Nonlinear stability of electro-visco-elastic Walters’ B type in porous media

Abstract

This paper investigates the nonlinear instability of a non-Newtonian fluid of the Walters’ B type. The fluids fill the regions inside and outside a vertical circular cylinder. An axial electric field of uniform strength is pervaded along the axis of the jet. The fluids are saturated in porous media. Typically, the nonlinear analysis is based on solving the linear governing equations of motion, then applying the convenient nonlinear boundary conditions. This methodology yields a nonlinear characteristic equation which governs the behavior of the interface deflection. As the nonlinear terms are omitted, a linear dispersion relation arises. Therefore, the stability criteria are analytically analyzed and numerically confirmed. The nonlinear approach depends on the multiple time scale technique together with the support of the Taylor theory. This approach resulted in a Ginzburg–Landau equation. Consequently, the stability criteria are achieved in both analytical and numerical analysis. Furthermore, by means of the expanded frequency analysis, a bounded approximate solution of the amplitude of the surface waves is accomplished. The homotopy perturbation method (MPM) is utilized to obtain an approximate distribution of the conducted artificial frequency. Additionally, the generating function of the interface is graphically represented. Several special cases are reported upon convenient data choices. Regions of stability and instability are addressed. In the stability profile, the electric field intensity is plotted versus the wave number. The influences of the parameters on the stability are identified. The nonlinear stability approach divides the phase plane into several parts of stability/instability.

Appendix

The coefficients that are appearing in Eq. (26) may be listed as follows:

$$c = \frac{{I_{0} (kR)\left( {2\mu^{\prime}_{1} k^{2} - \rho_{1} } \right)}}{{kI_{1} (kR)}} + \frac{{K_{0} (kR)\left( {2\mu^{\prime}_{2} k^{2} - \rho_{2} } \right)}}{{kK_{1} (kR)}} - \frac{{\mu^{\prime}_{2} - \mu^{\prime}_{1} }}{R},$$

$$a_{1} = \frac{{U_{1} \left( {\mu^{\prime}_{1} k^{2} (I_{0} (kR) + I_{2} (kR)) - \rho_{1} I_{0} (kR)} \right)}}{{k\,c\,I_{1} (kR)}} + \frac{{U_{2} \left( {\mu^{\prime}_{2} k^{2} (K_{0} (kR) + K_{2} (kR)) - \rho_{2} K_{0} (kR)} \right)}}{{k\,c\,K_{1} (kR)}},$$

$$a_{2} = \frac{T}{{R^{2} }},$$

$$b_{1} = \frac{{2(\mu_{1} - \mu_{2} )}}{c\,R} - \frac{{I_{0} (kR)\left( {2\mu_{1} k^{2} + \nu_{1} } \right)}}{{k\,c\,I_{1} (kR)}} - \frac{{K_{0} (kR)\left( {2\mu_{2} k^{2} + \nu_{2} } \right)}}{{k\,c\,K_{1} (kR)}},$$

$$c_{1} = - \frac{{\rho_{1} U_{1} I_{0} (kR)}}{{\,c\,I_{1} (kR)}} - \frac{{\rho_{2} U_{2} K_{0} (kR)}}{{\,c\,K_{1} (kR)}},$$

$$b_{2} = - \frac{{U_{1} \left( {\mu_{1} k^{2} (I_{0} (kR) + I_{2} (kR)) + \nu_{1} I_{0} (kR)} \right)}}{{k\,c\,I_{1} (kR)}} - \frac{{U_{2} \left( {\mu_{2} k^{2} (K_{0} (kR) + K_{2} (kR)) + \nu_{2} K_{0} (kR)} \right)}}{{k\,c\,K_{1} (kR)}},$$

$$c_{2} = - \frac{{I_{0} (kR)K_{0} (kR)\left( {\varepsilon_{1} - \varepsilon_{2} } \right)^{2} E_{0}^{2} }}{{\,c\,\left( {\varepsilon_{1} I_{1} (kR)K_{0} (kR) + \varepsilon_{2} I_{0} (kR)K_{1} (kR)} \right)}} - \frac{{\rho_{1} U_{1}^{2} I_{0} (kR)}}{{\,c\,I_{1} (kR)}} - \frac{{\rho_{2} U_{2}^{2} K_{0} (kR)}}{{\,c\,K_{1} (kR)}}.$$