Theoretical and Mathematical Physics, 191(1): 580–601 (2017) STABILITY CHARACTERISTICS OF PERIODIC STREAMING FLUIDS IN POROUS MEDIA S. A. Alkharashi ∗ and Y. Gamiel † We study the linear stability of a three-layer flow of immiscible liquids located in a periodic normal electric field. We consider certain porous media assumed to be uniform, homogeneous, and isotropic. We analytically and numerically simulate the system of linear evolution equations of such a medium. The linearized problem leads to a system of two Mathieu equations with complex coefficients of the damping terms. We study the effects of the streaming velocity, permeability of the porous medium, and the electrical properties of the flow of a thin layer (film) of liquid on the flow instability. We consider several special cases of such systems. As a special case, we consider a uniform electric field and solve the transition curve equations up to the second order in a small dimensionless parameter. We show that the dielectric constant ratio and also the electric field play a destabilizing role in the stability criteria, while the porosity has a dual effect on the wave motion. In the case of an alternating electric field and a periodic velocity, we use the method of multiple time scales to calculate approximate solutions and analyze the stability criteria in the nonresonance and resonance cases; we also obtain transition curves in these cases. We show that an increase in the velocity and the electric field promote oscillations and hence have a destabilizing effect. Keywords: linear stability, periodic electric field, porous media, Mathieu equation, streamline DOI: 10.1134/S0040577917040092 1. Introduction Our purpose here is to develop a mathematical model for the flow of a thin layer (film) of liquid bounded by two layers of another liquid in porous media. The fluids are subjected to a periodic electric field normal to the layers. There are currently many works devoted to hydrodynamic stability. For example, unsteady electro- hydrodynamic stability was studied in [1], where the stability of a basic flow of streaming fluids in an oblique periodic electric field was analyzed. In [2], the instability of a potential flow of a viscous liquid in a horizontal rectangular channel was studied, and an explicit dispersion relation was obtained in which the effects of surface tension and viscosity on the normal stress were taken into account but the effect of shear stresses was not. The influence of viscosity on the stability of a planar interface separating two incompressible superposed fluids of uniform densities in the case where the whole system is located in a uniform horizontal magnetic ∗ Quesna Technical College, Tanta Technical Commercial Institute, Ministry of Higher Education, Tanta, Egypt; Applied Sciences Department, College of Technological Studies, The Public Authority for Applied Education and Training, Adiliya, Kuwait, e-mail: sameh7977@yahoo.com. † Department of Engineering Mathematics and Physics, Faculty of Engineering, Tanta University, Tanta, Egypt. Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 191, No. 1, pp. 126–150, April, 2017. Original article submitted January 13, 2016; revised February 3, 2016. 580 0040-5779/17/1911-0580 c  2017 Pleiades Publishing, Ltd.