| المستخلص: |
In Applied Mathematics as well as Physics, the Euler’s and Navier-Stokes equations of motion describe the motion of fluids in absence and presence of viscosity, respectively. These equations arise from applying Newton’s second law to fluid motion together with the assumption that the fluid stress is the sum of diffusing viscous term (proportional to the gradient of velocity) and a pressure term. The equations are useful because they describe the physics of many topics of interest. They may be used to model the weather, ocean currents, water flow in a pipe and airflow around a wing. These equations in their simplified and full forms help with the design of aircraft, study of blood flow, design of power stations, analysis of pollution, and many other applications. Coupled with Maxwell’s equations, they can be used to model and study electro hydrodynamics and magneto hydrodynamics. The Euler and Navier-Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses. In most of these applications, we use these equations in cylindrical and spherical configurations rather than the Cartesian coordinates. Therefore, in this research project, we aim firstly to derive the complete forms of Euler and Navier-Stokes equations of motion in different coordinate systems. Secondly, we apply these equations to study the electro hydrodynamic instability problem to two superposed dielectric fluids in relative horizontal motion under the effect of rotation about the vertical axis using the perturbation technique using normal modes method. We end our study by solving the obtained characteristic equation numerically using Mathematica software, and then drawing some figures to discuss and analyze the stability or instability of the system under consideration.
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