Classes and Properties of Exact Solutions to the ThreeDimensional Incompressible Navier-Stokes Equations - Gunawan, Nugroho (2010)
The Navier-Stokes equations together with the continuity equation are one of the long standing problems in mathematical physics. They form a system of nonlinear partial differential equations that describe the fluid flow phenomena, whether laminar or turbulent. The nonlinearity of the equations is obscure which defies all conventional methods of analytical solution to the differential equations. The analytical methods are found to be very important to model physical phenomena. They form basic understanding of the phenomena at different circumstances, at least qualitatively. In addition to their physics, the analytical methods are also useful to find and extend the class of existence, uniqueness and regularity in the pure mathematics sense. This thesis introduces new analytical methods of finding solutions of the incompressible Navier-Stokes equations. The work is based on the criteria of wellposed problem which is then solved by the proposed special classes of the solution either qualitatively or quantitatively. Firstly, general qualitative properties of solutions to the three-dimensional incompressible flows are presented. The method rs performed from the implementation of vector analysis into the energy equation with the consideration of zero rate energy. Trivial solution is obtained from any initial-boundary value problems. For the cases of non trivial solution, the analyticity of the solutions is assumed to investigate the triviality at intersection regions. Some physical consequences due to violation of the trivial solutions are also performed with the application of the vorticity equations, which are related to the onset of turbulence. Therefore, non trivial solutions will also represent turbulence whether they have singularity or not. This hypothesis is supplemented by investigation on the solution in the special classes of v = v x and v = V + V x of the three-dimensional incompressible Navier-Stokes equations. Analysis is taken using the vorticity equations rather than the original Navier Stokes equations based on qualitative mathematical work. Results show that the corresponding problem admits a unique and regular solution because the original problems can be transformed to class of linear parabolic and elliptic equations. The first analytical solution is then produced using the four components coordinate transformation .; = kx + ly + mz- ct. While, the second solution is produced using the three components coordinate transformation .; = ly + mz- ct . Velocity vector in the solutions is based on the relation v =Vet>+ v x ct> where <t> is a potential function that is defined as <l>=P(x,~)R(~). The potential function is firstly substituted into the continuity equation. The solution for R is produced using a certain mathematical condition and the resultant expression is used sequentially in the Navier-Stokes equations to reduce the problem to the class of nonlinear ordinary differential equations in P terms. Here, more general solutions are also obtained based on the particular solutions of P . The two solutions are based on a zero and constant pressure gradients which are given to illustrate the applicability of the method. The third analytical solution utilises a potential function m the form ct> = P(x,y,q)R(y)s(.;) with the application of the transformed coordinate ~ = kz- dt). In this solution, the pressure term is presented in a general functional form. The solutions for R and S are obtained by imposing a certain mathematical condition. General solutions are then obtained based on the particular solutions of P where the equation is reduced to the form of linear differential equation. A method for finding closed-form solutions for general linear differential equations is proposed and uniqueness of the solution is proved and regularised. The fourth analytical solution is derived using the vorticity equation. The solution is produced by implementing a potential function in the form ct> = P(x,y,,;)R(y)s(.;) with the application of the transformed coordinate ~ = kz- c; (t) . The pressure is then solved by applying the velocity vector into the Navier-Stokes equations to complete the solutions. Two examples are given to illustrate the applicability of the theorem. The uniqueness of the solution is also proved. Validation against two laminar flow experiments and three different turbulent flow cases including numerical case are carried out and reported in this work. The flow cases used in the validation are laminar jet flow, turbulent jet flow, boundary layer flow, turbulent channel flow and combustion. Generally, the solution is able to follow the trends in the corresponding cases. Although the analytical solution is derived for non-reacting flows, it proved capable of reproducing trends of cases including combustion. ix