Electrohydrodynamic(EHD) studies the interaction between fluid mechanics and an electric field. The electric field plays important role in several practical problems chemical engineering and other related fields. Elhefnawy et al. [1] investigated the stability analysis of a finitely conducting cylinders in the presence of an axial electric field. Elcoot [2] has studied nonlinear analysis of viscous potential flow of capillary instability with presence axial electric field. Awasthi and Tamsir [3] studied capillary instability of a cylindrical interface under an axial electric field through porous media. They noticed that, axial electric field played stabilize on the stability of the system and the medium doesn't influence the behavior of electric field on capillary instability.

There are many elastico-viscous fluids which can't be characterized by Maxwell's or oldroyd's constitutive relations, e.g. Walters B${}^{\prime }$ fluid and Rivlin-Ericksen fluid. We are interested here in Rivlin-Ericksen model. Rivlin-Ericksen fluid is an important class of viscoelastic fluids in chemical engineering, industries and different geophysical situations. The usual viscoelastic and viscous terms in the equation of motion of Rivlin-Ericksen fluid are replaced by the resistive term $(\mu +{\mu }^{\prime }\frac{\partial }{\partial t}){\nabla }^2\mathbf{V}$ , where $\mu$ and ${\mu }^{\prime }$ are the dynamic viscosity and dynamic viscoelasticity of the Rivlin-Ericksen fluid. The study of viscoelastic fluids has become increasingly important in the past few years. Funda and Joseph [4] studied the capillary instability of an Oldroyd-B viscoelastic fluid surrounded by a viscous fluid with the help of viscoelastic potential flow theory. Abhilasha [5] studied the stability of Rivlin-Ericksen viscoelastic fluids in the presence of magnetic field. El-Sayed et al. [6] investigated the nonlinear EHD instability of two superposed Rivlin-Ericksen viscoelastic fluids in porous media in the presence of suspended particles. Recently, Moatimid and Zekry [7] studied the nonlinear EHD instability of a vertical cylindrical interface among two viscoelastic fluids. Dharamendra and Awasthi [8] examined the linear stability of an interface formed by viscoelastic fluid and viscous fluid, they found that the viscoelasticity induces stability.

The theory of viscous potential flow(VPF) has played a great role in studying different stability problems. The flow is considered irrotational in viscous potential flow, so the viscous term i.e. $\mu$ ${\nabla }^2\mathbf{V}$ in the Navier Stokes equation is identically zero when the vorticity is zero but the viscous stresses are not zero, where $\mu$ indicates viscosity and $\mathbf{V}$ indicates velocity of fluid flow. In viscous potential flow theory,tangential stresses are not considered and viscosity enters through normal stress balance [9]. The viscous potential flow analysis of capillary instability was studied by Funda and Joseph [10] and noticed that viscous potential flow is better approximation of the exact solution than the inviscid model. In recent years, Navier Stokes equation in the cylindrical coordinates has received much attention due to its wide range of applications in many fields, such as geophysical problems, modern technology and industries. Many of researchers focused on the solutions of nonlinear differential equations by numerical or analytical method due to its important role in many fields (see Refs. [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]).

The aim of this paper is to study the linear stability of two Rivlin-Ericksen viscoelastic fluids separated by a cylindrical interface in an annular configuration in presence of horizontal axial electric field for asymmetric disturbance. The dispersion equation of complex coefficients for asymmetric disturbance has been obtained. The effects of Reynolds number, inner fluid fraction, viscosity ratio, density ratio, axial electric field, viscoelastic ratio and uniform velocities of fluids on the growth rates are studied. A comparison between viscoelastic potential flow and inviscid potential flow has been achieved. Some limiting cases of results of this present work recovered the earlier studies. The problem finds its importance in geophysical situations.

Our system consists of two incompressible, irrational and viscoelastic Rivlin-Ericksen dielectric fluids separated by a cylindrical interface in annular configuration. The outer and inner fluids are streaming with a uniform velocities $U_2$ and $U_1$ along the z-axis respectively as shown in Fig. 1. The two viscoelastic Rivlin-Ericksen fluids are influenced by an axial electric field $E_0.$ The region of inner fluid is $r_{{}_1}<$ $r<R$ with thickness $h_1$ and the region of outer fluid is $R<$ $r<r_{{}_2}$ with thickness $h_2.$ In formulating the problem, the subscripts 1 and 2 denote parameters associated with inner and outer fluids, respectively.

The equations of motion and continuity can be written as [6]

where $\mu (=\rho \nu )$ is the fluids dynamic viscosity, ${\mu }^{\prime }(=\rho {\nu }^{\prime })$ is the dynamic viscoelastic, $\rho$ is the densities, ${\epsilon }_{}$ is the dielectric constant and $E_{}$ is the electric field.

We can write the disturbed interface as following:

where $\eta$ is the disturbance of the interface radius.

The velocity potentials ${\Phi }_j(r,\theta ,z,t)$ satisfy the Laplace equations, hence,

The quasi-static approximation is assumed to be valid for our problem, so the electric filed is irrotational. Thus we can write the electrical equations in the following form:

The total electric fields ${\mathbf{E}}_j(j=1,2)$ may be written as:

where ${\Psi }_j(r,\theta ,z,t)$ is the electric scalar potential function. Equations (5) and (6) yields

The unit perpendicular vector in the first order terms may be written as:

The solutions of electric and velocity scalar potential functions must satisfy the following linearization boundary conditions:

(1) Conditions on the walls are given by Moatimid et al. [23]

(2) The kinematic boundary condition is [23]

(3) The tangential electric field component is continuous at the interface, we obtain

By using Eqs. (6) and (8), then Eq. (12) yields

(4) The normal electric displacement component is continuous at the interface, hence [23]

Equation (15), on using Eqs. (6) and (8) reduced to

(5) The interfacial condition for conservation of momentum can be written as [24]

where $\parallel Y\parallel =Y_2-Y_1$ indicate the jump of the outer and internal fluid layers, respectively. $\sigma$ denote the surface tension.

By using the normal mode method, ${\Phi }_j,$ ${\Psi }_j$ and $\eta$ can be represented as follows [25]

where $A,k,w$ and $m$ denote initial amplitude, wave number, growth rate and azimuthal wave number respectively.

On using Eqs. (18) and (19) with boundary conditions (9) and (11), the general solution of Eq. (4) can be written as

where $c.c$ denote the complex conjugate of the preceding term and $H_1\left(kr\right)$ , $H_2\left(kr\right)$ are given by

The symbols $I_m$ and $K_m$ are first and second type of modified Bessel functions, respectively.

On solving Eq. (7) with helping of boundary conditions (10), (13) and (16), we obtain

where

By substituting from Eqs. (19), (20), (21), (24) and (25) into Eq. (17), we obtain the following dispersion relation

where

Note that, in absence of viscoelastic, i.e. when ${\mu }_1^{\prime }={\mu }_2^{\prime }=0$ and the electric field $(E_0=0),$ equation(28) reduces to the same dispersion relation derived earlier by Awasthi et al. [26]. Also, for axisymmetric disturbance, i.e. when $m=0$ and absence of viscoelastic we obtain the same dispersion relation equation obtained earlier by Awasthi and Tamsir [3].

To write the dispersion relation in non-dimensional form, introducing the following dimensionless groups

where Re is the Reynolds number and $\beta$ denotes inner fluid fraction. Hence the non-dimensional form of Eq. (28) is

where $\begin{array}{ccc}\hfill {\tilde{a}}_0& =& (H_1\left(\tilde{k}\tilde{R}\right)-\tilde{\rho }{H}_2\left(\tilde{k}\tilde{R}\right))-2{\tilde{k}}^{2}L({\tilde{\mu }}^{\prime }{F}_2\left(\tilde{k}\tilde{R}\right)-F_1\left(\tilde{k}\tilde{R}\right)),\hfill \\ & & \\ \hfill {\tilde{a}}_1& =& 2\tilde{k}(\tilde{\rho }{\tilde{U}}_2{H}_2\left(\tilde{k}\tilde{R}\right)-{\tilde{U}}_1{H}_1\left(\tilde{k}\tilde{R}\right))+2{\tilde{k}}^{3}L({\tilde{\mu }}^{\prime }{\tilde{U}}_2{F}_2\left(\tilde{k}\tilde{R}\right)-{\tilde{U}}_1{F}_1\left(\tilde{k}\tilde{R}\right)),\hfill \\ & & \\ \hfill {\tilde{b}}_1& =& -\frac{2{\tilde{k}}^2}{\text{Re}}(\tilde{\mu }{F}_2\left(\tilde{k}\tilde{R}\right)-F_1\left(\tilde{k}\tilde{R}\right)),\hfill \end{array}$

where $L=\frac{{\nu }_1^{\prime }}{h^2}$ denotes viscoelastic parameter. The analytical solution of Eq. (29) can be written as:

where

Hence, the imaginary part of Eq. (30) is given by

In this section, Eq. (31) is used to compute the variation of Imaginary part of growth rate ${\tilde{\omega }}_I$ with wave number $\tilde{k}$ using Mathematica software for various parameters included in the linear analysis. If ${\tilde{\omega }}_I$ $<0$ , the disturbance decays with time while if ${\tilde{\omega }}_I>0,$ the flow is unstable as disturbance grows. The case ${\tilde{\omega }}_I=0$ is the neutrally stable. For the purpose of numerical computation, the asymmetric case $(m=1)$ have been considered in Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11 and the radius of inner and outer cylinders are taken as $1cm$ and $2cm$ in Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11

Fig. 2 illustrates comparison between Viscoelastic potential flow(VPF) and Inviscid potential flow(IPF), it is clear that the growth of Rivlin-Ericksen viscoelastic fluid is lower than the growth of inviscid fluid, and therefore, the viscoelasticity has stabilizing effect on the system. Similar result was obtained by Dharamendra and Awasthi [8]. Fig. 3 shows the variation of growth rate ${\tilde{\omega }}_I$ given by Eq. (31) with wave number $\tilde{k}$ for different values of $\beta =(0.3,0.4,0.5),$ it is clear that, the growth rate curves increases by increasing $\beta$ . Hence, we get that the inner fluid fraction $\beta$ has destabilizing effect. this result is in agreement with previous work (Awasthi et al. [26]). Fig. 4 shows the influences of density fluids ratio on the growth rate ${\tilde{\omega }}_I$ . The following can be observed, the increase of $\tilde{\rho }$ leads to increase the growth rate curves. Hence, we conclude that density ratio of fluids play destabilizing influence. Fig. 5 shows the variation of growth rate ${\tilde{\omega }}_I$ with wave number $\tilde{k}$ for different values of electric field ${\tilde{E}}_0(=0,20,40)$ . In this figure we can observe that, the increase in the electric field parameter causes a decrease in the growth rate curves. therefore, the electric field parameter has a stabilizing nature. This result is compatible with previous studies obtained by Awasthi and Tamsir [3]. Fig. 6 compares between growth rate of asymmetric and axisymmetric disturbance, it can be observe from Fig. 6, that axisymmetric perturbation is more stable than the asymmetric perturbation in the wave number range $0\le \tilde{k}\le 2.6$ , while the asymmetric perturbation is more stable afterward. Hence, we conclude that asymmetric disturbance has important role on the stability of the system, i.e., it has destabilizing influence for small wave number and then stabilizing influence. Fig. 7 Shows the effect of dielectric constant ratio $\tilde{\epsilon }$ on the growth rate of disturbance. We observe that growth rate curve increases with dielectric constant ratio, hence we conclude that dielectric constant ratio has destabilizing effect. This result is consistent with previous studies Awasthi et al. [26]. Fig. 8 is depicted to indicate the effect of Reynolds number Re. It is evident from Fig. 8 that growth rate curve decreases as Reynolds number increases. Hence Reynolds number has stabilizing effect. Also, both of length scale and the characteristic velocity playing stabilizing role as Reynolds number decreases with decreasing length scale or characteristic velocity. Fig. 9 shows the effect of dynamic viscosity ratio $\tilde{\mu }$ on growth rate.It is clear that, the growth rates have the same value in the wave number range $0\le \tilde{k}\le 4.4$ for all dynamic viscosity ratio values, after which, growth rate curve increases with increasing the value of dynamic viscosity ratio. Hence dynamic viscosity ratio has a destabilizing effect. Similar result was verified earlier by El-Sayed et al. [28] and Moatimid and Amer [27] in their linear study. Fig. 10 illustrates the variation of growth rate curves for different values of the dynamic viscoelastic ratio ${\tilde{\mu }}^{\prime }(=0.05,0.07,0.09),$ it is clear that as dynamic viscoelastic ratio increases, growth rate decreases i.e. dynamic viscoelastic ratio has stabilizing influence. El-Sayed et al. [6] came to a similar result in their linear study. Also, Dharamendra and Awasthi [8] came to a similar conclusion that viscoelastic has stabilizing effect. Fig. 11 shows the behavior of growth rate curve for various values of uniform velocities, it is clear that as ${\tilde{U}}_1$ and ${\tilde{U}}_2$ increase, growth rate decreases. Hence uniform velocities have stabilizing effect.

In this article, the electrohydrodynamics Kelvin-Helmholtz instability of cylindrical flow has been investigated. the solution of the equations of motion via normal modes technique leads to a quadratic dispersion relation in the growth rate with complex coefficients. This problem examines the influence of various physical parameters on the instability of the flow. Stability is discussed theoretically and illustrated graphically using Mathematica software. The obtained results can be summarized as follows:

1. The asymmetric disturbances have important roles on the stability of the system, destabilizing effect and then stabilizing effect, separated by a critical wave number.

2. Each of dynamic viscoelastic ratio, uniform velocities, Reynolds number and electric field has a stabilizing effect on the system.

3. The dynamic viscosity ratio, density fluids ratio, dielectric constant ratio and inner fluid fraction have destabilizing effect on the system.

4. The system is more stable when the viscoelasticity is present than whenever it is absent.

The results obtained in this study offer a better understanding of linear analysis of Kelvin Helmholtz instability of cylindrical interface in the presence of axial electric field.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The researcher would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.