Suliman ELzobeir Shain Elmaleeh2023-01-222023-01-222022http://hdl.handle.net/123456789/18707A Thesis Submitted in Fulfillment of the Requirements for the Degree of PhD in MathematicsAbstract In this research, we studied the thermal stability of a fluid that flows between two parallel surfaces under the influence of rotaƟon and the magneƟc field. We started with the physical nature of the problem and formed the governing mathemaƟcal equaƟons of the moƟon, which are nonlinear parƟal differenƟal equaƟons, we used the method of linear stability analysis to determine its stability, and find the relaƟon between the non-dimensional parameters that govern the system, by using the Spectral Chebyshev Tau method. We started in the first chapter by studying the problem. In the second chapter we studied the problem under the effect of magneƟc field. In the third we studied the stability of rotaƟon, and in the fourth chapter the stability of coueƩe flow between two coaxial cylinders, and obtained analyƟcal soluƟon. In each cases of chapter two and three we obtained the numerical results which represents the relaƟonship between the non- dimensional numbers and comparing there results by the previous results that were obtained for the same problemCOUETTE FLOW AND BENARDThesis