Viscous-inviscid coupled problem with interfacial data.

摘要

The Navier-Stokes equation is the primary equation of computational fluid dynamics describing the flow/motion of fluids in $Rn$, (n = 2,3). These types of equations are often used in computations of aircraft and ship design, weather prediction, and climate modelling. By appropriate assumptions, it has been generalized to a system of equations known as the incompressible Navier-Stokes equations, see [13]. This important system has been studied for centuries by mathematicians, engineers and other scientists to explain and predict the behavior of the system under consideration, but still the understanding of the solutions to this system remains minimal. The challenge is to make substantial progress toward a mathematical theory which will solve the puzzle behind the Navier-Stokes equations. To make contributions to this mathematical theory, scientists have studied and derived many other systems from it. Among them is the viscous/inviscid coupled problem (VIC) introduced first by Xu Chuanju in his Ph.D dissertation [24].; The work presented in this dissertation involves Xu's [27] problem and focuses on three main objectives. The first one is to show the existence and uniqueness of the solution for the system, which results from the viscous/inviscid coupled problem when interfacial data (VIC-ID) are imposed. The second objective is to prove that the solution of this system can be obtained as a limit of solutions of two subproblems defined in different subdomains of the domain, this was originally done by Xu [27] using lifting operators and uniform relaxation parameter. In this dissertation it is done by using non-uniform relaxation parameters, avoiding the use of lifting operators. Finally, the last objective is to provide new exact solutions when some of the boundary conditions are not satisfied in the subdomains (weak boundary conditions) and all boundary conditions are satisfied in at least one of the subdomains (weaker boundary conditions) of the viscous/inviscid coupled problem. The new improvements presented in this dissertation demonstrate progress towards the existing theory of the VIC problem and therefore for the Navier-Stokes equations.