Electrohydrodynamic Instability of a Cylindrical Interface: Effect of the Buoyancy Thermo-Capillary in Porous Media

Abstract

Electrohydrodynamics (EHD) instability of a vertical cylindrical interface is tackled in the present study. The interface separates two viscous, homogeneous, porous, incompressible, and dielectric fluids which totate about the common cylindrical axis with different uniform angular velocities. A uniform axial electric field acts upon the considered system. Additionally, the influence of heat transfer is incorporated into the buoyancy term as well as the surface tension parameter, giving rise to the thermo-capillary effect. In this context, the viscous potential theory as well as the standard normal modes analysis are employed. The distributions of temperature, pressure, and velocity fields are evaluated. The linear stability approach resulted in an exceedingly complicated transcendental dispersion relation. The non-dimensional analysis revealed some physical Ohnesorge, Darcy, Rayleigh, Prandtle and Weber numbers. Actually, the dispersion relation has no closed form solution. Consequently, a numerical technique is utilized to display the stability profile. The relation between the growth rate and the wavenumber of the surface waves is constructed. The influences of various physical parameters on the stability profile are illustrated. It is found that the Ohnesorge number plays a dual role in the stability configuration.

Appendix

The constants that appearing in Eq. (20) may be listed as:

$${a_{11}} = \frac{{\Delta_1}}{\Delta }{T_0},\;\;\; {a_{12}} = \left( {\frac{{{T_a}\Delta - {\Delta_1}{I_m}({s_1}a)}}{{\Delta {K_m}({s_1}a)}}} \right){T_0},\;\;\; {a_{21}} = \frac{{\Delta_2}}{\Delta }{T_0},\qquad and \qquad {a_{22}} = \left( {\frac{{{T_b}\Delta - {\Delta_2}{I_m}({s_2}b)}}{{\Delta {K_m}({s_2}b)}}} \right){T_0},$$

where

$$\begin{aligned}{\Delta_1} =&\; \left( {s{k_f}\left({\frac{{{T_b}{K_m}({s_2}R)}}{{{K_m}({s_2}b)}} - \frac{{{T_a}{K_m}({s_1}R)}}{{{K_m}({s_1}a)}}} \right)\left( {\frac{{{I_m}({s_2}b)K_m^{^{\prime}}({s_2}R) - {K_m}({s_2}b)I_m^{^{\prime}}({s_2}R)}}{{{K_m}({s_2}b)}}} \right) } \right. \\& \left.- {\left( {\frac{{{T_b}s{k_f}K_m^{^{\prime}}({s_2}R)}}{{{K_m}({s_2}b)}} - \frac{{{T_a}K_m^{^{\prime}}({s_1}R)}}{{{K_m}({s_1}a)}}} \right)\left( {\frac{{{I_m}({s_2}b){K_m}({s_2}R) - {K_m}({s_2}b){I_m}({s_2}R)}}{{{K_m}({s_2}b)}}} \right)} \right),\end{aligned}$$

$$\begin{aligned} {\Delta_2} = \left( {\left( {\frac{{{T_b}s{k_f}K_m^{^{\prime}}({s_2}R)}}{{{K_m}({s_2}b)}} - \frac{{{T_a}K_m^{^{\prime}}({s_1}R)}}{{{K_m}({s_1}a)}}} \right)\left( {\frac{{{I_m}({s_1}R){K_m}({s_1}a) - {K_m}({s_1}R){I_m}({s_1}a)}}{{{K_m}({s_1}a)}}} \right)} \right. \hfill \\ \left. \,\,\,\, {-} {\left( {\frac{{{T_b}{K_m}({s_2}R)}}{{{K_m}({s_2}b)}} - \frac{{{T_a}{K_m}({s_1}R)}}{{{K_m}({s_1}a)}}} \right)\left( {\frac{{{K_m}({s_1}a)I_m^{^{\prime}}({s_1}R) - {I_m}({s_1}a)K_m^{^{\prime}}({s_1}R)}}{{{K_m}({s_1}a)}}} \right)} \right)\,\,\,\,, \hfill \\ \end{aligned}$$

$$\begin{aligned} \triangle=sk_f\left(\frac{I_m\left(s_2R\right)K_m\left(s_1a\right)-K_m\left(s_1R\right)I_m\left(s_1a\right)}{K_m\left(s_1a\right)}\right)\left(\frac{I_m\left(s_2b\right)K{_{m}^{\prime}}\left(s_2R\right)-K_m\left(s_2b\right)I{_{m}^{\prime}}\left(s_2R\right)}{K_m\left(s_2b\right)}\right)-\\ \left(\frac{I_m\left(s_2b\right)K_m\left(s_2R\right)-K_m\left(s_2b\right)I_m\left(s_2R\right)}{K_m\left(s_2b\right)}\right)\left(\frac{K_m\left(s_1a\right)I{_{m}^{\prime}}\left(s_1R\right)-I_m\left(s_1a\right)K{_{m}^{\prime}}\left(s_1R\right)}{K_m\left(s_1a\right)}\right) \cdot \end{aligned}$$

The constants that appearing in Eq. (21) may be listed as:

$$\begin{aligned}b_{11}=&\frac{\sigma_0}{R^2}\left(\frac{iR^2z^2Ra}{k\Pr}\right)\left(\frac{q_1^2}{s_1^2-q_1^2}\right)\frac{\Delta_{1a}}{\Delta_a}\eta_0,\qquad\qquad\qquad\qquad &b_{12}=-\frac{\sigma_0}{R^2}\left(\frac{iR^2z^2Ra}{k\Pr}\right)\left(\frac{q_1^2}{s_1^2-q_1^2}\right)\frac{\Delta_{2a}}{\Delta_a}\eta_0\;,\\ b_{21}=&\frac{\sigma_0}{R^2}\left(\frac{i\rho_0\beta R^2z^2Ra}{k\Pr}\right)\left(\frac{q_2^2}{s_2^2-q_2^2}\right)\frac{\Delta_{1b}}{\Delta_b}\eta_0,\qquad\qquad\mathrm{and}\qquad\qquad &b_{22}=-\frac{\sigma_0}{R^2}\left(\frac{i\rho_0\beta R^2z^2Ra}{k\Pr}\right)\left(\frac{q_2^2}{s_2^2-q_2^2}\right)\frac{\Delta_{2b}}{\Delta_b}\eta_0\;,\end{aligned}$$

where

$${\begin{aligned} {\Delta_{1a}} =&\; \left( {\frac{{{\omega_1}{s_1}R}}{{4i\sqrt {We} }}\left( {\frac{{\Delta_1}}{\Delta }I_m^{^{\prime}}({s_1}R) + \left( {\frac{{{T_a}\Delta - {\Delta_1}{I_m}({s_1}a)}}{{\Delta {K_m}({s_1}a)}}} \right)K_m^{^{\prime}}({s_1}R)} \right) + m\left( {\frac{{\Delta_1}}{\Delta }{I_m}({s_1}R) + \left( {\frac{{{T_a}\Delta - {\Delta_1}{I_m}({s_1}a)}}{{\Delta {K_m}({s_1}a)}}} \right){K_m}({s_1}R)} \right)} \right. \\& \left. { - \frac{{kR(s_1^2 - q_1^2)(\omega + im\sqrt {We} )(\omega_1^2 + 16We)\Pr }}{{4\sqrt {We} q_1^2{Z^2}Ra}}} \right)\left( {\frac{{{\omega_1}{q_1}aK_m^{^{\prime}}({q_1}a)}}{{4i\sqrt {We} }} + mK{}_m({q_1}a)} \right) - \left( {\frac{{{\omega_1}{q_1}RK_m^{^{\prime}}({q_1}R)}}{{4i\sqrt {We} }} + mK{}_m({q_1}R)} \right) \\& \times \left( {\frac{{{\omega_1}{s_1}a}}{{4i\sqrt {We} }}\left( {\frac{{\Delta_1}}{\Delta }I_m^{^{\prime}}({s_1}a) + \left( {\frac{{{T_a}\Delta - {\Delta_1}{I_m}({s_1}a)}}{{\Delta {K_m}({s_1}a)}}} \right)K_m^{^{\prime}}({s_1}a)} \right) + m{T_a}} \right)\;, \end{aligned}}$$

$$\begin{aligned} {\Delta_{2a}} =&\; \left( {\frac{{{\omega_1}{s_1}R}}{{4i\sqrt {We} }}\left( {\frac{{\Delta_1}}{\Delta }I_m^{^{\prime}}({s_1}R) + \left( {\frac{{{T_a}\Delta - {\Delta_1}{I_m}({s_1}a)}}{{\Delta {K_m}({s_1}a)}}} \right)K_m^{^{\prime}}({s_1}R)} \right) + m\left( {\frac{{\Delta_1}}{\Delta }{I_m}({s_1}R) + \left( {\frac{{{T_a}\Delta - {\Delta_1}{I_m}({s_1}a)}}{{\Delta {K_m}({s_1}a)}}} \right){K_m}({s_1}R)} \right)} \right. \\& \left. { - \frac{{kR(s_1^2 - q_1^2)(\omega + im\sqrt {We} )(\omega_1^2 + 16We)\Pr }}{{4\sqrt {We} q_1^2{Z^2}Ra}}} \right)\left( {\frac{{{\omega_1}{q_1}aI_m^{^{\prime}}({q_1}a)}}{{4i\sqrt {We} }} + mI{}_m({q_1}a)} \right) - \left( {\frac{{{\omega_1}{q_1}RI_m^{^{\prime}}({q_1}R)}}{{4i\sqrt {We} }} + mI{}_m({q_1}R)} \right) \\& \times \left( {\frac{{{\omega_1}{s_1}a}}{{4i\sqrt {We} }}\left( {\frac{{\Delta_1}}{\Delta }I_m^{^{\prime}}({s_1}a) + \left( {\frac{{{T_a}\Delta - {\Delta_1}{I_m}({s_1}a)}}{{\Delta {K_m}({s_1}a)}}} \right)K_m^{^{\prime}}({s_1}a)} \right) + m{T_a}} \right), \end{aligned}$$

$$\begin{aligned} {\Delta_a} =&\; \left( {\frac{{{\omega_1}{q_1}aI_m^{^{\prime}}({q_1}a)}}{{4i\sqrt {We} }} + mI{}_m({q_1}a)} \right)\left( {\frac{{{\omega_1}{q_1}RK_m^{^{\prime}}({q_1}R)}}{{4i\sqrt {We} }} + mK{}_m({q_1}R)} \right) \\& - \left( {\frac{{{\omega_1}{q_1}aK_m^{^{\prime}}({q_1}a)}}{{4i\sqrt {We} }} + mK{}_m({q_1}a)} \right) \times \left( {\frac{{{\omega_1}{q_1}RI_m^{^{\prime}}({q_1}R)}}{{4i\sqrt {We} }} + mI{}_m({q_1}R)} \right), \end{aligned}$$

$${\begin{aligned} {\Delta_{1b}} =&\; \left( {\frac{{{\omega_2}{s_2}R}}{{4i\sqrt {We} }}\left( {\frac{{\Delta_2}}{\Delta }I_m^{^{\prime}}({s_2}R) + \left( {\frac{{{T_b}\Delta - {\Delta_2}{I_m}({s_2}b)}}{{\Delta {K_m}({s_2}b)}}} \right)K_m^{^{\prime}}({s_2}R)} \right) + m\left( {\frac{{\Delta_2}}{\Delta }{I_m}({s_2}R) + \left( {\frac{{{T_b}\Delta - {\Delta_2}{I_m}({s_2}b)}}{{\Delta {K_m}({s_2}b)}}} \right){K_m}({s_2}R)} \right)} \right. \\& \left. { - \frac{{kR(s_2^2 - q_2^2)(\omega + im\Omega\sqrt {We} )(\omega_2^2 + 16\Omega^2 We)\Pr }}{{4\beta\Omega\sqrt {We} q_2^2{Z^2}Ra}}} \right)\left( {\frac{{{\omega_2}{q_2}bK_m^{^{\prime}}({q_2}b)}}{{4i\Omega\sqrt {We} }} + mK{}_m({q_2}b)} \right) - \left( {\frac{{{\omega_2}{q_2}RK_m^{^{\prime}}({q_2}R)}}{{4i\Omega\sqrt {We} }} + mK{}_m({q_2}R)} \right) \\& \times \left( {\frac{{{\omega_2}{s_2}b}}{{4i\Omega\sqrt {We} }}\left( {\frac{{\Delta_2}}{\Delta }I_m^{^{\prime}}({s_2}b) + \left( {\frac{{{T_b}\Delta - {\Delta_2}{I_m}({s_2}b)}}{{\Delta {K_m}({s_2}b)}}} \right)K_m^{^{\prime}}({s_2}b)} \right) + m{T_b}} \right), \end{aligned}}$$

$${\begin{aligned} {\Delta_{2b}} =&\; \left( {\frac{{{\omega_2}{s_2}R}}{{4i\Omega\sqrt {We} }}\left( {\frac{{\Delta_2}}{\Delta }I_m^{^{\prime}}({s_2}R) + \left( {\frac{{{T_b}\Delta - {\Delta_2}{I_m}({s_2}b)}}{{\Delta {K_m}({s_2}b)}}} \right)K_m^{^{\prime}}({s_2}R)} \right) + m\left( {\frac{{\Delta_2}}{\Delta }{I_m}({s_2}R) + \left( {\frac{{{T_b}\Delta - {\Delta_2}{I_m}({s_2}b)}}{{\Delta {K_m}({s_2}b)}}} \right){K_m}({s_2}R)} \right)} \right. \\& \left. { - \frac{{kR(s_2^2 - q_2^2)(\omega + im\Omega\sqrt {We} )(\omega_2^2 + 16\Omega^2 We)\Pr }}{{4\beta\Omega\sqrt {We} q_2^2{Z^2}Ra}}} \right)\left( {\frac{{{\omega_2}{q_2}bI_m^{^{\prime}}({q_2}b)}}{{4i\Omega\sqrt {We} }} + mI{}_m({q_2}b)} \right) - \left( {\frac{{{\omega_2}{q_2}RI_m^{^{\prime}}({q_2}R)}}{{4i\Omega\sqrt {We} }} + mI{}_m({q_2}R)} \right) \\& \times \left( {\frac{{{\omega_2}{s_2}b}}{{4i\Omega\sqrt {We} }}\left( {\frac{{\Delta_2}}{\Delta }I_m^{^{\prime}}({s_2}b) + \left( {\frac{{{T_b}\Delta - {\Delta_2}{I_m}({s_2}b)}}{{\Delta {K_m}({s_2}b)}}} \right)K_m^{^{\prime}}({s_2}b)} \right) + m{T_b}} \right), \end{aligned}}$$

and

$$\begin{aligned} \Delta_{b} =&\; \left( {\frac{{\omega_{2} q_{2} bI_{m}^{^{\prime}} (q_{2} b)}}{{4i\Omega \sqrt {We} }} + mI{}_{m}(q_{2} b)} \right)\left( {\frac{{\omega_{2} q_{2} RK_{m}^{^{\prime}} (q_{2} R)}}{{4i\Omega \sqrt {We} }} + mK{}_{m}(q_{2} R)} \right) \\&- \left( {\frac{{\omega_{2} q_{2} bK_{m}^{^{\prime}} (q_{2} b)}}{{4i\Omega \sqrt {We} }} + mK{}_{m}(q_{2} b)} \right) \times \left( {\frac{{\omega_{2} q_{2} RI_{m}^{^{\prime}} (q_{2} R)}}{{4i\Omega \sqrt {We} }} + mI{}_{m}(q_{2} R)} \right) \end{aligned}$$

The constants that appear in Eq. (29) may be written as: 

$$\begin{aligned} {c_{11}} = \left( {\frac{{ - i{E_0}(1 - \varepsilon ){f_2}}}{{{d_1}{f_2} - \varepsilon {d_2}{f_1}}}} \right){\eta_0}, \qquad\qquad\qquad\qquad\;\;\;\;\;\; {c_{12}} = \frac{I_m^{^{\prime}}(ka)}{{K_m^{^{\prime}}(ka)}}\left( {\frac{{i{E_0}(1 - \varepsilon ){f_2}}}{{{d_1}{f_2} - \varepsilon {d_2}{f_1}}}} \right){\eta_0},\\ {c_{21}} = \left( {\frac{{ - i{E_0}(1 - \varepsilon ){f_1}}}{{{d_1}{f_2} - \varepsilon {d_2}{f_1}}}} \right){\eta_0}, \qquad\qquad \mathrm{and} \qquad\qquad {c_{22}} = \frac{{{{I^{\prime}}_m}\left( {kb} \right)}}{{{{K^{\prime}}_m}\left( {kb} \right)}}\left( {\frac{{i{E_0}\left( {1 - \varepsilon } \right){f_1}}}{{{d_1}{f_2} - \varepsilon {d_2}f}}} \right){\eta_0}, \end{aligned}$$

where.

$$\begin{aligned} {f_1} = \frac{{{I_m}(kR)K_m^{^{\prime}}(ka) - {K_m}(kR)I_m^{^{\prime}}(ka)}}{K_m^{^{\prime}}(ka)}, \qquad\qquad\qquad\qquad\qquad\;\; {f_2} = \frac{{{I_m}\left( {kR} \right){{K^{\prime}}_m}\left( {kb} \right) - {K_m}\left( {kR} \right){{I^{\prime}}_m}\left( {kb} \right)}}{{{{K^{\prime}}_m}\left( {kb} \right)}}, \\ {d_1} = \frac{{{{I^{\prime}}_m}\left( {kR} \right){{K^{\prime}}_m}\left( {ka} \right) - {{K^{\prime}}_m}\left( {kR} \right){{I^{\prime}}_m}\left( {ka} \right)}}{{{{K^{\prime}}_m}\left( {ka} \right)}},\qquad\qquad and \qquad\qquad {d_2} = \frac{I_m^{^{\prime}}(kR)K_m^{^{\prime}}(kb) - K_m^{^{\prime}}(kR)I_m^{^{\prime}}(kb)}{{K_m^{^{\prime}}(kb)}}. \end{aligned}$$

The quantities that appear in Eq. (41) may be written as:

$${\begin{aligned} {Q_1} =& \frac{{{\omega_1}{R^2}}}{{4i\sqrt {We} }}\left( {q_1^2\left( {\frac{{{\Delta_{1a}}I_m^{^{\prime\prime}}({q_1}R) - {\Delta_{2a}}K_m^{^{\prime\prime}}({q_1}R)}}{{\Delta_a}}} \right) + s_1^2\left( {\frac{{\Delta_1}}{\Delta }I_m^{^{\prime\prime}}({s_1}R) + \left( {\frac{{{T_a}\Delta - {\Delta_1}{I_m}({s_1}a)}}{{\Delta {K_m}({s_1}a)}}} \right)K_m^{^{\prime\prime}}({s_1}R)} \right)} \right) \hfill \\& + mR\left( {{q_1}\left( {\frac{{{\Delta_{1a}}I_m^{^{\prime}}({q_1}R) - {\Delta_{2a}}K_m^{^{\prime}}({q_1}R)}}{{\Delta_a}}} \right) + {s_1}\left( {\frac{{\Delta_1}}{\Delta }I_m^{^{\prime}}({s_1}R) + \left( {\frac{{{T_a}\Delta - {\Delta_1}{I_m}({s_1}a)}}{{\Delta {K_m}({s_1}a)}}} \right)K_m^{^{\prime}}({s_1}R)} \right)} \right) \hfill \\& - m\left( {\left( {\frac{{{\Delta_{1a}}{I_m}({q_1}R) - {\Delta_{2a}}{K_m}({q_1}R)}}{{\Delta_a}}} \right) + \left( {\frac{{\Delta_1}}{\Delta }{I_m}({s_1}R) + \left( {\frac{{{T_a}\Delta - {\Delta_1}{I_m}({s_1}a)}}{{\Delta {K_m}({s_1}a)}}} \right){K_m}({s_1}R)} \right)} \right) \hfill \\ \end{aligned}}{\mathrm {and}}$$

$$\begin{aligned} {Q_2} =& \frac{{{\omega_2}{R^2}}}{{4i\Omega \sqrt {We} }}\left( {q_2^2\left( {\frac{{{\Delta_{1b}}I_m^{^{\prime\prime}}({q_2}R) - {\Delta_{2b}}K_m^{^{\prime\prime}}({q_2}R)}}{{\Delta_b}}} \right) + s_2^2\left( {\frac{{\Delta_2}}{\Delta }I_m^{^{\prime\prime}}({s_2}R) + \left( {\frac{{{T_b}\Delta - {\Delta_2}{I_m}({s_2}b)}}{{\Delta {K_m}({s_2}b)}}} \right)K_m^{^{\prime\prime}}({s_2}R)} \right)} \right) \hfill \\& + mR\left( {{q_2}\left( {\frac{{{\Delta_{1b}}I_m^{^{\prime}}({q_2}R) - {\Delta_{2b}}K_m^{^{\prime}}({q_2}R)}}{{\Delta_b}}} \right) + {s_2}\left( {\frac{{\Delta_2}}{\Delta }I_m^{^{\prime}}({s_2}R) + \left( {\frac{{{T_b}\Delta - {\Delta_2}{I_m}({s_2}b)}}{{\Delta {K_m}({s_2}b)}}} \right)K_m^{^{\prime}}({s_2}R)} \right)} \right) \hfill \\& - m\left( {\left( {\frac{{{\Delta_{1b}}{I_m}({q_2}R) - {\Delta_{2b}}{K_m}({q_2}R)}}{{\Delta_b}}} \right) + \left( {\frac{{\Delta_2}}{\Delta }{I_m}({s_2}R) + \left( {\frac{{{T_b}\Delta - {\Delta_2}{I_m}({s_2}b)}}{{\Delta {K_m}({s_2}b)}}} \right){K_m}({s_2}R)} \right)} \right) \hfill \\ \end{aligned}$$