2
1.4
1.6
1.8
2.0
1
19.525
23.0678
27.005
31.306
35.944
40.894
2
17.871
19.476
21.258
23.213
25.333
27.613
3
19.109
20.146
21.282
22.518
23.853
25.285
4
20.731
21.510
22.354
23.262
24.236
25.274
5
22.379
23.013
23.692
24.418
25.189
26.008
n3
17.772
19.468
21.028
22.479
23.843
25.133
num.^{a}
19.54
19.48
21.30
23.28
25.21
25.14
aAveraged numerical results each to be compared with corresponding (bold) asymptotic value in the same column.
5. 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.
Appendix
The case of the neglect of the Hall and ion-slip effects {m=0,n=1} can be obtained by setting γ=0. The azimuthal velocity component G vanishes identically, while for F and V we get the expansions(A.1a)F~E+β-113α2[E-E2]+β-2172α4×[E(-8Vsα+23)+E2(8Vsα-32)+E39]+⋯,(A.1b)V~1α[(Vsα-2)+2E]-β-113α3(1-E)2+β-2172α5[E3(8Vsα-20)+E(-16Vsα+46)hhhhhhhhhh+E2(8Vsα-32)+6E3]+⋯
where α is determined from(A.3a)1~α2+β-1(Vsα-2)-13α2β-2+⋯.
This asymptotic relation can be used to determine an expansion for α of form (5), through substituting and equating coefficients of like powers of β. The result is
(A.3b)α~1+β-1(1-12Vs)+β-2(18Vs2-13)+⋯.
It is to be noted that the sinusoidal behavior disappears. Expansions (A.1a) and (A.1b) involve negative exponentials (e-iαη,i=1,2,…) only.
Further, introduction of expansion (A.3b) into expansions (A.1a), (A.1b), (A.2a), and (A.2b), then expanding, leads to expansions that correspond to the expansions of Ariel [5], who treated the case of no mass transfer, Vs=0, through coordinate straining.
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