Abstract The problem of nonlinear instability of interfa-cial waves between two immiscible conducting cylindricalfluids of a weak Oldroyd 3-constant kind is studied. The sys-tem is assumed to be influenced by an axial magnetic field,where the effect of surface tension is taken into account. Theanalysis, based on the method of multiple scale in both spaceand time, includes the linear as well as the nonlinear effects.This scheme leads to imposing of two levels of the solv-ability conditions, which are used to construct like-nonlinearSchrödinger equations (l-NLS) with complex coefficients.These equations generally describe the competition betweennonlinearity and dispersion. The stability criteria are theoret-ically discussed and thereby stability diagrams are obtainedfor different sets of physical parameters. Proceeding to thenonlinear step of the problem, the results show the appear-ance of dual role of some physical parameters. Moreover,these effects depend on the wave kind, short or long, exceptfor the ordinary viscosity parameter. The effect of the fieldon the system stability depends on the existence of viscosityand differs in the linear case of the problem from the non-linear one. There is an obvious difference between the effectof the three Oldroyd constants on the system stability. Newinstability regions in the parameter space, which appear dueto nonlinear effects, are shown.
K. Zakaria · M. A. Sirwah · A. Assaf (B)Department of Mathematics, Faculty of Science,Tanta University, Tanta, Egypte-mail: [email protected]
1 Introduction
We study the flow of a fluid which is electrically conductingand moving in a magnetic field. The study of flows of thistype is known as Magnetohydrodynamics or MHD for short.The simplest example of an electrically conducting fluid is aliquid metal, for example, mercury or liquid sodium. How-ever, the major use of MHD is in plasma physics. A plasmais a hot, ionized gas containing electrons and ions. There aretwo serious technological applications of MHD, that mayboth become very important in the future. First, strong mag-netic fields may be used to confine rings or columns of hotplasma that (it is hoped) will be held in place long enough forthermonuclear fusion to occur and for net power to be gen-erated. In the second application, which is directed towarda similar goal, liquid metals are driven through a magneticfield in order to generate electricity. The study of magne-tohydrodynamics is also motivated by its widespread appli-cation to the description of space (within the solar system)and astrophysical plasmas (beyond the solar system). SomeMHD problems were discussed in Refs. [1–5]. The instabil-ity of the plane interface between two superposed fluids witha relative horizontal velocity is called the Kelvin–Helmholtzinstability. The Kelvin–Helmholtz instability due to shearflow in stratified fluids has attracted the attention of manyresearchers because of its determinant influence on the sta-bility of planetary and stellar atmospheres and in practicalapplications. The study of the Kelvin–Helmholtz instabilityhas a long history in hydrodynamics. Malik and Singh [6]investigated the nonlinear Kelvin–Helmholtz properties of(2 + 1) dimensional wave packets propagating at the interfaceof two superposed ferrofluids. They considered that the flu-ids are moving with uniform speeds parallel to the commoninterface and subjected to a tangential magnetic field. Theyderived a nonlinear equation which governs the evolution of
the amplitude of the system. The effect of a time-dependentacceleration in the presence of a tangential magnetic fieldon the nonlinear Kelvin–Helmholtz instability has been dis-cussed by El-Dib [7].
Many magnetohydrodynamic problems of practical inter-est involving fluids as a working medium have attractedengineers, physicists and mathematicians as well. Theseproblems pose challenges to cope with the non-linearity ofthe governing equations, field coupling, and complex bound-ary conditions. Further, using Newtonian fluid models toanalyse, predict and simulate the behaviour of viscoelasticfluids has been widely adopted in industries. However, theflow characteristics of viscoelastic fluids are quite differentfrom those of Newtonian fluids. This suggests that in prac-tical applications the behaviour of viscoelastic fluids cannotbe represented by that of Newtonian fluids. Hence, it is nec-essary to study the flow behaviour of viscoelastic fluids inorder to obtain a thorough cognition and improve the utiliza-tion in various manufactures. In this paper, the magnetohy-drodynamic flow of an Oldroyd 3-constant fluid is tackled.The fluid is conducting and the effect of an applied mag-netic field is considered. The presented analysis is of inter-est because the theoretical study of magnetohydrodynamic(MHD) channel flows has widespread applications in design-ing cooling systems with liquid metals, MHD generators,accelerators, pumps and flow meters. Owing to the couplingof the equations of fluid mechanics and electrodynamics, theequations governing MHD non-Newtonian flows are rathercumbersome, and analytic solutions are rare. Various authors[8–13] considered an Oldroyd 3-constant model. Recently,Baris [14] examined the hydrodynamic flow of an Oldroyd6-constant fluid in the absence of magnetic field. The problemdealing with the steady and slow flow in the wedge betweentwo intersecting planes, one of them is fixed and the otheris moving, was analysed. Using truncated series expansionsand a polar coordinate system, the governing equations ofthe problem were solved analytically subject to the relevantboundary conditions. Strictly speaking, such approximatesolutions are valid only in the domain far from the cornerformed by the two planes. In Ref. [15], the magnetohydro-dynamic flow of an Oldroyd 6-constant fluid is tackled. Itis concerned with steady Couette, Poiseuille and general-ized Couette flows. The fluid is electrically conducting bya transverse magnetic field. The non-linear developed gov-erning equation is solved analytically and numerically sub-ject to appropriate boundary conditions. A parametric studyof the physical parameters involved in the problems suchas the applied magnetic field and the material constants isconducted. The obtained results are illustrated graphically toshow salient features of the solutions. Numerical results showthat the applied magnetic field tends to reduce the flow veloc-ity. Depending on the choice of the material parameters, thefluid exhibits shear-thickening or shear-thinning behaviour.
In the recent years a great deal of interest has been focusedon understanding the rheological effects occurring in the flowof non-Newtonian fluids through porous media. This prob-lem appears to be, at this time, of special interest in oil reser-voir engineering, where an increasing interest is being shownin the possibility of improving oil recovery efficiency fromwater flooding projects through mobility control with non-Newtonian displacing fluids. Consequently, it has becomeessential to have an adequate understanding of the rheo-logical effect of non-Newtonian displacing and displacedfluids in an oil displacement mechanism. Many technolog-ical processes involve the parallel flow of fluids of differ-ent viscosity, elasticity and density through porous media.Such flows exist in packed bed reactors in the chemicalindustry, petroleum engineering, boiling in porous mediaand in many other processes. Should the interface betweenthe two fluids become unstable, a substantial increase in theresistance to the flow would result. This increase in resis-tance, in turn, may cause flooding in counter current packedchemical reactors and dryout in boiling porous media. Inthe same vein, in petroleum production engineering, suchinstabilities lead to emulsion formation. Hence, the knowl-edge of the conditions for the onset of instability will enableus to predict the limiting operation conditions of the aboveprocesses. The flow through porous media is of consider-able interest for petroleum engineers and in geophysical fluiddynamicists. The instability of a plane interface between twosuperposed fluids of different densities, whenever occurs, iscalled the Rayleigh–Taylor instability. The problem of theKelvin–Helmholtz instability of Oldroydian viscoelastic flu-ids in a porous medium finds its use in chemical technologicaland geophysical fluid dynamics. It is also assumed to be moresuitable for use in the oil industry. In Ref. [16], the Kelvin–Helmholtz instability of two semi-infinite Oldroydian flu-ids in a porous medium has been considered. The systemis influenced by a vertical electric field. Stability diagramsare obtained for different sets of physical parameters. Newinstability regions in the parameter space, which appear dueto nonlinear effects, are shown. Recently, Hayat et al. [17–26]studied different types of the problem of MHD Oldroydianfluid flow. In Ref. [17], some different patterns of an Oldroyd6-constant fluid flow were investigated analytically such asCouette and Poiseuille flows. It was concluded that the effectof non-Newtonian parameters on the flow appears only inthe case of unsteady state. An incompressible conductingOldroyd-B fluid contained between two non-conducting par-allel plates was demonstrated in Ref. [18] as the upper platewas assumed to oscillate in its plane. The system was ini-tially stressed by a uniform transverse magnetic field. Theanalytic solution of the boundary value problem was con-structed and then the velocity field was determined ana-lytically and hence investigated numerically. On the otherhand, a study of the two-dimensional peristaltic motion of an
MHD instability of interfacial waves between two immiscible incompressible cylindrical fluids 499
Oldroyd-B fluid in the limiting case of long wavelength andlow frequency was performed in Ref. [19] where the analyt-ical solution, up to the second order, was obtained. It wasdeduced that these solutions are valid for all values of bothReynolds and Weissenberg numbers. In Ref. [20], the prob-lem of MHD flow of an Oldroyd-B fluid in a circular pipethrough porous medium was discussed. An exact solutionis constructed by means of the fractional calculus approach.The numerical results showed that the effect of the permeabil-ity of the porous medium on the flow is opposite to that of themagnetic field. Moreover, such study is extended to includeHall effect on the flow in Refs. [21–23]. The problem ofMHD flow of an Oldroyd 6-constant fluid between concen-tric cylinders was performed in Ref. [24]. The differentialequations of motion together with the appropriate bound-ary conditions are obtained and consequently solved numer-ically. The results revealed that the continuous increasing inthe applied magnetic field leads to decreasing in the fluidvelocity. In Ref. [25], the authors demonstrated the steadyflow of an Oldroyd 8-constant conducting fluid between twocoaxial cylinders through a porous medium analytically bymeans of the homotopy analysis method (HAM) and numer-ically by the finite difference method. HAM was exploitedto solve the non-linear boundary value problems of both theOldroyd 6-constant fluid and fluid-slip in Ref. [26]. It wasfound that the velocity increases with the increase of the slipcoefficient in the case of Poiseuille flow model, while thisvelocity decreases in the vicinity of the moving plate andincreases near the fixed plate with the increment of the slipcoefficient in the Couette flow case.
A generalization of the nonlinear instability for Oldro-dyian viscoelastic cylindrical conducting flow in the pres-ence of an axial magnetic field is the goal of this study. In thispaper, the nonlinear boundary conditions are formulated, andthen a multiple-scale analysis is presented which involves theexpansion of all physical quantities at the deformed interface.The resulting problem in defining an adjoint operator and thesubsequent consequences, in order to obtain the stability cri-teria, are the main purpose of this paper. It is organized asfollows: in Sect. 2, the governing equations and the nonlinearboundary conditions are emphasized and the multiple-scaletechnique and the related solvability conditions is introduced.The linear solutions, according to the nonlinear boundaryconditions, and the linear stability analysis are recapitulatedin Sect. 3. Section 4 is devoted to the derivation of the nonlin-ear evolution equation governing the complex amplitude ofthe system. Consequently, the stability criteria are discussed.
2 Definition of the problem
Consider two incompressible immiscible perfectly conduct-ing streaming fluids between two cylinders. Also, consider
an injection velocity through the two cylinders. The motionis limited to two dimensions. The two fluids are separatedwith the interface r = R. The inner fluid occupies the spacea(1) < r < R whereas R < r < a(2) is taken by the outerfluid. The inner and outer fluids are moving with uniformvelocities u(1)
0 and u(2)0 along the z axis, respectively. The sys-
tem is subjected to a uniform axial magnetic field (0, 0, h0)
permeating both fluids. The inner fluid is inviscid, while theouter one has an Oldroydian viscoelastic property at the inter-face surface.
The distance and the time are made dimensionless usingR and R/u0, respectively, where u0 is the injection velocityand R is the radius of the unperturbed surface. The quan-tities ρ(1) and ρ(2) are the densities of the inner and outerfluids, respectively. The quantities a(1) and a(2) are the radiiof the inner and outer cylinders, respectively. The velocitiesare made dimensionless by u0.
The basic equations that govern the motion with the bound-ary conditions for the two fluids are:
where j = 1 and 2 correspond to the inner and outer regions,respectively,
T (2)mn = µ
(1 + λ1
(∂
∂t+ u(2) · ∇
))−1
×(
1 + λ2
(∂
∂t+ u(2) · ∇
)) (∂u(2)
m
∂xn+ ∂u(2)
n
∂xm
)
is the viscoelastic shear stress tensor, ρ = ρ(2)/ρ(1), W =τ/ρ(1) Ru2
0 is the dimensionless Weber number, µ = 12 qµ is
the viscosity coefficient, τ is the surface tension, u( j)m is the
velocity vector, xm is the position vector, h( j) equals the mag-netic induction divided by the square root of the product ofdensity and permeability which has been called the magneticfield for simplification, n is the outward-drawn normal, η isthe elevation of the free surface above the unperturbed level,
π( j) = p( j)
ρ( j) + 12 h( j)2
is the total pressure, λ1 is the stressrelaxation time considered a small quantity so the nonlinear
terms in it can be neglected and λ2 is the strain retardationtime.
We expand all boundary conditions at the interface r = 1+η(z, t) about r = 1 using Taylor’s series. To investigate thenonlinear finite interfacial waves, we use the multiple scalemethod in the time t and the dimension z , since tn = qnt ,zn = qnz, n = 0, 1, 2. The elevation then takes the form
η(z0, z1, z2, t0, t1, t2, q)
=3∑
�=1
q�η�(z0, z1, z2, t0, t1, t2) + O(q4),
and
f ( j)(r, z0, z1, z2, t0, t1, t2, q)
= f ( j)0 +
3∑�=1
q� f ( j)� (r, z0, z1, z2, t0, t1, t2) + O(q4)
stands for any of the physical quantities π( j), u( j)r , u( j)
z , h( j)r
or h( j)z , with u( j)
r0 = h( j)r0 = 0, u( j)
z0 = u( j)0 and h( j)
z0 = h0.The parameter q represents a small dimensionless parametercharacterizing the steepness ratio of the wave. The uniformlyvalid solutions of the resulting equations depend on removingthe secular terms which give the following equations for thesecond and the third-order, respectively:
kµ((
λ1 − λ2)β0 + i
(β1 + λ1λ2β2
))A
−iSω
∂ A
∂t1+ iSk
∂ A
∂z1= 0, (9)
−iSω
∂ A
∂t2+ 1
2Sωω
∂2 A
∂t12 + (iβ7 + β8
)∂ A
∂t1
+iSk∂ A
∂z2+ (
iβ0 + β1) ∂ A
∂z1+ 1
2Skk
∂2 A
∂z12
−Skω
∂2 A
∂z1∂t1+ β4 A2 A = 0, (10)
where
Sk = ∂S
∂k, Sω = ∂S
∂ω, Skk = ∂2S
∂k2 ,
Sωω = ∂2S
∂ω2 , Skω = ∂2S
∂k∂ω,
β7 = −µ(λ1 − λ2
)β8, β8 = µ
(λ1λ2β9 + β
),
β0 = −µ(λ1 − λ2
)β5, β1 = µ
(β6 + λ1λ2β7
),
and the values of the coefficients involved in these equationsare lengthy and are available from the authors upon request.In the special case λ1 = λ2 = 0, the nonsecularity conditions
for the second and the third-order take the forms, respectively,
kµβ1 A − Sω
∂ A
∂t1+ Sk
∂ A
∂z1= 0,
−iSω
∂ A
∂t2+ 1
2Sωω
∂2 A
∂t12 + µβ∂ A
∂t1+ iSk
∂ A
∂z2
+µβ6∂ A
∂z1+ 1
2Skk
∂2 A
∂z12 − Skω
∂2 A
∂z1∂t1+ β4 A2 A = 0,
which are similar to those obtained by Zakaria [27]. Whilein the special case µ = λ1 = λ2 = 0, the nonsecularityconditions for the second and the third-order take the forms,respectively ,
−Sω
∂ A
∂t1+ Sk
∂ A
∂z1= 0,
−iSω
∂ A
∂t2+ 1
2Sωω
∂2 A
∂t12 + iSk∂ A
∂z2+ 1
2Skk
∂2 A
∂z12
−Skω
∂2 A
∂z1∂t1+ β4 A2 A = 0,
which are similar to those obtained by Gill [28].
3 The linear theory
The solution of the linear theory depends on neglecting thenonlinear terms from the equations of motion as well as fromthe boundary conditions [29]. The progressive wave solutionof the first-order problem with respect to the lowest-ordervariables z0, t0 yields
MHD instability of interfacial waves between two immiscible incompressible cylindrical fluids 501
α( j)4 = α
( j)1 α
( j)2 , α
( j)5 = α
( j)1 α
( j)3 ,
α( j)6 = α
( j)4
α( j)0
, α( j)7 = α
( j)5
α( j)0
,
since
α( j)0 =ku( j)
0 − ω, α( j)1 = α
( j)0
α( j)0
2 − k2h20
, = kz0 − ωt0,
A(t1, t2, z1, z2) is the amplitude of the propagating waves andwill be determined in the following sections, k is the wavenumber, which is assumed to be real and positive, i = √−1and ω is the growth rate.
In various instability phenomena in fluids, the stabilityof a basic flow is often described by a single dimensionlessstability parameter, say the stream velocity or the viscositynumber. The dispersion relation then will be
S(k, ω) = α3ω2 + α2ω + α1 = 0, (17)
where
α1 = 1
α(1)8 α
(2)8
(kρα
(2)9
(h2
0 − u(2)0
2)α
(1)8
+((k2 − 1
)Wα
(1)8 − kα
(1)9
(h2
0 − u(1)0
2))α
(2)8
),
α2 = 2ρα(2)9 u(2)
0
α(2)8
− 2α(1)9 u(1)
0
α(1)8
,
α3 = 1
k
(α
(1)9
α(1)8
− α(2)9
α(2)8
),
since α( j)8 = I1(k)K1(ka( j)) − K1(k)I1(ka( j)), α
( j)9 =
I1(ka( j))K0(k) + I0(k)K1(ka( j)). It is concluded, fromEq. (17), that the system will be linearly stable if the growthrate has only real values , which in turn leads to the stabilitycondition
α22 − 4α1α3 > 0, or (u(1)
0 − u(2)0 )2 < uc, (18)
where
uc = ρα1 + α2
ρkα1α2
(W
(k2 − 1
)α3 + kh2
0(ρα1 + α2)), (19)
since
α1 = α(2)9 α
(1)8 , α2 = −α
(1)9 α
(2)8 , α3 = α
(1)8 α
(2)8 .
As special cases of Eq. (17) we consider now the non stream-ing case, when u(1)
0 = u(2)0 = 0, the case of the absence of
the magnetic field and the case which takes the two condi-tions together. In case of the absence of the magnetic field,the stability condition takes the form
(u(1)0 − u(2)
0 )2 <ρα1 + α2
ρkα1α2
(W
(k2 − 1
)α3
).
In the non streaming case, the stability condition takes the
formα
(1)8 α
(2)8 (k2−1)W
α(2)8 α
(1)9 −α
(1)8 α
(2)9 ρ
− h02k > 0, which is similar to that
obtained by Lee [30]. If we consider h0 = 0 in the nonstreaming case, the stability condition will then take the form
α(1)8 α
(2)8 (k2 − 1) > 0, and α
(2)8 α
(1)9 − α
(1)8 α
(2)9 ρ > 0,
or
α(1)8 α
(2)8 (k2 − 1) < 0, and α
(2)8 α
(1)9 − α
(1)8 α
(2)9 ρ < 0.
We study in the different parts of Fig. 1, of the linear the-ory, the influence of the magnetic field h0, the dimensionlessnumber W and the thickness of the fluid layer between thetwo cylinders. In all parts, the transition curve is drawn asa relation between the wave number k and the square of thevelocity difference (u(1)
0 − u(2)0 )2. The resulting curve repre-
sents the critical curve, at uc = (u(1)0 −u(2)
0 )2. The area under
the curve represents the stable region uc < (u(1)0 − u(2)
0 )2,whereas the area above the curve represents the unstableregion uc > (u(1)
0 −u(2)0 )2. In all figures, the value ρ = 1.1 is
taken. In Fig. 1a, the transition curve of the dispersion rela-tion (17) is drawn at W = 0.1, a(1) = 0.1 and a(2) = 1.1. Thevalues 1.5, 2 and 2.5 are taken separately for the magneticfield h0. In Fig. 1b, the transition curve is drawn at h0 = 1.5,W = 0.1 and a(2) = 1.1. The values 0.1, 0.5 and 0.9 aretaken separately for the radius a(1). In Fig. 1c, the transitioncurve is drawn at h0 = 1.5, a(1) = 0.1 and a(2) = 1.1. Thevalues 0.1, 1 and 2 are taken separately for the number W .
In order to investigate the effect of the magnetic field h0
on in the linear problem, one can notice the three curves ofFig. 1a. It is showed that the lower area S increases as thevalue of h0 increases, and the upper area U decreases in thesame direction. Therefore, h0 represents a stabilizing effectin Fig. 1a. So h0 plays a stable role in the linear problem.
To know the effect of the thickness of the fluid layerbetween the two cylinders on the stability of the linear prob-lem, one can compare the various curves in Fig. 1b, and it isshowed that the increase in the inner cylinder radius a(1) pro-duces a corresponding increase in the unstable area U and,in the same time, a corresponding decrease in the stable areaS. Therefore the parameter a(1) has an instabilizing effect incase of Fig. 1b. Then one can say that the radius of the innercylinder a(1) plays an unstable role in the linear problem,i.e. the thickness of the fluid layer between the two cylindersplays a stable role in the linear problem.
To discuss the effect of the dimensionless number W onthe system stability, one can compare the curves in Fig. 1c.The decrease of the lower area S and the increase of theupper area U due to the increase of the number W in therange 1 ≤ k ≤ 10 will be clear, while the contrast occurs inthe range 0 < k < 1. Then the parameter W plays a dualrole in the linear case of the stability problem depending onthe wave kind, whether it is long or short.
Fig. 1 The curves represent the transition value as in Eq. (17) for a system having ρ = 1.1, a(1) = 0.1, a(2) = 1.1, W = 0.1 and h0 = 1.5. In theparts (a), (b) and (c) three different values are taken for h0, a(1) and W , respectively
Hence, it is concluded that in the linear problem, both themagnetic field h0 and the thickness of the fluid layer betweenthe two cylinders play a stable role, while the dimensionlessnumber W plays a dual role in the system stability.
4 Stability of nonlinear waves
This section is devoted to the derivation of the nonlinearevolution equation governing the amplitude. Consequently,the stability criteria are discussed. In all of the figures whichstudy the nonlinear problem, the values 0.1, 1.1, 0.001, 1.1,6, and 3.7 are taken as fixed values for the parameters a(1),a(2), q, ρ, u(1)
0 and u(2)0 , respectively.
4.1 Stability away from the neutral curve
Away from the marginal state (Sω �= 0), one can simplifyEqs. (9) and (10) to take the following form:
−(
i
(qβ7
Sω
+ 1
)+ qβ8
Sω
)∂ A
∂t− 1
2ω′′ ∂2 A
∂z2 + (iγ0 + γ1
)A
+(i(γ2 − ω′) + γ3
)∂ A
∂z− q2β4
Sω
A2 A = 0, (20)
where
ω′ = dω
dk= −Sk/Sω,
ω′′ = d2ω
dk2 = 1
Sω
(−Skk + 2Skω Sk
Sω
− Sωω Sk2
Sω2
),
γ0 = − q2k2µ2(λ1 − λ2
)β0
(β1 + λ1λ2β2
)Sωω
S3ω
,
γ1 = q2k2µ2((
λ1−λ2)β0+β1+λ1λ2β2
)((λ2−λ1
)β0+β1+λ1λ2β2
)Sωω
2S3ω
,
γ2 = q(kµ
(λ1 − λ2
)β0
(Sωω Sk − Sω Skω
) − β0 S2ω
)S3ω
,
γ3 = − q(β1
(Sω
)2 + kµ(Sωω Sk − Sω Skω
)(β1 + λ1λ2β2
))S3ω
.
4.1.1 The stationary solution
In this section the stability for the stationary solution ofEq. (20) is examined. This solution is assumed to be A = a,
where a2 = γ1Sω/(q2β4).
The stability criterion of the steady state can be determinedby superposing small perturbations on the unperturbed solu-tion by letting
A(z, t) = (a + a1(z, t))exp{iδ1(z, t)}, (21)
where a1(z, t) and δ1(z, t) are small real quantities. The con-ditions for a to be real are
γ1Sω/β4 > 0, and λ1 = λ2.
Substitute Eq. (21) into Eq. (20), neglect the nonlinear termsin a1(z, t) and δ1(z, t), and hence seek the solutions of theperturbed functions in the form
where a1 and δ1 are real quantities. Then � will satisfy thefollowing dispersion relation
�2 + (γ6 + iγ7
)� + γ4 + iγ5 = 0, (23)
where
γ4 = k21 Sω
((k2
1(ω′′)2 + 2γ1ω′′ − 4
(ω′ − γ2
)2 − 4γ 23
)Sω − 6a2q2ω′′β4
)4((
β27 + β2
8
)q2 + 2Sωβ7q + S2
ω
) ,
γ5 = k1 Sω
(Sωγ0
(γ2 − ω′) + ((
ω′′k21 + γ1
)Sω − 3a2q2β4
)γ3
)(β2
7 + β28
)q2 + 2Sωβ7q + S2
ω
,
γ6 = β8(3a2q3β4 − q
(ω′′k2
1 + γ1)Sω
) − γ0 Sω
(Sω + qβ7
)(β7
2 + β82)q2 + 2Sωβ7q + S2
ω
,
γ7 = 2k1 Sω
(Sω
(ω′ − γ2
) + qβ7(ω′ − γ2
) − qβ8γ3)
(β7
2 + β82)q2 + 2Sωβ7q + S2
ω
.
The eigenvalue � can be written as �r + i�i , where the realpart is the growth rate and the imaginary one is the angularfrequency. If �r < 0, the state is stable, but if �r > 0, itis unstable. According to Zakaria [27], the roots of Eq. (23)have negative real parts and hence the system will be stableif and only if
γ6 > 0, −γ52 + γ6(γ7γ5 + γ4γ6) ≥ 0, (24)
provided that γ1Sω/β4 > 0 and λ1 = λ2.In a convenient form, condition (24) is transformed to the
where b1(z, t) and δ1(z, t) are small real quantities. Substi-tuting Eq. (28) into Eq. (20) and neglecting the nonlinearterms in b1(z, t) and δ1(z, t) such that
where b1 and δ1 are real quantities, we have the characteristicequation
�22 + (
γ0 + iγ1)�2 + γ8 + iγ9 = 0, (30)
where
γ8 = (k2
3 Sω
(2( − q
(3qβ4b2 + �1β7
)−(
�1 − γ1 + 3k2(ω′ − γ2
))Sω
)ω′′
−4((
ω′ − γ2)2 + γ 2
3
)Sω − Sω
(3k2
2 − k23
)ω′′2))
/(4((
β27 + β2
8
)q2 + 2Sωβ7q + S2
ω
)),
γ9 = (k3Sω
( − 2q�1β8(ω′ + k2ω
′′ − γ2)
−(2γ0
(ω′ + k2ω
′′ − γ2) + ((
k22 − 2k2
3
)ω′′
+2(�1 − γ1
))γ3
)Sω − 2q
(3qβ4b2 + �1β7
)γ3
))/(
2((
β27 + β2
8
)q2 + 2Sωβ7q + S2
ω
)),
γ0 = [qβ8
(6b2q2β4 − (
2k2ω′ + (k2
2 + 2k23)ω′′
+2γ1 − 2k2γ2)Sω
) − 2(γ0 + k2γ3
)Sω
(Sω + qβ7
)]/[
2((
β27 + β2
8
)q2 + 2Sωβ7q + S2
ω
),
γ1 = [2k3Sω
(Sω
(ω′ + k2ω
′′ − γ2)
+qβ7(ω′ + k2ω
′′ − γ2) − qβ8γ3
)]/[(
β72 + β8
2)q2 + 2Sωβ7q + S2ω
].
None of the roots of Eq. (30) has positive real part and hencethe system will be stable if and only if
γ0 > 0, −γ92 + γ0(γ9γ1 + γ8γ0) ≥ 0, (31)
subject to
(−2β7q�1 + Sω(2γ1 − 2�1
+k2(−2γ2 + 2ω′ + k2ω′′)))/β4 > 0.
In a convenient form, condition (31) is transformed to thefollowing:With
(−2β7q�1 + Sω(2γ1 − 2�1 + k2(−2γ2 + 2ω′
+k2ω′′)))/β4 > 0,
we have
ϑ3k32 + ϑ4 > 0,(
β72 + β8
2)q2 + 2Sωβ7q + S2
ω > 0,
ϑ5k36 + ϑ6k3
4 + ϑ7k32 + ϑ8 ≥ 0,
(32)
or
ϑ3k32 + ϑ4 < 0,(
β72 + β8
2)q2 + 2Sωβ7q + S2
ω < 0,
ϑ5k36 + ϑ6k3
4 + ϑ7k32 + ϑ8 ≥ 0,
(33)
where ϑ3, ϑ4, ϑ5, ϑ6, ϑ7 and ϑ8 are functions of the parame-ters of the problem except for k3.
In Figs. 2–5, the transition curve of Eq. (30) is drawn atk = 2 in the plane k3 − W . In the following we take intoconsideration the parameters h0, µ, λ1 and λ2. In Fig. 2, thethree parameters µ, λ1 and λ2 are fixed while h0 changesamong the values 1.5, 1.75 and 2, respectively. In Fig. 3 thefixed parameters are h0, λ1 and λ2, while µ takes the values0.7, 1.5 and 5. In Fig. 4 the parameter λ1 varies among thevalues 3 × 10−6, 0.07 and 0.1, whereas the remaining para-meters are fixed. Finally, in Fig. 5 the parameter λ2 changesamong the values 3 × 10−6, 0.07 and 0.3, whereas the otherparameters do not change. In Fig. 2 we notice that the areas Uand S2 decrease whereas the area S1 increases as h0 increasesfrom 1.50 to 2.00. In Fig. 3 we can find that the areas U andS2 increase whereas the area S1 decreases, as µ increases
123
504 K. Zakaria et al.
Fig. 2 The curves represent the transition value as in Eq. (30) for a system having a(1) = 0.1, a(2) = 1.1, q = 0.001, ρ = 1.1, u(1)0 = 6,
u(2)0 = 3.7, µ = 0.7, λ1 = 3 × 10−6, λ2 = 0.3 and k = 2 with three different values of h0
Fig. 3 The same system ofFig. 2 is considered but withh0 = 1.5 and with threedifferent values of µ
Fig. 4 The same system ofFig. 2 is considered but withh0 = 1.5 and with threedifferent values of λ1
from 0.7 to 5.0. It is shown in Fig. 4 that the areas U andS2 decrease whereas the area S1 increases, as λ1 increasesfrom 3×10−6 to 0.1. In Fig. 5, as the parameter λ2 increases,the unstable area U increases and the stable areas S1 and S2
decreases. Hence h0, λ1 and µ play a dual role on the systemstability in case of short waves, while the parameter λ2 hasan instabilizing effect in this case. To investigate the effectof the wave length on the system stability, the value k = 0.1which indicates the case of long waves is taken in Figs. 6–9.In these figures we start with the values h0 = 1.5, µ = 0.7,λ1 = 3×10−6 and λ2 = 0.3. In Fig. 6, the field only changesamong three values; namely 1.5, 1.75 and 2. In Fig. 7 the vis-cosity µ only changes among the values 0.7, 1.5 and 5. InFig. 8, λ1 only changes among the values 3×10−6, 0.07 and0.1. In Fig. 9, λ2 only changes among the values 3 × 10−6,0.07 and 0.3. Also by comparing the areas in Figs. 2–5 andthe corresponding ones in Figs. 6–9, respectively, we noticethe increase of the stable area and the decrease of the unsta-ble area with the increase of the wave number. Also h0, µ,λ1 and λ2 play a dual role on the system stability in case oflong waves.
In what follows, the special case of the absence ofviscosity is considered. Setting µ = 0 in Eq. (30) and hencereplacing −i�2 by �2 we have the following dispersionrelation
�22 + γ1�2 − γ0 = 0, (34)
where {γ0, γ1} = {γ8, γ1} at µ = 0. We recall that the con-dition of stability in our case is that the discriminate of thequadratic Eq. (34) is positive, i.e.
4γ0 + γ 21 > 0,
or in a convenient form:
ω′′ > 0, (4�1 − 4k2ω′ − 2k2
2ω′′ + k32ω′′) > 0, (35)
or
ω′′ < 0, (4�1 − 4k2ω′ − 2k2
2ω′′ + k32ω′′) < 0. (36)
In addition, we notice that all terms containing λ1 or λ2 whenµ = 0 disappear.
In Fig. 10, the transition curve of Eq. (34) is plotted atk = 2 as a relation between W on the x-axis and k2
3 onthe y-axis. Only the field is changed among 1.5, 1.75 and2. In Fig. 10 we notice that as h0 increases, the stable areasincreases. Hence we see that h0 plays a stable role on thesystem stability in case of short waves. To discuss the effectof the wavelength on the stability of the system, the valuek = 0.1 which refers to the case of long waves is taken inFig. 11. In this figure we also change the field h0 only amongthree values 1.5, 1.75 and 2. Comparing the areas in Fig. 10and the corresponding ones in Fig. 11, we notice the increaseof the stable area and the decrease of the unstable area withthe increase of the wave number. Also h0 plays a stable roleon the system stability in case of long waves.
123
MHD instability of interfacial waves between two immiscible incompressible cylindrical fluids 505
Fig. 5 The same system ofFig. 2 is considered but withh0 = 1.5 and with threedifferent values of λ2
Fig. 6 The same system ofFig. 2 is considered but withk = 0.1
Fig. 7 The same system ofFig. 3 is considered but withk = 0.1
Fig. 8 The same system ofFig. 4 is considered but withk = 0.1
Fig. 9 The same system ofFig. 5 is considered but withk = 0.1
4.2 Stability on the neutral curve
On the marginal state (Sk �= 0), we can simplify Eqs. (9) and(10) to take the following form:
i∂ A
∂z− 1
2k′′ ∂2 A
∂t2 + (iγ2 + γ3
)A
+(iγ4 + γ5
)∂ A
∂t− q2β4
SkA2 A = 0, (37)
where
k′ = dk
dω= −Sω/Sk,
k′′ = d2k
dω2 = 1
Sk
(2SkωSω
Sk− Skk S2
ω
S2k
− Sωω
),
γ2 = 1
S3k
(q2kµ
(((λ1 − λ2
)β0β1 − (
β1 + λ1λ2β2)β0
)Sk
−kµ(λ1 − λ2
)β0
(β1 + λ1λ2β2
)Skk
)),
γ3 = 1
2S3k
(q2kµ
( − kµ((
λ1 − λ2)β0
+β1 + λ1λ2β2)Skk
((λ1 − λ2
)β0 − β1 − λ1λ2β2
)−2Sk
((λ1 − λ2
)β0β0 + (
β1 + λ1λ2β2)β1
))),
γ4 = 1
S3k
(q( − β7S2
k + kµSkω
(λ1 − λ2
)β0Sk
+Sω
( − β0)Sk − kµ
(λ1 − λ2
)β0SωSkk
)) + k′,
γ5 = − 1
S3k
(q(β8S2
k + Sωβ1Sk
+ kµ(Sk Skω − SωSkk
)(β1 + λ1λ2β2
))).
123
506 K. Zakaria et al.
Fig. 10 The curves represent the transition value as in Eq. (34) for asystem having the values a(1) = 0.1, a(2) = 1.1, q = 0.001, ρ = 1.1,u(1)
0 = 6, u(2)0 = 3.7, k2 = 1 and k = 2 with three different values of
h0
Fig. 11 The same system of Fig. 10 is considered but with k = 0.1
In this position the stability for the stationary solution ofEq. (37) is examined. This solution is assumed to be
A = a,
where a2 = Sk γ3/(q2β4). The stability criterion of the steadystate can be determined by superposing small perturbationson the unperturbed solution by letting
A(z, t) = (a + a1(z, t))exp{iδ1(z, t)}, (38)
where a1(z, t) and δ1(z, t) are small real quantities. The con-ditions for a to be real are
Sk γ3/β4 > 0 and λ1 = λ2. (39)
Substituting Eq. (38) into Eq. (37), neglecting the nonlinearterms in a1(z, t) and δ1(z, t), and hence seeking the solutionsof the perturbed functions in the form
where a1 and δ1 are real quantities. Then � will satisfy thefollowing dispersion relation
�4 + β4�3 + β3�
2 + (β1 + iβ2)� + β0 = 0, (41)
where
β0 = −4k21
k′′2 ,
β1 = 4Sk γ2γ4 + 4(γ3Sk − 3a2q2β4
)γ5
k′′2Sk,
β2 = 8k1γ4
k′′2 ,
β3 = 2((
3a2q2β4/Sk − γ3)k′′ + 2
(γ 2
4 + γ 25
))k′′2 ,
β4 = −4γ5
k′′ .
The eigenvalue � can be written as �r + i�i , where the realpart is the growth rate and the imaginary one is the angularfrequency. If �r < 0 the state is stable, but if �r > 0 it isunstable. The roots of Eq. (41) have negative real parts if andonly if
β4 > 0,
β3β24 − β1β4 > 0,
β4(β31 − 2β3β4β
21 − (β2
2 + β0β3)β34
+(β1β23 + β0β1)β
24 ) > 0,
(−β20 β2
2 β84 − β0(β
31 − 2β3β4β
21
−(β22 + β0β3)β
34 + (β1β
23 + β0β1)β
24 )
×(β0β34 + β1(β1β4 − β3β
24 ))β2
4 ) > 0,
(42)
subject to Sk γ3/β4 > 0 and λ1 = λ2.
Here, the stability for the periodic solution of Eq. (37) isexamined. This solution is assumed to be
A(z, t) = b exp{i(k2z − �1t)}, (43)
where
b2 = − Sk
2q2β4(−k′′�2
1 − 2γ4�1 + 2k2 − 2γ3),
�1 = λ2/λ5.
The stability criterion of our state can be determined by super-posing small perturbations on the unperturbed solution byletting
where b1(z, t) and δ1(z, t) are small real quantities. Substi-tuting Eq. (44) into Eq. (37) and neglecting the nonlinearterms in b1(z, t) and δ1(z, t) such that
MHD instability of interfacial waves between two immiscible incompressible cylindrical fluids 507
where b1 and δ1 are real quantities, we have the characteristicequation
�24 + γ�2
3 + γ9�22 + (γ7 + iγ8)�2 + γ6 = 0, (46)
where
γ6 = −4k23
k′′2 ,
γ7 = 1
k′′2Sk
(4Sk γ2
(�1k′′ + γ4
) + 2( − 6b2β4q2
−(k′′�2
1 + 2k2 − 2γ3)Sk
)γ5
),
γ = −4γ5
k′′ , γ8 = 8k3(�1k′′ + γ4
)k′′2 ,
γ9 = 1
k′′2Sk
(6b2k′′β4q2 + Sk
(3�2
1(k′′)2
+2(k2 − γ3 + 3�1γ4
)k′′ + 4
(γ 2
4 + γ 25
))).
The roots of Eq. (46) have negative real parts and hence thesystem will be stable if and only if
γ > 0, γ9γ2 − γ7γ > 0,
γ (γ 37 − 2γ9γ γ 2
7 − (γ 28 + γ6γ9)γ
3
+(γ7γ29 + γ6γ7)γ
2) > 0,
(−γ 26 γ 2
8 γ 8 − γ6(γ37 − 2γ9γ γ 2
7 − (γ 28 + γ6γ9)γ
3
+(γ7γ29 + γ6γ7)γ
2)(γ6γ3 + γ7(γ7γ − γ9γ
2))γ 2) > 0,
(47)
provided that
(k′′�21 + 2γ4�1 − 2k2 + 2γ3)Sk/β4 > 0.
In Figs. 12–15, the transition curve of Eq. (46) is plottedin the W − k3 plane at k = 2. In Fig. 12, the parameter h0 ischanged while the parameters µ, λ1 and λ2 are fixed. Thesevalues of h0 are 1.5, 1.75 and 2. In Fig. 13, the parametersh0, λ1 and λ2 are fixed, whereas the parameter µ changes
Fig. 12 The curves represent the transition value as in Eq. (46) for asystem having the values a(1) = 0.1, a(2) = 1.1, q = 0.001, ρ = 1.1,u(1)
Fig. 13 The same system of Fig. 12 is considered but with h0 = 1.5and with three different values of µ
Fig. 14 The same system of Fig. 12 is considered but with h0 = 1.5and with three different values of λ1
Fig. 15 The same system of Fig. 12 is considered but with h0 = 1.5and with three different values of λ2
among the values 0.7, 1.5 and 5. In Fig. 14 the parametersµ, h0 and λ2 are fixed, while λ1 changes among the values3×10−6, 0.07 and 0.1. In Fig. 15 the only changed parameteris λ2. Its values are 3 × 10−6, 0.07 and 0.3, respectively. InFig. 12 we notice that as the parameter h0 increases, the areaS increases. In Fig. 13 we notice that the increase of theparameter µ increases the stable area S. In Fig. 14 we notice
123
508 K. Zakaria et al.
Fig. 16 The same system of Fig. 12 is considered but with k = 0.1
Fig. 17 The same system of Fig. 13 is considered but with k = 0.1
that the increase of λ1 decreases the stable area S. In Fig. 15we notice that the increase of λ2 increases the stable area inthis case. Hence h0, λ2 and µ play a stable role on the systemstability in case of short waves, while the parameter λ1 has aninstabilizing effect in this case. To investigate the effect of thewave length on the system stability, the value k = 0.1 whichindicates the case of long waves is taken in Figs. 16–19. Inthese figures we start with the values h0 = 1.5, µ = 0.7,λ1 = 3 × 10−6 and λ2 = 0.3. In Fig. 16, the field changesonly among 1.5, 1.75 and 2. In Fig. 17 the viscosity µ changesonly among the values 0.7, 1.5 and 5. In Fig. 18, λ1 changesonly among the values 3 × 10−6, 0.07 and 0.1. In Fig. 19, λ2
changes only among the values 3 × 10−6, 0.07 and 0.3. Alsoby comparing the areas in Figs. 12–15 and the correspondingones in Figs. 16–19, we notice the increase of the stable areaand the decrease of the unstable area with the increase of thewave number. Also h0, λ2 and µ play a stable role on thesystem stability in case of long waves, while the parameterλ1 has an instabilizing effect in this case.
In the following, the special case of the absence of viscos-ity is considered. Let us replace −i�2 by �2 in Eq. (46) tohave the following dispersion relation
�42 − γ3�
22 − γ2�2 + γ6 = 0, (48)
Fig. 18 The same system of Fig. 14 is considered but with k = 0.1
Fig. 19 The same system of Fig. 15 is considered but with k = 0.1
where {γ2, γ3} = {γ8, γ9} when µ = 0. Thus in order toobtain the stability criterion, all roots of the last dispersionrelation must be real together. In Figs. 20–22, the four rootsof Eq. (48) are plotted in case of short waves as functions ofthe variable W , and the field is changed among the values1.5, 1.75 and 2. In Figs. 20–22, the solid curve, the dottedcurve, the dotted-dashed curve and the dashed curve repre-sent the four roots of Eq. (48). We recall that in our specialcase the system is stable when all of the four roots of Eq. (48)appear together on the plot, while the system is unstable if atleast two of the roots are complex. In addition, we notice thatall terms containing λ1 or λ2 when µ = 0. Investigating thecurves shown in Fig. 20 we notice that the four curves appeartogether in two ranges S1 and S2; namely 1 < W < 3.6 and4.2 < W < 4.5, but this does not happen in the remainingranges. So the system stability takes place in the two rangesS1 and S2, whereas the instability occurs in the other ranges.Noticing the curves shown in Fig. 21 we see that the fourcurves appear together in two ranges S1 and S2; namely 1.5 <
W < 5.3 and 6.25 < W < 6.85, but this does not happenin the remaining ranges. So the system stability takesplace in the two ranges S1 and S2, whereas the instabilityoccurs in the other ranges. Looking at the curves shownin Fig. 22 we notice that the four curves appear together
123
MHD instability of interfacial waves between two immiscible incompressible cylindrical fluids 509
Fig. 20 The curves represent the roots of Eq. (48) for a system havinga(1) = 0.1, a(2) = 1.1, q = 0.001, ρ = 1.1, u(1)
0 = 6, u(2)0 = 3.7,
k2 = 1 and k = 2 with h0 = 1.5
Fig. 21 The same system of Fig. 20 is considered but with h0 = 1.75
in two ranges S1 and S2; namely 2.25 < W < 7.25 and8.75 < W < 9.5, but this does not happen in the remain-ing ranges. So the system stability takes place in the tworanges S1 and S2, whereas the instability occurs in the otherranges. In Figs. 20–22, we notice that the ranges of stabilityincrease with the increase of the field value h0. We also noticean extreme approach between the two upper curves, whichare the solid curve and the dot-dashed one, in Figs. 20–22.Furthermore, we show an approach between the two lowercurves, which are the dotted curve and the dashed one, inthe same figures too. Hence h0 plays a stable role on thesystem stability in case of short waves. To investigate theeffect of the wave length on the system stability, we take thevalue 0.1 for k in Figs. 23–25. Through Figs. 23–25, the onlychanged para-meter is the field, which takes the values 1.5,1.75 and 2. We notice the coincidence between the two uppercurves, which are the solid curve and the dot-dashed curvein Figs. 23–25; as well as the extreme approach between thetwo lower curves, which are the dotted curve and the dashedone in the same figures.
Fig. 22 The same system of Fig. 20 is considered but with h0 = 2
Fig. 23 The same system of Fig. 20 is considered but with k = 0.1
Fig. 24 The same system of Fig. 21 is considered but with k = 0.1
Studying the curves shown in Fig. 23 we notice that thefour curves appear together in the range 1 < W < 10 but thisdoes not happen outside this range. So the system stabilitytakes place in this range, whereas the instability occurs inthe other ranges. Investigating the curves shown in Fig. 24we notice that the four curves appear together in the range1.25 < W < 10 but this does not happen outside this range.So the system stability takes place in this range, whereas
123
510 K. Zakaria et al.
Fig. 25 The same system of Fig. 22 is considered but with k = 0.1
the instability occurs in the other ranges. Looking at thecurves shown in Fig. 25 we notice that the four curves appeartogether in the range 1.5 < W < 10 but this does not happenoutside this range. So the system stability takes place in thisrange, whereas the instability occurs in the other ranges. InFigs. 23–25, we notice that the ranges of stability decreasewith the increase of the field value h0. Comparing the rangesin Figs. 20–22 and the corresponding ones in Figs. 23–25,we notice the decrease of the stable range and the increaseof the unstable range with the increase of the wave number.Also h0 plays an unstable role on the system stability in caseof long waves.
Table 1 summarizes the results of all figures. In order tostudy the effect of the wave kind in case of periodic-awaywe compare Figs. 2–5 with Figs. 6–9, respectively. Thenwe show that the behaviour of the parameters h0, µ and λ1
does not change with the increase of the wave number in thiscase, whereas the effect of λ2 changes. In order to study theeffect of the wave kind in case of periodic-on we compareFigs. 12–15 with Figs. 16–19, respectively. Then we showthat the behaviour of the parameters h0, µ, λ1 and λ2 doesnot change with the increase of the wave number in this case.In order to study the effect of the wave kind in case of invis-cid periodic-away we compare Fig. 10 with Fig. 11. Thenwe show that the behaviour of the parameter h0 does notchange with the increase of the wave number in this case. Inorder to study the effect of the wave kind in case of inviscidperiodic-on we compare the effect of h0 in Figs. 20–22 withits effect in Figs. 23–25, respectively. Then we conclude thatthe behaviour of the parameter h0 changes with the increaseof the wave number in this case. So we can conclude thatthe effect of the parameters h0 and λ2 on the system stabilitydepends generally on the wave kind, while the behaviour ofthe parameters µ and λ1 does not depend on the wave kindin the nonlinear problem.
To study the effect of approaching the marginal curvein case of periodic long waves we compare Figs. 6–9 with
Figs. 16–19, respectively, then we notice the change of theparameters h0, µ, λ1 and λ2 effect as we approach themarginal curve. In case of periodic short waves we com-pare Figs. 2–5 with Figs. 12–15, respectively, then we noticethe change of the parameters h0, µ, λ1 and λ2 effect as weapproach the marginal curve. Therefore we can say that theparameters h0, µ, λ1 and λ2 effect on the system stabilityin the nonlinear case depends generally on the extent ofapproaching the critical curve.
To study the effect of viscosity absence in case of periodic-away long waves we compare Fig. 6 and Fig. 11 and wenotice that the behaviour of the parameter h0 changes with theabsence of viscosity. To study the effect of viscosity absencein case of periodic-on long waves we compare Fig. 16 withFigs. 23–25, then we notice that the behaviour of the para-meter h0 changes with the absence of viscosity. To study theeffect of viscosity absence in case of periodic-away shortwaves we compare Fig. 2 and Fig. 10 and we notice that thebehaviour of the parameter h0 changes with the absence ofviscosity. To study the effect of viscosity absence in case ofperiodic-on short waves we compare Fig. 12 with Figs. 20–22, then we notice that the behaviour of the parameter h0 doesnot change with the absence of viscosity. So, in general wecan see that the effect of h0 on the system stability dependson the existence of viscosity. Looking at the table, we canclearly see that the effect of h0 on the system stability dif-fers in the linear case of the problem from the nonlinear one.Considering the table carefully, we can see that there is anobvious difference between the effect of the three constantsµ, λ1 and λ2 on the system stability in the nonlinear case.
5 Conclusions
Using the theories of conducting and Oldroydian fluids, weanalyzed the behaviour of the nonlinear interfacial wavesbetween two cylindrical immiscible liquids in the presenceof an axial magnetic field. The multiple scale method wascarried out to investigate the stability criteria away and on themarginal state of the linear theory. The evolution equationsgoverning the nonlinear elevation’s amplitude away and onthe marginal state were studied using the modulation concept.Different numerical examples were considered.
According to the linear step of the problem, we see clearlythat both the magnetic field h0 and the thickness of the fluidlayer between the two cylinders play a stable role, whereasthe dimensionless number W plays a dual role in the systemstability depending on the wave number. Proceeding to thenonlinear step of the problem, we see the appearance of dualrole of some physical parameters. New instability regions inthe parameter space, which appear due to nonlinear effects,are shown.
123
MHD instability of interfacial waves between two immiscible incompressible cylindrical fluids 511
Table 1 An abstract of theresults
S indicates the stable role,U indicates the unstable role andD indicates the dual role
Parameter Linear Nonlinear
Short waves Long waves
Viscous Inviscid Viscous Inviscid
Away On Away On Away On Away On
h0 S D S S S D S S U
Thickness S – – – – – – – –
W U – – – – – – – –
µ – D S – – D S – –
λ1 – D U – – D U – –
λ2 – U S – – D S – –
From the concluded conditions in the stationary cases wenotice that the two parameters λ1 and λ2 must be equal inthese cases. In Figs. 12–15 and Figs. 16–19 we notice theexistence of one stable area instead of two areas in Figs. 2–5and Figs. 6–9. Generally it is obvious that in the cases whichare away from the marginal state, two stable areas appearwhile in the cases which are on the marginal state, there isonly one stable area. Also, we can conclude that the effectof the parameters h0 and λ2 on the system stability dependsgenerally on the wave kind (short or long), while the behav-iour of the parameters µ and λ1 does not depend on the wavekind in the nonlinear problem. Furthermore, we can say thatthe effect of the parameters h0, µ, λ1 and λ2 on the systemstability in the nonlinear case depends generally on the extentof approaching the critical curve. In general we can concludethat the effect of h0 on the system stability depends on theexistence of viscosity. We can clearly see that the effect of h0
on the system stability differs in the linear case of the prob-lem from the nonlinear one. Finally, we can see that there isan obvious difference between the effect of the three para-meters µ, λ1 and λ2 on the system stability in the nonlinearcase.
Appendix
Equations of motion:1-Order of q
∂π( j)1
∂r− ω
∂u( j)r1
∂− k
∂h( j)r1
∂h0 + k
∂u( j)r1
∂u( j)
0 = 0, (A1)
k∂π
( j)1
∂− ω
∂u( j)z1
∂− k
∂h( j)z1
∂h0 + k
∂u( j)z1
∂u( j)
0 = 0, (A2)
−ω∂h( j)
r1
∂+ ku( j)
0∂h( j)
r1
∂− k
∂u( j)r1
∂h0 = 0, (A3)
−ω∂h( j)
z1
∂+ ku( j)
0
∂h( j)z1
∂− k
∂u( j)z1
∂h0 = 0, (A4)
1
r
∂(ru( j)r1 )
∂r+ k
∂u( j)z1
∂= 0, (A5)
1
r
∂(rh( j)r1 )
∂r+ k
∂h( j)z1
∂= 0, (A6)
ω∂η1
∂− ku( j)
0∂η1
∂+ u( j)
r1 = 0, (A7)
h(2)r1 − h(1)
r1 = 0, (A8)
−W
(∂2η1
∂2 k2 + η1
)− (ρπ
(2)1 − π
(1)1 ) = 0, (A9)
u( j)r1 = 0. (A10)
2-Order of q2
∂π( j)2
∂r− ω
∂u( j)r2
∂− k
∂h( j)r2
∂h0 + k
∂u( j)r2
∂u( j)
0
−(−∂u( j)
r1
∂t1+ ∂h( j)
r1
∂z1h0+ ∂h( j)
r1
∂rh( j)
r1 + k∂h( j)
r1
∂h( j)
z1
−∂u( j)r1
∂z1u( j)
0 − ∂u( j)r1
∂ru( j)
r1 −k∂u( j)
r1
∂u( j)
z1
)=0, (A11)
k∂π
( j)2
∂− ω
∂u( j)z2
∂− k
∂h( j)z2
∂h0 + k
∂u( j)z2
∂u( j)
0
−(
−∂π( j)1
∂z1− ∂u( j)
z1
∂t1+ ∂h( j)
z1
∂z1h0 + ∂h( j)
z1
∂rh( j)
r1
+k∂h( j)
z1
∂h( j)
z1 − ∂u( j)z1
∂z1u( j)
0 − ∂u( j)z1
∂ru( j)
r1
−k∂u( j)
z1
∂u( j)
z1
)= 0, (A12)
−ω∂h( j)
r2
∂+ ku( j)
0∂h( j)
r2
∂− k
∂u( j)r2
∂h0
−(−∂h( j)
r1
∂t1+ ∂u( j)
r1
∂z1h0+ ∂u( j)
r1
∂rh( j)
r1 +k∂u( j)
r1
∂h( j)
z1
123
512 K. Zakaria et al.
−∂h( j)r1
∂z1u( j)
0 − ∂h( j)r1
∂ru( j)
r1 −k∂h( j)
r1
∂u( j)
z1
)=0, (A13)
−ω∂h( j)
z2
∂+ ku( j)
0
∂h( j)z2
∂− k
∂u( j)z2
∂h0
−(
−∂h( j)z1
∂t1+ ∂u( j)
z1
∂z1h0 + ∂u( j)
z1
∂rh( j)
r1 + k∂u( j)
z1
∂h( j)
z1
−∂h( j)z1
∂z1u( j)
0 − ∂h( j)z1
∂ru( j)
r1 − k∂h( j)
z1
∂u( j)
z1
)= 0, (A14)
1
r
∂(ru( j)r2 )
∂r+ ∂u( j)
z1
∂z1+ k
∂u( j)z2
∂= 0, (A15)
1
r
∂(rh( j)r2 )
∂r+ ∂h( j)
z1
∂z1+ k
∂h( j)z2
∂= 0, (A16)
ω∂η2
∂− ku( j)
0∂η2
∂+ u( j)
r2
−(
∂η1
∂t1− ∂u( j)
r1
∂rη1+ ∂η1
∂z1u( j)
0 +k∂η1
∂u( j)
z1
)=0, (A17)
(h(2)r2 −h(1)
r2 )−(
(h(2)z1 − h(1)
z1 )k∂η1
∂
−η1∂(h(2)
r1 −h(1)r1 )
∂r
)=0, (A18)
−(
W
(1
2
(∂η1
∂z0
)2
− η21 + 2
∂2η1
∂z1∂z0
)
+µ
(∂u(2)
r1
∂r− λ1
(∂2u(2)
r1
∂r∂t0
+(
∂3u(2)r1
∂r∂z20
u(2)0
2 + 2∂3u(2)
r1
∂r∂z0∂t0u(2)
0
+∂3u(2)r1
∂r∂t20
)λ2 + ∂2u(2)
r1
∂r∂z0u(2)
0
)
+(
∂2u(2)r1
∂r∂t0+ ∂2u(2)
r1
∂r∂z0u(2)
0
)λ2
)+ ∂(ρπ
(2)1 − π
(1)1 )
∂rη1
)
−W
(∂2η2
∂z20
+ η2
)− (ρπ
(2)2 − π
(1)2 ) = 0, (A19)
u( j)r2 = 0. (A20)
3-Order of q3
∂π( j)3
∂r− ω
∂u( j)r3
∂− k
∂h( j)r3
∂h0 + k
∂u( j)r3
∂u( j)
0
−(
−∂u( j)r1
∂t2− ∂u( j)
r2
∂t1+ ∂h( j)
r1
∂z2h0 + ∂h( j)
r2
∂z1h0
+∂h( j)r2
∂rh( j)
r1 + ∂h( j)r1
∂rh( j)
r2 + ∂h( j)r1
∂z1h( j)
z1 + k∂h( j)
r2
∂h( j)
z1
+k∂h( j)
r1
∂h( j)
z2 − ∂u( j)r1
∂z2u( j)
0 − ∂u( j)r2
∂z1u( j)
0 − ∂u( j)r2
∂ru( j)
r1
−∂u( j)r1
∂ru( j)
r2 − ∂u( j)r1
∂z1u( j)
z1
−k∂u( j)
r2
∂u( j)
z1 − k∂u( j)
r1
∂u( j)
z2
)= 0, (A21)
k∂π
( j)3
∂− ω
∂u( j)z3
∂− k
∂h( j)z3
∂h0 + k
∂u( j)z3
∂u( j)
0
−(
−∂π( j)1
∂z2− ∂π
( j)2
∂z1
−∂u( j)z1
∂t2− ∂u( j)
z2
∂t1+ ∂h( j)
z1
∂z2h0 + ∂h( j)
z2
∂z1h0
+∂h( j)z2
∂rh( j)
r1 + ∂h( j)z1
∂rh( j)
r2
+∂h( j)z1
∂z1h( j)
z1 + k∂h( j)
z2
∂h( j)
z1 + k∂h( j)
z1
∂h( j)
z2
−∂u( j)z1
∂z2u( j)
0 − ∂u( j)z2
∂z1u( j)
0 − ∂u( j)z2
∂ru( j)
r1 − ∂u( j)z1
∂ru( j)
r2
−∂u( j)z1
∂z1u( j)
z1 − k∂u( j)
z2
∂u( j)
z1 − k∂u( j)
z1
∂u( j)
z2
)= 0, (A22)
−ω∂h( j)
r3
∂+ ku( j)
0∂h( j)
r3
∂− k
∂u( j)r3
∂h0 −
(−∂h( j)
r1
∂t2− ∂h( j)
r2
∂t1
+∂u( j)r1
∂z2h0 + ∂u( j)
r2
∂z1h0 + ∂u( j)
r2
∂rh( j)
r1 + ∂u( j)r1
∂rh( j)
r2
+∂u( j)r1
∂z1h( j)
z1 + k∂u( j)
r2
∂h( j)
z1 + k∂u( j)
r1
∂h( j)
z2 − ∂h( j)r1
∂z2u( j)
0
−∂h( j)r2
∂z1u( j)
0 − ∂h( j)r2
∂ru( j)
r1 − ∂h( j)r1
∂ru( j)
r2 − ∂h( j)r1
∂z1u( j)
z1
−k∂h( j)
r2
∂u( j)
z1 − k∂h( j)
r1
∂u( j)
z2
)= 0, (A23)
−ω∂h( j)
z3
∂+ ku( j)
0
∂h( j)z3
∂− k
∂u( j)z3
∂h0 −
(−∂h( j)
z1
∂t2− ∂h( j)
z2
∂t1
+∂u( j)z1
∂z2h0 + ∂u( j)
z2
∂z1h0 + ∂u( j)
z2
∂rh( j)
r1 + ∂u( j)z1
∂rh( j)
r2
+∂u( j)z1
∂z1h( j)
z1 + k∂u( j)
z2
∂h( j)
z1 + k∂u( j)
z1
∂h( j)
z2 − ∂h( j)z1
∂z2u( j)
0
−∂h( j)z2
∂z1u( j)
0 − ∂h( j)z2
∂ru( j)
r1 − ∂h( j)z1
∂ru( j)
r2 − ∂h( j)z1
∂z1u( j)
z1
−k∂h( j)
z2
∂u( j)
z1 − k∂h( j)
z1
∂u( j)
z2
)= 0, (A24)
1
r
∂(ru( j)r3 )
∂r+
(∂u( j)
z1
∂z2+ ∂u( j)
z2
∂z1
)+ k
∂u( j)z3
∂= 0, (A25)
123
MHD instability of interfacial waves between two immiscible incompressible cylindrical fluids 513
1
r
∂(rh( j)r3 )
∂r+
(∂h( j)
z1
∂z2+ ∂h( j)
z2
∂z1
)+ k
∂h( j)z3
∂= 0, (A26)
ω∂η3
∂− ku( j)
0∂η3
∂+ u( j)
r3
−(
−1
2
∂2u( j)r1
∂r2 η21 + k
∂η1
∂
∂u( j)z1
∂rη1
+∂η1
∂t2+ ∂η2
∂t1+ ∂u( j)
r2
∂r(−η1) − ∂u( j)
r1
∂rη2
+∂η1
∂z2u( j)
0 + ∂η2
∂z1u( j)
0 + ∂η1
∂z1u( j)
z1
+k∂η2
∂u( j)
z1 + k∂η1
∂u( j)
z2
)= 0, (A27)
(h(2)r3 − h(1)
r3 ) −(
1
2
(h(2)
r1 − h(1)r1
)(k∂η1
∂
)2
−1
2
∂2(h(2)r1 − h(1)
r1 )
∂r2 η21
−∂(h(2)r2 − h(1)
r2 )
∂rη1 + k
∂η1
∂
∂(h(2)z1 − h(1)
z1 )
∂rη1
−∂(h(2)r1 − h(1)
r1 )
∂rη2 + ∂η1
∂z1(h(2)
z1 − h(1)z1 )
+k∂η2
∂(h(2)
z1 − h(1)z1 ) − k
∂η1
∂(h(2)
z2 − h(1)z2 )
)= 0, (A28)
−(
1
2
∂2(ρπ(2)1 − π
(1)1 )
∂r2 η21 + ∂(ρπ
(2)2 − π
(1)2 )
∂rη1
+W
(η3
1 − 1
2
(∂η1
∂z0
)2
η1 − 2η2η1 − 3
2
∂2η1
∂z20
(∂η1
∂z0
)2
+∂2η1
∂z21
+ ∂η1
∂z1
∂η1
∂z0+ ∂η1
∂z0
∂η2
∂z0+ 2
∂2η1
∂z2∂z0+ 2
∂2η2
∂z1∂z0
)
−µ
(−∂u(2)
r2
∂r− λ2
∂η1
∂t0
∂2u(2)r1
∂r2 − ∂η1
∂z0
(∂u(2)
r1
∂z0+ ∂u(2)
z1
∂r
−λ1
(∂2u(2)
r1
∂z0∂t0+ ∂2u(2)
z1
∂r∂t0+ η1
∂2u(2)r1
∂r2 − λ2u(2)r1
∂2u(2)r1
∂r2
−λ2∂2u(2)
r1
∂r∂t1− λ2u(2)
z1∂2u(2)
r1
∂r∂z0
−λ2u(2)0
∂2u(2)r1
∂r∂z1− λ2η1
∂3u(2)r1
∂r2∂t0
−λ2u(2)0 η1
∂3u(2)r1
∂r2∂z0− λ2
∂2u(2)r2
∂r∂t0− λ2u(2)
0∂2u(2)
r2
∂r∂z0
+((
∂3u(2)r1
∂z03 − 2
∂3u(2)r1
∂r2∂z0+ ∂3u(2)
z1
∂r∂z02
)u(2)
02
+2
(− ∂3u(2)
r1
∂r2∂t0+ ∂3u(2)
r1
∂z02∂t0
+ ∂3u(2)z1
∂r∂z0∂t0
)u(2)
0 + ∂3u(2)r1
∂z0∂t02
+ ∂3u(2)z1
∂r∂t02
)λ2 +
(−∂2u(2)
r1
∂r2 + ∂2u(2)r1
∂z02 + ∂2u(2)
z1
∂r∂z0
)u(2)
0
)
+(
∂2u(2)r1
∂z0∂t0+ ∂2u(2)
z1
∂r∂t0+
(−∂2u(2)
r1
∂r2 + ∂2u(2)r1
∂z02
+ ∂2u(2)z1
∂r∂z0
)u(2)
0
)λ2
)+
((∂η1
∂t0+ u(2)
r1
)∂2u(2)
r1
∂r2
+∂2u(2)r1
∂r∂t1+ ∂2u(2)
r2
∂r∂t0+ ∂3u(2)
r1
∂r2∂t0η1
+((
∂2η1
∂z02
∂2u(2)r1
∂r2 + ∂3u(2)r2
∂r∂z02 + 2
∂3u(2)r1
∂r∂z0∂z1
+ ∂4u(2)r1
∂r2∂z02 η1
)u(2)
02 +
((∂u(2)
r1
∂z0+ 2
∂2η1
∂z0∂t0
)∂2u(2)
r1
∂r2
+∂u(2)z1
∂z0
∂2u(2)r1
∂r∂z0+ 2
(∂η1
∂t0+ u(2)
r1
)∂3u(2)
r1
∂r2∂z0+ 2
∂3u(2)r1
∂r∂z0∂t1
+2∂3u(2)
r1
∂r∂z1∂t0+ 2
∂3u(2)r2
∂r∂z0∂t0+ 2
∂4u(2)r1
∂r2∂z0∂t0η1
+ 2∂3u(2)
r1
∂r∂z02 u(2)
z1
)u(2)
0
+(
∂2η1
∂t02 + ∂u(2)r1
∂t0
)∂2u(2)
r1
∂r2 + ∂u(2)z1
∂t0
∂2u(2)r1
∂r∂z0
+2
(∂η1
∂t0+ ur1
)∂3u(2)
r1
∂r2∂t0+ ∂3u(2)
r2
∂r∂t02 + 2∂3u(2)
r1
∂r∂t0∂t1
+ ∂4u(2)r1
∂r2∂t02 η1 + 2∂3u(2)
r1
∂r∂z0∂t0u(2)
z1
)λ2
+(
∂2u(2)r1
∂r∂z1+ ∂2u(2)
r2
∂r∂z0+ ∂3u(2)
r1
∂r2∂z0η1
)u(2)
0
+ ∂2u(2)r1
∂r∂z0u(2)
z1
)λ1
)+ ∂(ρπ
(2)1 − π
(1)1 )
∂rη2
)
−W
(∂2η3
∂z02 + η3
)− (ρπ
(2)3 − π
(1)3 ) = 0, (A29)
u( j)r3 = 0. (A30)
References
1. Pushkar, E.A.: Gasdynamic analogies in problems of the obliqueinteraction of MHD shock waves. Fluid Dyn. Translated fromIzvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti iGaza 36(6), 989–1003 (2001)
2. El-Sayed, M.F.: Magnetohydrodynamic stability of two stream-ing superposed viscoelastic conducting fluids. Z. Naturforsch.56a, 416–424 (2001)
3. Ilin, K.I., Trakhinin, Y.L., Vladimirov, V.A.: The stability of steadyMHD flows with current-vortex sheets. Phys. Plasmas 10, 2649–2658 (2003)
4. Blokhin, A., Trakhinin, Yu.: Stability of Strong Discontinuitiesin Magnetohydrodynamics and Electrohydrodynamics. Nova Sci-ence Publishers, New York (2003)
5. Shirota, T.: Regularity of solutions to mixed problems of linearized.[M.H.D. equations]. Nara Women’s University. Koukyuroku Math.1, 1–42 (1994) (in Japanese)
6. Malik, S.K., Singh, M.: Nonlinear focusing and the Kelvin–Helmholtz instability in ferrofluid/non-magnetic fluid systems.Phys. Fluids 31, 1069–1073 (1988)
7. El-Dib, Y.O.: Nonlinear stability of Kelvin–Helmholtz waves inmagnetic fluids stressed by a time-dependent acceleration and atangential magnetic field. J. Plasma Phys. 55, 219–234 (1996)
8. Rajagopal, K.R., Bhatnagar, R.K.: Exact solutions for some simpleflows of an Oldroyd-B fluid. Acta Mech. 113, 233–239 (1995)
9. Rajagopal, K.R.: On an exact solution for the flow of an Oldroyd-Bfluid. Bull. Tech. Univ. Istanbul 47, 617–623 (1996)
10. Pontrelli, G., Bhatnagar, R.K.: Flow of a viscoelastic fluid betweentwo rotating circular cylinders subject to suction or injection. Int.J. Numer. Methods Fluids 24, 337–349 (1997)
11. Hayat, T., Siddiqui, A.M., Asghar, S.: Some simple flows of anOldroyd-B fluid. Int. J. Eng. Sci. 39, 135–147 (2001)
12. Hayat, T., Hutter, K., Asghar, S., Siddiqui, A.M.: MHD flows ofan Oldroyd-B fluid. Math. Comput. Modell. 36, 987–995 (2002)
14. Baris, S.: Flow of an Oldroyd 6-constant fluid between intersectingplanes, one of which is moving. Acta Mech. 147, 125–135 (2001)
15. Wang, Y., Hayat, T., Hutter, K.: On non-linear magnetohydrody-namic problems of an Oldroyd 6-constant fluid. Int. J. Nonlin.Mech. 40, 49–58 (2005)
16. Moatimid, G.M., El-Dib, Y.O.: Nonlinear Kelvin–Helmholtz insta-bility of Oldroydian viscoelastic fluid in porous media. PhysicaA 333, 41–64 (2004)
17. Hayat, T., Khan, M., Ayub, M.: Exact solutions of flow problemsof an Oldroyd-B fluid. Appl. Math. Comput. 151, 105–119 (2004)
18. Hayat, T., Nadeem, S., Asghar, S.: Hydromagnetic couette flow ofan Oldroyd-B fluid in a rotating system. Int. J. Eng. Sci. 42, 65–78(2004)
19. Hayat, T., Wang, Y., Hutter, K., Asghar, S., Siddiqui, A.M.: Peri-staltic transport of an Oldroyd-B fluid in a planar channel. Math.Problems Eng. 4, 347–376 (2004)
20. Khan, M., Hayat, T., Asghar, S.: Exact solution for MHD flow of ageneralized Oldroyd-B fluid with modified Darcys law. Int. J. Eng.Sci. 44, 333–339 (2006)
21. Hayat, T., Hussain, M., Khan, M.: Hall effect on flows of anOldroyd-B fluid through porous medium for cylindrical geome-tries. Comput. Math. Appl. 52, 269–282 (2006)
22. Khan, M., Maqbool, K., Hayat, T.: Influence of Hall current onthe flows of a generalized Oldroyd-B fluid in a porous space. ActaMech. 184, 1–13 (2006)
23. Hayat, T., Khan, S.B., Khan, M.: The influence of Hall current onthe rotating oscillating flows of an Oldroyd-B fluid in a porousmedium. Nonlin. Dyn. 47, 353–362 (2007)
24. Hayat, T., Khan, M., Wang, Y.: Non-Newtonian flow between con-centric cylinders. Commun. Nonlin. Sci. Numer. Simul. 11, 297–305 (2006)
25. Hayat, T., Khan, M., Sajid, M., Ayub, M.: Steady flow of an Oldroyd8-constant fluid between coaxial cylinders in a porous medium.J. Porous Media 9(8), 709–722 (2006)
26. Hayat, T., Khan, M., Ayub, M.: The effect of the slip condi-tion on flows of an Oldroyd 6-constant fluid. J. Comput. Appl.Math. 202, 402–413 (2007)
27. Zakaria, K.: Nonlinear instability of a liquid jet in the presence ofa uniform electric field. Fluid Dyn. Res. 26, 405–420 (2000)
28. Gill, G.K., Trehan, S.K.: Nonlinear instabilities in superposedfluids in the presence of adsorption. Int. J. Eng. Sci. 34(2), 213–226 (1996)
29. Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability.Oxford University Press, Oxford (1961)
30. Lee, D.S.: Nonlinear waves on the surface of a magnetohydrody-namic fluid column. Z. Naturforsch. 56a, 585–595 (2001)
