Swirling Capillary Instability of Rivlin–Ericksen Liquid with Heat Transfer and Axial Electric Field

Abstract

The mutual influences of the electric field, rotation, and heat transmission find applications in controlled drug delivery systems, precise microfluidic manipulation, and advanced materials’ processing techniques due to their ability to tailor fluid behavior and surface morphology with enhanced precision and efficiency. Capillary instability has widespread relevance in various natural and industrial processes, ranging from the breakup of liquid jets and the formation of droplets in inkjet printing to the dynamics of thin liquid films and the behavior of liquid bridges in microgravity environments. This study examines the swirling impact on the instability arising from the capillary effects at the boundary of Rivlin–Ericksen and viscous liquids, influenced by an axial electric field, heat, and mass transmission. Capillary instability arises when the cohesive forces at the interface between two fluids are disrupted by perturbations, leading to the formation of characteristic patterns such as waves or droplets. The influence of gravity and fluid flow velocity is disregarded in the context of capillary instability analyses. The annular region is formed by two cylinders: one containing a viscous fluid and the other a Rivlin–Ericksen viscoelastic fluid. The Rivlin–Ericksen model is pivotal for comprehending the characteristics of viscoelastic fluids, widely utilized in industrial and biological contexts. It precisely characterizes their rheological complexities, encompassing elasticity and viscosity, critical for forecasting flow dynamics in polymer processing, food production, and drug delivery. Moreover, its applications extend to biomedical engineering, offering insights crucial for medical device design and understanding biological phenomena like blood flow. The inside cylinder remains stationary, and the outside cylinder rotates at a steady pace. A numerically analyzed quadratic growth rate is obtained from perturbed equations using potential flow theory and the Rivlin–Ericksen fluid model. The findings demonstrate enhanced stability due to the heat and mass transfer and increased stability from swirling. Notably, the heat transfer stabilizes the interface, while the density ratio and centrifuge number also impact stability. An axial electric field exhibits a dual effect, with certain permittivity and conductivity ratios causing perturbation growth decay or expansion.

1. Introduction

Heat and mass transmission’s influence on capillary instability finds applications across various fields. In microfluidics, controlling the heat and mass transfer at interfaces enables the precise manipulation of droplet formation and mixing, crucial for lab-on-a-chip devices. In material processing industries, an understanding of these effects provides benefits in designing advanced manufacturing techniques, optimizing phase transitions, and enhancing the product quality in areas such as polymer processing and thin film coating technologies. Additionally, in biological systems, managing the heat and mass transfer impacts cell cultures, drug delivery systems, and tissue engineering processes, facilitating controlled and efficient biological reactions. In Ref. [1], the impact of heat transmission on the stability of annular viscous–viscous fluid configurations was examined. Investigating interfacial instability with heat transfer, one of the authors of this paper focused [2] on the interface connecting a viscous liquid and an Oldroyd B-type fluid. Furthermore, in Ref. [3], we studied the viscous–Rivlin–Ericksen fluid interface in an annulus while analyzing the heat/mass transportation between phases.

Swirling exerts a significant impact on capillary instability, altering the delicate balance between surface tension, fluid flow, and interfacial dynamics. When a swirling motion is introduced to a fluid system, it modifies the fluid’s velocity distribution and flow patterns, which in turn affect the development of capillary instabilities at liquid interfaces. This swirling motion can either stabilize or amplify instabilities depending on the specific conditions and parameters involved. In Ref. [4], the interplay of non-dimensional parameters contributing to flow instability, primarily stemming from the dynamics involved in mass/heat transportation was investigated. To analyze the system’s stability, the study in Ref. [5] took into consideration the swirl and mass/heat transfer as significant factors. The authors also used the ratio among the conduction and evaporation warmth fluxes to characterize heat transportation at the interface. In Ref. [6], the stability of enclosed swirling layers with heat/mass transfer and offered a fresh perspective on gas velocity oscillations was investigated. When the liquid was above the vapor, Ref. [7] investigated the swirl impact on mass/heat transfer at the liquid–vapor interface. When both cylinders were rotating, the liquid–vapor interface in an annulus and studied its instability was considered [8]. In Refs. [9,10,11], we investigated the stability dynamics of diverse interfaces experiencing heat/mass transfer in cylindrical geometries. Recently, the swirling effect on the instability of interfaces formed with Rivlin–Ericksen and Walter’s B fluid inclusive of mass/heat transport was investigate [12,13].

The application of an electric field alongside heat/mass transmission significantly influences capillary instability by altering surface tension gradients and fluid behavior. At the interface, the electric field interacts with mass transfer and heat, influencing the system’s stability. Capillary instabilities may be enhanced or suppressed by this combined effect. Research conducted by Abdel Rauf Elhefnawy [14] delved into the consequences of an oblique electric field on the boundary between two streaming fluids. Meanwhile, Mohamed Fahmy El-Sayed [15] explored the ramifications of an electric field applied perpendicular to the flow of fluids within a permeable medium. Additionally, El-Sayed [16,17] scrutinized the interface between two streaming liquids when subjected to electric fields, considering both scenarios with and without the involvement of heat and mass transport.

Absolute instability is crucial in various fluid dynamic applications, as it indicates the occurrence of unstable behavior, regardless of perturbation magnitude or frequency. Understanding absolute instability helps predict abrupt transitions in flow patterns and design systems with improved stability and performance. Its significance lies in providing insights into critical conditions where small disturbances can trigger large-scale flow disturbances, impacting engineering processes, such as combustion, propulsion, and boundary layer control. In Ref. [18], the absolute and convective instabilities of two-dimensional wakes created behind a flat plate and in the vicinity of the trailing edge of a thin wedge-shaped aerofoil in both incompressible and compressible fluidswere investigated. Hosne Jasmine and Jitesh Gajjar [19] examined the stability characteristics of a conducting fluid flow over a rotating disk under the influence of a uniform magnetic field applied perpendicular to the disk. The study assumed that the magnetic field remained unaffected by the fluid motion. Mustafa Türkyılmazoglu [20] revisited the linear absolute/convective instability mechanisms governing the incompressible Von Karman’s boundary layer flow over a rotating disk. Türkyılmazoglu [21] employed a long-wavelength asymptotic approach to scrutinize the region of absolute instability within the compressible rotating disk boundary layer flow. This analytical method allows for a detailed examination of the dynamics of the flow field, particularly focusing on the onset of absolute instability and its implications for the overall stability of the system.

The impact of an electric field in the axial direction on the instability of viscous liquids’ interface with mass/heat transmission was examined [22,23]. It was observed that the interface was more stable when the electric field was present in the system. In Ref. [24], one of the authors of this paper explored the electro-hydrodynamic instability when the medium was porous, and the interface was allowing a heat/mass transfer. In Ref. [25], the surface charge effect on the capillary stability along with the mass/heat transmission and electric field was studied. Recently, we [26] explored the control of the electric field in the axial direction on the stability of the Rivlin–Ericksen/viscous fluid interface. In this study, the fluids lay in the annulus and the cylinders were not rotating.

The collective consequence of the electric field, rotation, and heat transmission significantly influences the capillary instability by altering the surface tension gradients, fluid motion patterns, and temperature distribution, amplifying or suppressing instability phenomena in various systems such as liquid films or droplets. Understanding their interplay enhances control over instabilities, which is crucial in applications like material processing, and biological systems. Investigations into the mutual effects of rotation, heat transmission, and electric power on the instability of the viscous fluid/Rivlin–Ericksen interface in a cylindrical configuration have been conducted. The two fluids are located in the annular area between two rigid cylinders, the outer of which revolves at a continuous angular velocity while the inner cylinder stays stationary. Heat and mass transfers amid the phases are made possible by the interface. The fluids have different permittivity and conductivity, and an axially applied constant electric field is used. We establish and numerically analyze a quadratic equation describing the growth parameter of disturbances. Section 2 provides details on the modeling, including the boundary conditions and interface. The linear form of the perturbed equations is derived in Section 3, and the dispersion association is explained in Section 4. The obtained consequences are discussed in Section 5, and a brief summary of the key outcomes is given in Section 6.

2. Modeling of the Physical Problem

2.1. Physical Formulation

The formulation can be understood by considering an annular region formed by a solid interior cylinder (of radius $a$) and an exterior cylinder (of radius $b$). The interior cylinder is stationary, while the exterior cylinder is moving with a linear velocity of $r\Omega$, where $\Omega$ denotes its angular velocity. The diagram of the problem is shown in Figure 1. The annular area is occupied with a viscous liquid enclosed by the Rivlin–Ericksen (RE) liquid, and these fluids are separated by cylindrical boundary $r=R$. Therefore, the viscous fluid-consuming density (${\rho }_v$), viscosity (${\mu }_v$), electrical conductivity (${\sigma }_{ v}$), and permittivity (${\epsilon }_{ v}$) inhabit the region $a<r<R$, while RE viscoelastic fluid-having density (${\rho }_r$), viscosity (${\mu }_r$),viscoelasticity (${{\mu }^{\prime }}_r$), electrical conductivity (${\sigma }_{ r}$), and permittivity (${\epsilon }_{ r}$) occupy the region $R<r<b$. The fluids experience a constant electric field $E$ directed uniformly along the axial direction. The viscous layer maintains temperature $T_v$, while the viscoelastic layer maintains temperature $T_r$ ($T_r>T_v$). Initially, both fluid phases are in heat equilibrium and $\gamma$ will denote the surface tension.

With the RE fluid model, the governing equations for the mass and momentum can be inscribed as [12,16]

$$\begin{array}{l}\nabla \cdot {\mathit{q}}_r=0,\\ {\rho }_r\left(\frac{\partial {\mathit{q}}_r}{\partial t}+({\mathit{q}}_r\cdot \nabla ){\mathit{q}}_r+2\left(\mathbf{\Omega }\times {\mathit{q}}_{\mathit{r}}\right)\right)=-\nabla \left(P_r-\frac{1}{2}{\epsilon }_r{\overrightarrow{\mathit{E}}}_{\mathit{r}}^2\right)+\nabla \cdot {\mathit{\tau }}_{\mathit{r}},\end{array}$$

(1)

where $P_r$ is the pressure, ${\overrightarrow{\mathit{E}}}_r$ is the electric field vector, and ${\mathit{q}}_r=(u_r,v_r)$ is the velocity in the viscoelastic fluid phase, $\mathbf{\Omega }$ represents the rotation vector, and $t$ denotes the time.

In the case of RE fluid, the stress tensor ${\mathit{\tau }}_{\mathit{r}}$ can be represented by utilizing both the velocity vector and the deformation rate tensor, as illustrated below [26]:

$${\mathit{\tau }}_{\mathit{r}}={\mu }_r{\mathit{D}}_1+{\alpha }_1{\mathit{D}}_2+{\alpha }_2{\mathit{D}}_1^2.$$

(2)

Here, ${\alpha }_1$ and ${\alpha }_2$ denote the normal stress moduli, and tensors ${\mathit{D}}_1$ and ${\mathit{D}}_2$ are defined as

$${\mathit{D}}_1=\nabla {\mathit{q}}_{\mathit{r}}+{(\nabla {\mathit{q}}_{\mathit{r}})}^T; {\mathit{D}}_2=\frac{D{\mathit{D}}_1}{Dt}+(\nabla {\mathit{q}}_{\mathit{r}}){\mathit{D}}_1+{\mathit{D}}_1{(\nabla {\mathit{q}}_{\mathit{r}})}^T.$$

(3)

The electric field E employed in this inquiry is applied to both liquids in the z-axis direction and is symbolized by the vector field

$$\overrightarrow{\mathit{E}}=E\widehat{\mathit{z}},$$

(4)

where $\widehat{\mathit{z}}$ is the unit axial vector.

For the viscous fluid phase, the governing equations for the conservation of mass and conservation of momentum are inscribed as [14]

$$\begin{array}{l}\nabla \cdot {\mathit{q}}_v=0,\\ {\rho }_v\left(\frac{\partial {\mathit{q}}_v}{\partial t}+({\mathit{q}}_v\cdot \nabla ){\mathit{q}}_v\right)=-\nabla \left(P_v-\frac{1}{2}{\epsilon }_v{\overrightarrow{\mathit{E}}}_{\mathit{v}}^2\right)+{\mu }_v{\nabla }^2{\mathit{q}}_v,\end{array}$$

(5)

where $P_v$ is the pressure, ${\overrightarrow{\mathit{E}}}_v$ is the electric field, and ${\mathit{q}}_v=(u_v,v_v)$ is the velocity in the viscous fluid phase.

In the realm of electrodynamics, the common assumption is that the quasi-static approximation holds true. As a result, the electrical equations are typically considered to be

$$\nabla \cdot (\sigma \overrightarrow{\mathit{E}})=0,\nabla \times \overrightarrow{\mathit{E}}=0,$$

(6)

where $\sigma$ represents electrical conductivity and the Del operator is denoted by $\nabla$.

2.2. Boundary Conditions

In viscous potential flow theory, the absence of enforcing the no-slip condition allows for the consideration of fluid behavior beyond the constraints of viscous effects. However, the boundaries $r=a,b$ are rigid, and the normal components of velocities are inherently set to zero due to the immobility of the boundary, providing a basis for modeling the flow behavior near solid boundaries in a potential flow analysis:

$$\begin{array}{l}{u}_v=0 \mathrm{at} r=a,\\ u_r=0 \mathrm{at} r=b.\end{array}$$

(7)

In electrostatics, boundary conditions ensure physical and mathematical accuracy in modeling electric fields. For rigid boundaries, like metallic surfaces, the axial component of the electric potential must be zero. This ensures no net flow of the electric charge across the boundary, which is crucial for maintaining consistency. These conditions are fundamental for accurate analyses in various applications, such as electric circuits and semiconductor devices,

$$\begin{array}{l}\overrightarrow{\mathit{n}}\times {\overrightarrow{\mathit{E}}}_v=0 \mathrm{at} r=a,\\ \overrightarrow{\mathit{n}}\times {\overrightarrow{\mathit{E}}}_r=0 \mathrm{at} r=b,\end{array}$$

(8)

where, $\overrightarrow{\mathit{n}}$ is the unit outward normal at the interface.

2.3. Interfacial Conditions

The following mathematical expression of the interfacial condition demonstrates the mass conservation at the interface:

$${\rho }_v\left({\mathit{q}}_v\cdot \nabla f+\frac{\partial f}{\partial t}\right)={\rho }_r\left({\mathit{q}}_r\cdot \nabla f+\frac{\partial f}{\partial t}\right),$$

(9)

where $f=f(r,\theta ,z,t)$ denotes the interface equation.

If the fluids are thermally conducting with thermal diffusivity $D_T$, the equation of diffusion can be written as

$$\frac{\partial T}{\partial t}+\overrightarrow{q}\cdot \nabla T=D_T{\nabla }^{2}T.$$

(10)

We assume that the inner fluid has temperature $T_v$ and the outer fluid has temperature $T_r$, and the fluids are in thermodynamics equilibrium in the basic state, and the saturation temperature is equal to the interface temperature $T_{\mathrm{int}}$. Here, we assume that the heat transfer across the interface is dominated by the latent heat, and the conservation of energy (10) across the interface takes the form as follows

$$L{\rho }_v\left({\mathit{q}}_v\cdot \nabla f+\frac{\partial f}{\partial t}\right)=H\left(R+\xi \right)=\frac{{\kappa }_r\left(T_{\mathrm{int}}-T_r\right)}{\left(R+\xi \right)\left(\ln a-\ln \left(R+\xi \right)\right)}-\frac{{\kappa }_v\left(T_v-T_{\mathrm{int}}\right)}{\left(R+\xi \right)\left(\ln \left(R+\xi \right)-\ln b\right)}.$$

(11)

Here, $L$ represents the latent heat liberated during a phase change, whereas $H(R+\xi )$ signifies the net heat flow originating from the interface where the variable $\xi$ symbolizes the difference between the interface radius and the interface in stable state.

Expanding $H(R+\xi )$ in the equilibrium form and taking only linear terms reveal the following:

$$H(R+\xi )\approx H(R)+\xi H^{\prime }(R).$$

(12)

The expression

$$\overrightarrow{\mathit{n}}\times {\overrightarrow{\mathit{E}}}_v=\overrightarrow{\mathit{n}}\times {\overrightarrow{\mathit{E}}}_r,$$

(13)

denotes the preservation of the tangential part of the electric field across the interface.

Although there is dissatisfaction in the general flow at the interface, the movement of both bulk liquid substances on either side of the surface effectually balances the generation of charge within a physical component. The equation

$${\sigma }_v\left(\overrightarrow{\mathit{n}}\cdot {\overrightarrow{\mathit{E}}}_v\right)={\sigma }_r\left(\overrightarrow{\mathit{n}}\cdot {\overrightarrow{\mathit{E}}}_r\right),$$

(14)

illustrates the normal electric dislocation at the interface.

As soon as $P_r$ and $P_v$ represent the pressures in the RE and viscous liquid phases, respectively, the interfacial normal stress is in equilibrium at the system’s interface:

$$\begin{array}{l}{P}_r-P_v-2\overrightarrow{\mathit{n}}\left({\mu }_r+{{\mu }^{\prime }}_r\frac{\partial }{\partial t}\right)\cdot \nabla \otimes {\mathit{q}}_r\cdot \overrightarrow{\mathit{n}}+2{\mu }_v\overrightarrow{\mathit{n}}\cdot \nabla \otimes {\mathit{q}}_v\cdot \overrightarrow{\mathit{n}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{1}{2}\left[{\epsilon }_r\left(E_{rn}^2-E_{rt}^2\right)-{\epsilon }_v\left(E_{vn}^2-E_{vt}^2\right)\right]=\gamma \nabla \cdot \overrightarrow{\mathit{n}}.\end{array}$$

(15)

Here, $E_{rn}, E_{rt}$ are the normal components of ${\overrightarrow{\mathit{E}}}_r$ and $E_{vn}, E_{vt}$ are the normal components of ${\overrightarrow{\mathit{E}}}_v.$

3. Stability Analysis

3.1. Basic State

In the state of equilibrium $(r=R)$, the fluids in both phases are stationary, i.e., $(u_r,v_r)=\left(0,0\right)$ and $(u_v,v_v)=\left(0,0\right)$. There is no net flow of fluids between the phases, and the pressures in each phase are given by $\left(P_v-\frac{1}{2}{\epsilon }_v{\overrightarrow{\mathit{E}}}_v^2\right)=$ constant and $\nabla \left(P_r-\frac{1}{2}{\epsilon }_r{\overrightarrow{\mathit{E}}}_r^2\right)=2R{\Omega }^2\widehat{\mathit{r}}$ with $\widehat{\mathit{r}}$ denoting the unit vector along radial direction. This state of balance is maintained until an external force or disturbance acts upon the system.

3.2. Perturbed State

Small asymmetric disturbances disrupt the fundamental equilibrium state $(r=R)$, consequently leading to the manifestation of the following interface equation as

$$r=R+\xi \left(z,t\right) \mathrm{i}.\mathrm{e}., f(r,\theta ,z,t)=r-R-\xi \left(\theta ,z,t\right)=0.$$

(16)

The outer unit normal vector is defined as

$$\overrightarrow{\mathit{n}}=\frac{\left(\widehat{\mathit{r}}-\frac{1}{r}\frac{\partial \xi }{\partial \theta }\widehat{\mathit{\theta }}-\frac{\partial \xi }{\partial z}\widehat{\mathit{z}}\right)}{\{{1+{\left(\frac{1}{r}\frac{\partial \xi }{\partial \theta }\right)}^2+{\left(\frac{\partial \xi }{\partial z}\right)}^2\}}^{1/2}},$$

(17)

in terms of radial unit vector $\widehat{\mathit{r}}$, azimuthal unit vector $\widehat{\mathit{\theta }}$, and axial unit vector $\widehat{\mathit{z}}$.

Mathematically, the velocity can be depicted as the potential function’s gradient due to the application of potential flow theory in this study, and therefore, ${\mathit{q}}_{\mathit{v}}=\nabla {\varphi }_v; {\mathit{q}}_{\mathit{r}}=\nabla {\varphi }_r$.

When considering incompressible fluids, the continuity equation is structured as $\nabla \cdot \mathit{q}=0$ due to the consistent density throughout the process. Hence,

$${\nabla }^2{\varphi }_v={\nabla }^2{\varphi }_r=0.$$

(18)

The derivation of the electric field is feasible from the scalar electric potential function as it is presumed that the problem adheres to the quasi-static estimation, denoted by $\psi (r,z,t)$. Hence,

$${\overrightarrow{\mathit{E}}}_j=E\widehat{\mathit{z}}-\nabla {\psi }_j;(j=v,r).$$

(19)

For Gauss’s law to remain valid, the electric potentials must satisfy Laplace’s equation, expressed as follows:

$${\nabla }^2{\psi }_v={\nabla }^2{\psi }_r=0.$$

(20)

3.2.1. Boundary Conditions

As the flow is irrotational in both the phases, Equation (7) takes the form

$$\begin{array}{l}\frac{\partial {\varphi }_v}{\partial r}=0 \mathrm{at} r=a,\\ \frac{\partial {\varphi }_r}{\partial r}=0 \mathrm{at} r=b.\end{array}$$

(21)

Equation (8) in perturbed form can be expressed as

$$\begin{array}{l}\frac{\partial {\psi }_v}{\partial z}=0 \mathrm{at} r=a,\\ \frac{\partial {\psi }_r}{\partial z}=0 \mathrm{at} r=b.\end{array}$$

(22)

3.2.2. Interfacial Conditions

In this study, we focus on axisymmetric disruptions. To scrutinize the stability of the system, let us introduce axisymmetric disruptions to Equations (9), (11), (13), (14), and (15), isolating solely the linear terms. Consequently, the resulting equations at the interface read

$${\rho }_v\left(-\frac{\partial \xi }{\partial t}+\frac{\partial {\varphi }_v}{\partial r}\right)={\rho }_r\left(-\frac{\partial \xi }{\partial t}+\frac{\partial {\varphi }_r}{\partial r}\right),$$

(23)

$${\rho }_v\left(-\frac{\partial \xi }{\partial t}+\frac{\partial {\varphi }_v}{\partial r}\right)=\alpha \xi ,$$

(24)

$$\frac{\partial {\psi }_v}{\partial z}=\frac{\partial {\psi }_r}{\partial z},$$

(25)

$${\sigma }_v\left(E\frac{\partial \xi }{\partial z}+\frac{\partial {\psi }_v}{\partial r}\right)={\sigma }_r\left(E\frac{\partial \xi }{\partial z}+\frac{\partial {\psi }_r}{\partial r}\right),$$

(26)

$$\begin{array}{l}\left({\rho }_r\frac{\partial {\varphi }_r}{\partial t}-{\rho }_{r}R{\Omega }^2\xi +2\left({\mu }_r+{{\mu }^{\prime }}_r\frac{\partial }{\partial t}\right)\frac{{\partial }^2{\varphi }_r}{\partial r^2}+E{\epsilon }_r\frac{\partial {\psi }_r}{\partial z}\right)-({\rho }_v\frac{\partial {\varphi }_v}{\partial t}+2{\mu }_v\frac{{\partial }^2{\varphi }_v}{\partial r^2}\\ +E{\epsilon }_v\frac{\partial {\psi }_v}{\partial z})=-\gamma \left(\frac{{\partial }^2\xi }{\partial z^2}+\frac{1}{R^2}\frac{{\partial }^2\xi }{\partial {\theta }^2}+\frac{\xi }{R^2}\right).\end{array}$$

(27)

Here, $\alpha =\frac{H^{\prime }(R)}{L}$ represents the heat transfer coefficient and H’(R) represents the value of dH/dr at r = R.

The process of normal modes [23,24,25] is employed to solve the governing equations, allowing for the determination of the elevation of the interface:

$$\xi ={\xi }^{\prime } \exp \left[i\left(kz+n\theta -\omega t\right)\right]+ \mathrm{c}.\mathrm{c}.,$$

(28)

where ``c.c.’’ stands for complexconjugate.

Here, ${\xi }^{\prime }$ is the amplitude of perturbations, $k$ denotes the wave number, $n$ is an integer, and $\omega$ represents the growth rate.

Upon resolving Equations (18) and (20) with the application of boundary circumstances (23)–(26), one can formulate the expressions of potential functions as follows:

$${\varphi }_v=\frac{1}{k}\left(-i\omega +\frac{\alpha }{{\rho }_v}\right)A_v(kr){\xi }^{\prime } \exp \left[i\left(kz+n\theta -\omega t\right)\right]+ \mathrm{c}.\mathrm{c}.,$$

(29)

$${\varphi }_r=\frac{1}{k}\left(-i\omega +\frac{\alpha }{{\rho }_r}\right)A_r(kr){\xi }^{\prime } \exp \left[i\left(kz+n\theta -\omega t\right)\right]+ \mathrm{c}.\mathrm{c}.,$$

(30)

$${\psi }_v=\frac{i ({\sigma }_r-{\sigma }_v) E g_r(k) }{{\sigma }_v{g}_r(k) G_v(k)-{\sigma }_r{g}_v(k) G_r(k)}\left(I_n(ka)K_n(kr)-I_n(ka)K_n(kr)\right){\xi }^{\prime } \exp \left[i\left(kz+n\theta -\omega t\right)\right],$$

(31)

$${\psi }_r=\frac{i ({\sigma }_r-{\sigma }_v) E g_v(k) }{{\sigma }_v{g}_r(k) G_v(k)-{\sigma }_r{g}_v(k) G_r(k)}\left(I_n(kr)K_n(kb)-I_n(kb)K_n(kr)\right) {\xi }^{\prime } \exp \left(i\left(kz+n\theta -\omega t\right)\right),$$

(32)

where

$$A_v(kR)=\frac{I_n(kr){K^{\prime }}_n(ka)-{I^{\prime }}_n(ka)K_n(kr)}{{I^{\prime }}_n(kR){K^{\prime }}_n(ka)-{I^{\prime }}_n(ka){K^{\prime }}_n(kR)},A_r(kR)=\frac{I_n(kr){K^{\prime }}_n(kb)-{I^{\prime }}_n(kb)K_n(kr)}{{I^{\prime }}_n(kR){K^{\prime }}_n(kb)-{I^{\prime }}_n(kb){K^{\prime }}_n(kR)},$$

$$\begin{array}{l}{g}_v(k)=I_n(ka)K_n(kR)-I_n(kR)K_n(ka),G_v(k)={I^{\prime }}_n(kR)K_n(ka)+I_n(ka){K^{\prime }}_n(kR),\\ g_r(k)=I_n(kb)K_n(kR)-I_n(kR)K_n(kb),G_r(k)={I^{\prime }}_n(kR)K_n(kb)-I_n(kb){K^{\prime }}_n(kR),\end{array}$$

where I_n and K_n are the modified Bessel’s functions and ${I^{\prime }}_n(kR)$ and ${K^{\prime }}_n(kR)$ represent the values of derivatives $d{I}_n(kr)/dr$ and $d{K}_n(kr)/dr$ at $r=R$, respectively.

4. Dispersion Relation

Using the expressions of $\xi , {\varphi }_v, {\varphi }_r,{\psi }_v$, and ${\psi }_r$ from Equations (29)–(32) in Equation (27) and solving results in

$$B_0{\omega }^2+i{B}_1\omega -B_2=0,$$

(33)

where

$$B_0={\rho }_v{A}_v(kR)-{\rho }_r{A}_r(kR)-2{{\mu }^{\prime }}_r{k}^2{B}_r(kR),$$

$$B_1=\alpha \left(A_v(kR)-A_r(kR)\right)+2k^2\left({\mu }_v{B}_v(kR)-{\mu }_r{B}_r(kR)\right)-2\frac{{{\mu }^{\prime }}_r}{{\rho }_r}\alpha k^2{B}_r(kR),$$

$$\begin{array}{l}{B}_2=2\alpha k^2\left(\frac{{\mu }_v}{{\rho }_v}{B}_v(kR)-\frac{{\mu }_r}{{\rho }_r}{B}_r(kR)\right)+\frac{\gamma k}{R^2}\left(k^2{R}^2+n^2-1\right)+{\rho }_{r}R{\Omega }^{2}k\\ \hspace{1em}\hspace{1em}\hspace{1em}-\frac{k^2{E}^2{g}_v(k)g_r(k)({\epsilon }_v-{\epsilon }_r)({\sigma }_v-{\sigma }_r)}{{\sigma }_v{g}_r(k) G_v(k)-{\sigma }_r{g}_v(k) G_r(k)},\end{array}$$

$$B_i(kR)=\left(1+\frac{n^2}{k^2{R}^2}\right)A_i(kR)-\frac{1}{kR},(i=v,r).$$

To compute the growth rate in the dispersion relation, a transformation is utilized by rewriting the variable $\omega =i {\omega }_0$, which leads to the following equation:

$$B_0{\omega }_0^2+B_1{\omega }_0+B_2=0.$$

(34)

After employing the Routh–Hurwitz principles in Equation (34), the stability condition can be expressed as $B_0>0,B_1>0,B_2>0$.

For a special k, considering the features of modified Bessel functions, it can be inferred that $A_r(kR)<0$ and $A_v(kR)>0$. Given the positive values of the viscosities and densities of the fluids, $B_0>0$ and $B_1>0.$ Consequently, based on the stability condition, this suggests that $B_2>0$. Then:

$$\begin{array}{l}2\alpha k\left(\frac{{\mu }_v}{{\rho }_v}{B}_v(kR)-\frac{{\mu }_r}{{\rho }_r}{B}_r(kR)\right)+\frac{\gamma }{R^2}\left(k^2{R}^2+n^2-1\right)+{\rho }_{r}R{\Omega }^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{k{E}^2{g}_v(k)g_r(k)({\epsilon }_v-{\epsilon }_r)({\sigma }_v-{\sigma }_r)}{{\sigma }_v{g}_r(k) G_v(k)-{\sigma }_r{g}_v(k) G_r(k)}=0.\end{array}$$

(35)

Our analysis makes it possible to evaluate the stability of the system, detecting instability when wave numbers exceed a significant (critical) value and stability when they fall below it. Furthermore, Equation (35) yields the critical assessment of the transition of the electric force. Therefore, whether the electric force exceeds or remains under this critical value determines whether the system is stable. The following conclusions can be drawn from an implicit implication of the expression (34).

• If the electric force is absent, Equation (34) reproduces the equation obtained in Ref. [12].
• In the absence of rotation, one achieves the expression from Ref. [26].
• If both the electric field and rotation are absent, the expression from Ref. [3] is achieved.

We use the special length and the special velocity to get rid of the units in Equation (34) and create a dimensionless arrangement. This enables us to use dimensionless variables to represent the equation $h=b-a$:

$${\overline{B}}_0{N}^2+{\overline{B}}_{1}N+{\overline{B}}_2=0,$$

(36)

where

$${\overline{B}}_0=\rho A_v(\widehat{k}\widehat{R})-A_r(\widehat{k}\widehat{R})-2{\lambda }_r{\widehat{k}}^2{B}_r(\widehat{k}\widehat{R}),$$

$${\overline{B}}_1=\widehat{\alpha }\left(A_v(\widehat{k}\widehat{R})-A_r(\widehat{k}\widehat{R})\right)+2{\widehat{k}}^2\frac{\mathrm{Ca}}{\mathrm{We}}\left(\mu B_v(\widehat{k}\widehat{R})-B_r(\widehat{k}\widehat{R})\right)-2{\lambda }_r\widehat{\alpha }{\widehat{k}}^2{B}_r(\widehat{k}\widehat{R}),$$

$$\begin{array}{l}{\overline{B}}_2=2\widehat{\alpha }{\widehat{k}}^2\frac{\mathrm{Ca}}{\mathrm{We}}\left(\kappa B_v(\widehat{k}\widehat{R})-B_r(\widehat{k}\widehat{R})\right)+\frac{\widehat{k}}{{\widehat{R}}^2}\frac{1}{We}\left({\widehat{k}}^2{\widehat{R}}^2-1\right)+\widehat{R}\widehat{k}\mathrm{Ce}\\ \hspace{1em}\hspace{1em}\hspace{1em}-\frac{{\widehat{k}}^2{\widehat{E}}^2{g}_v(\widehat{k})g_r(\widehat{k})(1-\epsilon )(1-\sigma )}{\sigma g_r(\widehat{k}) G_v(\widehat{k})-g_v(\widehat{k}) G_r(\widehat{k})}.\end{array}$$

Here, the hat on top of the quantities indicates the variable is dimensionless, $\mathrm{Ca}=\frac{{\mu }_{r}V}{\gamma }$ is the capillary number, while $\mathrm{We}=\frac{{\rho }_r{V}^{2}h}{\gamma }$ is for the Weber number, $\mathrm{Ce}=\frac{{\rho }_r{\Omega }^2{h}^3}{\gamma }$ is the centrifuge number, μ, ρ, ε, and σ are the ratios of viscosities, densities, permittivities, and electrical conductivities defined as $\mu =\frac{{\mu }_v}{{\mu }_r},\rho =\frac{{\rho }_v}{{\rho }_r},\epsilon =\frac{{\epsilon }_v}{{\epsilon }_r},\sigma =\frac{{\sigma }_v}{{\sigma }_r}$, ${\lambda }_r$ denotes dimensionless viscoelasticity coefficient, and $\widehat{\alpha }=\frac{\alpha h}{V{\rho }_r}$ is the dimensionless heat transfer coefficient.

Expressed in a dimensionless manner, the the neutral stability criterion reads

$$\begin{array}{l}2\widehat{k}\widehat{\alpha }\frac{\mathrm{Ca}}{\mathrm{We}}\left(\kappa B_v(\widehat{k}\widehat{R})-B_r(\widehat{k}\widehat{R})\right)+\frac{1}{{\widehat{R}}^2}\frac{1}{We}\left({\widehat{k}}^2{\widehat{R}}^2+n^2-1\right)+\widehat{R}\mathrm{Ce}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{\widehat{k}{\widehat{E}}^2{g}_v(\widehat{k})g_r(\widehat{k})(1-\epsilon )(1-\sigma )}{g_v(\widehat{k}) G_r(\widehat{k})-\sigma g_r(\widehat{k}) G_v(\widehat{k})}=0.\end{array}$$

(37)

For the values of $n$ ≥ 1, no a positive value of $k$ exists indicating the absence of instability in asymmetric perturbations. However, an instability can potentially occur in case of asymmetric modes. Therefore, in numerical computations, only axisymmetric disturbances are taken into account, ensuring that asymmetric disturbances, which may lead to instability, are not missed.

5. Results and Discussion

In this Section, we outline the computational procedures for symmetric disturbances utilizing Equations (36) and (37). Equation (36) is an algebraic equation of two degrees yielding two distinct values when solved for specific inputs. To show the growth rate in this analysis, we plot the larger value. In contrast, Equation (37) is given as an implicit equation, and the wavenumber can be found more straightforward by utilizing the Newton–Raphson method. Stability is indicated by the area overhead of the wave-number curve, whereas instability is indicated by the area below. The following set of values is used to perform the numerical computations [12]:

$$\begin{array}{l}\mathrm{Ca}=0.007, \mathrm{Ce}=0.2, \mathrm{We}=0.7, \widehat{\alpha }=0.1, \rho =0.01,\\ {\lambda }_r=0.2, \mu =0.001, \epsilon =0.2, \sigma =0.3, \widehat{E}=2.\end{array}$$

Figure 2 and Figure 3 display a contrast between the earlier findings [12,26] and our current results. The numerical results obtained for the problem considered here are presented in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10. These numerical results provide a comprehensive explanation of the physical implications, emphasizing the power of diverse non-dimensional characteristics on the growth of perturbations.

5.1. Comparison with Earlier Results

In Ref. [26], we studied the dynamics of a viscous–RE fluid interface when fluids are imperiled to an identical electric force in the axial path using the irrotational theory governing viscous–viscoelastic fluids. Their research concentrated on the behavior of fluids trapped in an annular enclosure formed by two rigid, stationary cylinders. Our current study, in contrast, keeps the exterior cylinder rotating continuously at a continuous angular velocity while maintaining the interior cylinder’s fixed position. The comparison between the extreme disturbance growth found in our current exploration and the results of Ref. [26] is shown in Figure 2. Interestingly, disruptions grow more slowly in swirling outer cylinder scenarios than in stationary outer cylinder scenarios. This difference suggests that the outer cylinder’s whirling motion contributes to stabilization.

Centrifugal force within the RE fluid phase is significantly increased in the presence of a swirling outer cylinder. While the viscous liquid inclines to converge towards the center, this strength causes the liquid to transport away from the epicenter of curvature at greater velocities. The centrifugal force causes disturbances, but these forces counteract each other and stabilize the interface. As a result, the addition of the swirl motion serves as a barrier to the expansion of disturbances.

In Ref. [12], the authors performed an instability inquiry into the swirling viscous–RE liquid interface taking heat/mass transportation into account, but they did not take the electric field effect into account. In contrast, the current study takes into account the mass/heat transfer in conjunction with the impact of the electric power on the swirling RE–viscous liquid interface. Figure 3 displays a contrast between the present outcomes and the findings of Ref. [12], with the goal of analyzing the impact of the electric power.

From examining the conductivity ratio and permittivity ratio values one concludes that the existence of an electric force can either promote or obstruct the development of perturbations. Specifically, when these ratios are less than 1, the electric field induces instability. Conversely, varying permittivity and conductivity ratios lead to a stabilizing effect of the electric force. The characteristic nature of the electric force at the interface of viscous–RE liquids, whether singular or dual, is contingent upon these values of conductivity and permittivity ratios. These observations are consistent with findings in Ref. [12].

5.2. Main Results

Figure 4 illustrates the interface stability concerning various capillary number values. A noticeable trend emerges, indicating that with an increase in the capillary number, the stable area also expands, signifying a stabilizing weight of the capillary number. Notably, the capillary number, derived from the ratio of RE fluid viscosity to surface tension, plays a significant role. The viscosity of the RE liquid contributes to stabilization, while surface tension promotes perturbation growth.

As the capillary number rises, the behavior of the RE fluid tends towards increased viscosity. This heightened viscosity hinders fluid movement, progressively fostering instability. Conversely, surface tension behaves in a manner that promotes disintegration and instability by intensifying pressures at the fluid’s surface.

Figure 5 provides profound insights into the significant impact of the centrifuge number on the delicate balance of interface stability. Beneath the curve lies the precarious domain of instability, while above it unfolds the reassuring realm of stability. In this meticulous investigation, the inner cylinder remains stationary, while its outer counterpart undergoes continuous rotation. As the centrifuge number steadily increases, a noticeable expansion of the stable domain becomes apparent, revealing a potent stabilizing force intrinsic to fluid dynamics’ intricate fabric.

The centrifuge number, reflecting the intricate balance between rotational forces and surface tension’s tenacity, heralds this transformative revelation. With the escalating centrifuge number, one of two phenomena unfolds: either the whirlwind velocity intensifies, or the firm grasp of surface tension weakens. The relentless increase in centrifugal force, driven by the spiraling vortex, exerts a formidable counterforce against burgeoning perturbations. The core of this stabilizing phenomenon accentuates the amplification of aerodynamic effects. These effects, in stark opposition to the disturbances incited by the unyielding grip of surface tension, lead to a noticeable reduction in perturbation magnitude at the interface. Consequently, the once-aroused perturbations at the interface diminish significantly.

Figure 6 delineates the behavior of perturbation growth curves across varying values of the heat transfer coefficient $\widehat{\alpha }=0.001,0.01,0.02, \mathrm{and} 0.03$. As the heat transfer parameter increases, a noticeable decline in perturbation growth is observed. This inverse relationship between heat transmission and growth rate suggests the stabilizing effect of the heat transfer parameter. Similar effects of heat transport were observed in the studies conducted earlier in Refs. [2,3]. The rationale behind this phenomenon lies in the mechanisms of evaporation and condensation.

Initially, the RE liquid exhibits greater temperatures associated withthe viscous liquid, prompting evaporation in the exterior area and condensation in the interior region. With increased heat transmission, the evaporation–condensation processes intensify, contributing significantly to the system’s stabilization.

Figure 7 illustrates the influence of the permittivity ratio on the interface of viscous–RE fluid and the constant strength of the electric field. The reduction in growth of perturbations when increasing the permittivity ratio highlights its stabilizing effect. The permittivity ratio of fluids inversely correlates with the RE liquid’s permittivity and directly with that of the viscous fluid. Consequently, while the permittivity of the RE liquid destabilizes the interface, the permittivity of the viscous liquid contributes to stability.

On the other hand, Figure 8 displays the fluid conductivity ratio effect on the interface stability. It indicates the stabilizing nature of the conductivity quotient of fluids by establishing a decrease in perturbation growth with an increase in the conductivity ratio. This ratio, derived from the RE liquid’s conductivity compared to the viscous fluid, affects the system’s stability significantly. While the lower conductivity of the RE liquid decreases the interface’s stability, the higher conductivity of the viscous liquid contributes positively to stability.

Figure 9 demonstrates how the Weber number ($\mathrm{We}$) impacts the range of stability. An increase in the Weber number expands the stable region, indicating enhanced stability at the interface. Therefore, the Weber number emerges as a critical determinant influencing system stability.

Fluid density interacts with inertia force, and a higher fluid density delays instabilities while slowing down disturbance propagation. This dynamic contributes to the observed increased stability, especially in the context of the high-density RE fluid interface. Conversely, surface tension exhibits an inverse relationship with the Weber number, thereby destabilizing the system in this analysis. Surface tension generates surface forces contributing to breakup phenomena. As surface tension amplifies, the magnitude of these forces increases. This leads to earlier instances of instability.

In Figure 10, the influence of viscoelasticity is demonstrated inthe interface between RE and viscous fluids. With increased viscoelasticity, there is a noticeable decrease in perturbation growth, whether or not the outer cylinder is swirling. This observation suggests that swirling does not modify the inherent environment of viscoelasticity. However, in the incidence of heat transmission, the advancement of perturbations diminishes, highlighting the role of heat transfer in impeding instability at the interface.

Viscoelasticity arises from temporary links formed among particles, such as fibers. In various polymers, elongated molecules create transient bonds with adjacent molecules, contributing to their viscoelastic behavior. As viscoelasticity strengthens, these intermolecular connections become more robust, hindering the propagation of disturbances. Hence, heightened viscoelasticity exerts a stabilizing influence. Similar findings were reported by us earlier [3].

6. Conclusions

In thecurrent investigation, we conducted an analysis of linear stability concerning a rotating interface that involves viscous–RE fluids while an axial electric field is present. Our study considers simultaneous heat and mass transports occurring at this interface. The outcome of our analysis provides an expression that represents an analytical relationship expressed as a quadratic polynomial related to the growth rate parameter. We illustrate this relationship through graphical representations of the marginal state and growth rate parameters, offering comprehensive insights into the influence of various physical parameters. The significant conclusions drawn from our study are as follows.

• The axial electric force demonstrates a dual outcome on the interface.
• The swirling motion of the outer cylinder restrains the growth of perturbations.
• The enhanced heat transfer at the interface reinforces the stability of the arrangement.
• The viscoelasticity inherent in the RE fluid showcases stabilizing characteristics.
• The density of the RE liquid contributes to system stability, while surface tension acts as a destabilizing factor.
• Both the permittivity ratio and conductivity ratio of liquids area stabilizing character forthe interface.

Future Scope

This problem can be examined within the framework of the porous media for a deeper exploration of how medium porosity influences the stability of the system. The influence of the magnetic field can also be included. Moreover, a non-linear analysis can be employed to investigate more complex stability scenarios.