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

Abstract

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 𝑢_𝑠 that is proportional to the radial distance 𝑟 from its axis of symmetry 𝑧. Specifically, 𝑢_𝑠=𝜔𝑟, where 𝜔 is a constant of proportionality. The disk 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 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/𝑒𝑛_𝑒), and ion-slip coefficient 𝑘(=1/(1+𝑛_𝑒/𝑛_𝑎)𝜅_𝑎𝑖), all of which are considered constant. Respectively, 𝑛_𝑒 and 𝑛_𝑎 are the number densities for the electrons and neutral particles, 𝜅_𝑎𝑖 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]∇⋅𝐕=0,𝜌𝐕⋅∇𝐕=𝜇∇²𝐕−∇𝑝+𝐉×𝐁, (1a)(1b) 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 𝐉=𝜎[𝐕×𝐁−ℎ𝐉×𝐁−𝑘𝐁×(𝐉×𝐁)], (1c)

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 𝑢_,𝑟+𝑢𝑟+𝑤_,𝑧𝜌=0,𝑢𝑢_,𝑟−𝑣²𝑟+𝑤𝑢_,𝑧=𝜇𝑢_,𝑟𝑟+𝑢𝑟_,𝑟+𝑢_,𝑧𝑧−𝑝_,𝑟−𝜎𝐵²𝑛²+𝑚²(),𝜌𝑛𝑢−𝑚𝑣𝑢𝑣_,𝑟+𝑢𝑣𝑟+𝑤𝑣_,𝑧=𝜇𝑣_,𝑟𝑟+𝑣𝑟_,𝑟+𝑣_,𝑧𝑧−𝜎𝐵²𝑛²+𝑚²(),𝜌𝑛𝑣+𝑚𝑢𝑢𝑤_,𝑟+𝑤𝑤_,𝑧=𝜇𝑤_,𝑟𝑟+1𝑟𝑤_,𝑟+𝑤_,𝑧𝑧−𝑝_,𝑧, (2a)(2b)(2c)(2d)

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=𝑤_𝑠. 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:2𝐹+𝐻^′𝐹=0,_′′−𝐻𝐹^′−𝐹²+𝐺²()𝐺−𝛽𝑛𝐹−𝑚𝐺=0,_′′−𝐻𝐺^′()𝑄−2𝐹𝐺−𝛽𝑛𝐺+𝑚𝐹=0,_′=𝐻^′′−𝐻𝐻^′,𝐹(0)𝐺(0)𝐻(0)=1,=0,=𝐻_𝑠,𝐹(∞)𝐺(∞)𝑄(∞)∼0,∼0,∼0, (3a)(3b)(3c)(3d)(3e)(3f)(3g)(3h)(3i)(3j)

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 𝛽𝑛𝐹. 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 𝐻_𝑠 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 𝛽𝑚𝐹. This requires 𝐺^′′∼𝛽𝑚𝐹, leading to 𝐺=𝒪(𝛽⁰).

New scaled variables 𝜂=𝛽^1/2𝜁 and 𝑉=𝛽^1/2𝐻 are introduced, transforming Problem (3a), (3b), (3c), (3d), (3e), (3f), (3g), (3h), (3i), and (3j) to the form2𝐹+𝑉^′𝐹=0,_′′−𝑛𝐹+𝑚𝐺=𝛽⁻¹𝑉𝐹^′+𝐹²−𝐺²𝐺_′′,−𝑛𝐺−𝑚𝐹=𝛽⁻¹𝑉𝐺^′𝑄+2𝐹𝐺,_′=𝑉^′′−𝛽⁻¹𝑉𝑉^′,𝐹(0)𝐺(0)𝑉(0)=1,=0,=𝑉_𝑠,𝐹(∞)𝐺(∞)𝑄(∞)∼0,∼0,∼0, (4a)(4b)(4c)(4d)(4e)(4f)(4g)(4h)(4i)(4j)

where, now, the dashes denote differentiation with respect to 𝜂.

We expand the flow variables in powers of 𝛽⁻¹ in the form 𝑍∼𝑍₀+𝛽⁻¹𝑍₁+𝛽⁻²𝑍₂+⋯, (5) where 𝑍 stands for 𝐹, 𝐺, 𝑉, and 𝑄. The problems for 𝑍_𝑛, 𝑛=0,1,2,3,…, are linear. For 𝑍₀, we get the solutions [11]𝐹₀=𝑒^−𝛼𝜂𝐺cos𝛾𝜂,₀=−𝑒^−𝛼𝜂𝑉sin𝛾𝜂,₀=𝑉_𝑠−2𝛼+𝑒^−𝛼𝜂𝛾sin𝛾𝜂−𝛼cos𝛾𝜂𝛼²+𝛾²,𝑄₀=−2𝑒^−𝛼𝜂cos𝛾𝜂, (6a)(6b)(6c)(6d)

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:𝑛∼𝛼²−𝛾²+𝛽⁻¹𝛼𝑣₀+𝛽⁻²𝛼𝑣₁+⋯,𝑚∼2𝛼𝛾+𝛽⁻¹𝛾𝑣₀+𝛽⁻²𝛾𝑣₁+⋯,𝐹∼𝐸𝐶+𝛽⁻¹𝐸𝐶𝑓₁+𝑆𝑔₁−𝐸²𝑓₁+𝛽⁻²𝐸𝐶𝑓₂+𝑆𝑔₂+𝐸²𝑐+𝐸³𝐶𝑎+𝑆𝑏+⋯,𝐺∼−𝐸𝑆+𝛽⁻¹𝐸𝐶𝑔₁−𝑆𝑓₁−𝐸²𝑔₁+𝛽⁻²𝐸𝐶𝑔₂−𝑆𝑓₂+𝐸²𝑐̃+𝐸³𝐶𝑏̃𝑣𝑎+𝑆+⋯,𝑉∼₀+2𝜆𝐸−𝛾S+𝛼𝐶+𝛽⁻¹×𝑣₁+2𝜆𝐸𝐶𝛾𝑔₁+𝛼𝑓₁+𝑆𝛼𝑔₁−𝛾𝑓₁−𝐸²𝑓₁𝛼+𝛽⁻²𝑣₂+2𝜆𝐸𝐶𝛾𝑔₂+𝛼𝑓₂+𝑆𝛼𝑔₂−𝛾𝑓₂+𝐸²𝑐𝛼+𝐸³𝐶𝑎+𝑆𝑏̂+⋯, (7a)(7b)(7c)(7d)(7e)

where, for conciseness, 𝐸=𝑒^−𝛼𝜂,𝛼𝐶=cos𝛾𝜂,𝑆=sin𝛾𝜂,𝜆=²+𝛾²⁻¹,𝑣₀=𝑉_s−2𝜆𝛼, (8a)(8b)(8c)(8d)(8e)𝑓₁, 𝑔₁, and 𝑣₁ are obtained from 3𝛼²+𝛾²𝑓₁+2𝛼𝛾𝑔₁=𝜆𝛼²−𝛾²,−2𝛼𝛾𝑓₁+3𝛼²+𝛾²𝑔₁𝑣=2𝜆𝛼𝛾,₁=−2𝜆𝛾𝑔₁+𝛼𝑓₁+𝑓₁𝛼, (8f)(8g)(8h)𝑐 and 𝑐̃ satisfy 3𝛼²+𝛾²𝑐+2𝛼𝛾𝑐̃=𝑣₀𝛾𝑔₁+𝛼𝑓₁−2𝜆𝛼²−𝛾²𝑓₁,−2𝛼𝛾𝑐+3𝛼²+𝛾²𝑐̃=𝑣₀𝛼𝑔₁−𝛾𝑓₁−4𝜆𝛼𝛾𝑓₁, (8i)(8j)𝑎, 𝑏, 𝑎, and 𝑏̃ satisfy 8𝛼²𝑎−6𝛼𝛾𝑏+2𝛼𝛾𝑎=−1−4𝜆𝛼²𝑓₁,6𝛼𝛾𝑎+8𝛼²𝑏+2𝛼𝛾𝑏̃=1−4𝜆𝛼²𝛾𝑓₁𝛼−2𝑔₁,−2𝛼𝛾𝑎+8𝛼²𝑎−6𝛼𝛾𝑏̃=−21−2𝜆𝛼²𝑔₁+𝛾𝑓₁𝛼,−2𝛼𝛾𝑏+6𝛼𝛾𝑎+8𝛼²𝑏̃=−4𝜆𝛼𝛾𝑔₁+𝑓₁, (8k)(8l)(8m)(8n)𝑎 and 𝑏̂ satisfy 3𝛼𝑎−𝛾𝑏̂=2𝛾̂𝑎,𝑎+3𝛼𝑏=2𝑏, (8o)(8p) and finally 𝑓₂, 𝑔₂, and 𝑣₂ are obtained from 𝑓₂=−𝑐−𝑔𝑎,₂̃𝑣=−𝑐−𝑎,₂=−2𝜆𝛾𝑔₂+𝛼𝑓₂−𝑐𝛼−𝑎. (8q)(8r)(8s)

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, by𝐹^′(0)∼−𝛼+𝛽⁻¹𝛼𝑓₁+𝛾𝑔₁+𝛽⁻²−𝛼𝑓₂+𝛾𝑔₂−2𝛼𝑐−3𝛼𝑎+𝛾𝑏𝐺+⋯,_′(0)∼−𝛾+𝛽⁻¹𝛼𝑔₁−𝛾𝑓₁+𝛽⁻²−𝛼𝑔₂−𝛾𝑓₂̃𝑏̃𝑉(∞)−2𝛼𝑐−3𝛼𝑎+𝛾+⋯,∼𝑣₀+𝛽⁻¹𝑣₁+𝛽⁻²𝑣₂+⋯=𝑉_∞. (9a)(9b)(9c)

The expansion for the pressure can be obtained from 1𝑄=−2𝐹+2𝛽⁻¹𝑉²_∞−𝑉² (9d)

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, and𝑉_𝑠=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.

Figure 1 

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 𝑉_𝑠, 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+(𝑚²+𝑛²)^1/2}]^1/2, 𝛾₀=𝑚/2𝛼₀, 𝜆₀=(𝛼²₀+𝛾²₀)⁻¹, 𝑣₀₀=𝑉_s−2𝜆₀𝛼₀, 𝛼₁=−(1/2)𝑣₀₀, 𝛾₁=0, 𝜆₁=−2𝜆²₀(𝛼₀𝛼₁+𝛾₀𝛾₁), 𝑣₀₁=−2(𝜆₀𝛼₁+𝜆₁𝛼₀), 𝑓₁₀=𝜆₀(𝑛𝜃₁−𝑚𝜃₂)/𝜃₃, 𝑔₁₀=𝜆₀(𝑛𝜃₂+𝑚𝜃₁)/𝜃₃, and 𝑣₁₀=−𝜆₀(𝑛𝑓₁₀+𝑚𝑔₁₀)/𝛼₀, where 𝜃₁=3𝛼²₀+𝛾²₀, 𝜃₂=𝑚, and 𝜃₃=𝜃²₁+𝜃²₂. The following expansions are also obtained:𝐹^′(0)=−𝛼₀+𝛽⁻¹−𝛼₁+𝛼₀𝑓₁₀+𝛾₀𝑔₁₀+𝒪𝛽⁻²,𝐺^′(0)=−𝛾₀+𝛽⁻¹−𝛾₁−𝛾₀𝑓₁₀+𝛼₀𝑔₁₀+𝒪𝛽⁻²,𝑉(∞)=𝑣₀₀+𝛽⁻¹𝑣₀₁+𝑣₁₀+𝒪𝛽⁻². (10a)(10b)(10c)

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 𝑉_𝑠.

Figure 2 

Radial surface shear 𝐹′(0) versus 𝛽, with different values of 𝑛; 𝑚=1, 𝑉𝑠=0.

Figure 3 

Azimuthal surface traction −𝐺′(0) versus 𝛽, with different values of 𝑚; 𝑛=1, 𝑉𝑠=0.

Figure 4 

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 𝑚𝜏=4𝜋(1/2)𝑛+²+𝑛^21/21/2𝑚+𝒪𝛽⁻² (11) which shows that, to 𝒪(𝛽⁻²), 𝜏 is independent of 𝑉_𝑠. 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=𝑛√3. 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, 𝑉_𝑠=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.

Table 1 

Period of sinusoidal oscillations 𝜏 for different values of 𝑚 and 𝑛.

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 expansions𝐹∼𝐸+𝛽⁻¹13𝛼²𝐸−𝐸²+𝛽⁻²172𝛼⁴×𝐸−8𝑉_s𝛼+23+𝐸²8𝑉_s𝛼−32+𝐸³91+⋯,𝑉∼𝛼𝑉_s𝛼−2+2𝐸−𝛽⁻¹13𝛼³(1−𝐸)²+𝛽⁻²172𝛼⁵8𝑉_s𝛼−20+𝐸−16𝑉_s𝛼+46+𝐸²8𝑉_s𝛼−32+6𝐸³+⋯ (A.1a)(A.1b)

with𝐹^′(0)1∼−𝛼+𝛽3𝛼⁻¹+𝛽⁻²136𝛼³−4𝑉_s𝑉(∞)∼1𝛼+7+⋯,𝛼𝑉_s−1𝛼−23𝛼³𝛽⁻¹+𝛽⁻²118𝛼⁵2𝑉_s𝛼−5+⋯, (A.2a)(A.2b)

where 𝛼 is determined from1∼𝛼²+𝛽⁻¹𝑉_s−1𝛼−23𝛼²𝛽⁻²+⋯. (A.3a) 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 𝛼∼1+𝛽⁻¹11−2𝑉_𝑠+𝛽⁻²18𝑉_𝑠²−13+⋯. (A.3b)

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, 𝑉_𝑠=0, through coordinate straining.