A Method For Stokes Flow In The Symplectic System

There are many problems that can be reduced to viscous fluid problem in the chemical industry, environment engineering, physical chemistry, biomechanics, geophysics as well as meteorology etc. The research on this subject has been ongoing for almost a century. And because of its extensive applications, it has developed into an active research field. Stokes flow is the main and typical flow model of viscous fluid. The traditional method solves this problem in Euclidean space under the Lagrange system, which involves solving higher orders of partial differential equations and faces the difficulty of handling the boundary conditions. So it is necessary to investigate a new and efficient solving method.Based on the principle of energy dissipation, The Lagrange function, Hamiltonian action, is derived from the constitutive equations of incompressible Newtonian fluid and Stokes equations. Then the Hamiltonian function can be obtained. Finally, the canonical (dual) equations are obtained by appling the variation principle, and the symplectic system (Hamiltonian system) is introduced into plane and space Stokes flow problems. In the system, the fundamental problem is reduced to eigenvalue and eigensolution problem. Because of the completeness of eigensolution space and adjoint relationships of the symplectic ortho-normalization for eigensolutions, the solution can be can be expanded by a linear combination of the eigensolutions. Under the given boundary conditions, the expansion coefficients can be obtained, and then the semi-analytical solution of the problem. The close method of the symplectic system is presented.Two and three dimensional problems are investigated, and so is non-steady low Reynolds number flow. The main research work is as follows: first, the symplectic system is introduced into two-dimensional Stokes flow problem. The lid-driven flow, shear flow as well as channel flow is studied, and the analytical expressions of the solutions are also given. After the numerical computation, the velocities, stresses and pressures of the flow are obtained, meanwhile, the streamline patterns and velocity vectors are also plotted. Based on the numerical analysis, the flow mechanism, flow characteristics and the end effects etc. are also revealed. Second, the symplectic system is introduced into three-dimensional Stokes flow problem, and the zero eigenvalue solutions and nonzero eigenvalue solutions are given. The special adjoint relationships of the symplectic ortho-normalization is investigated, and the mode of every nonzero eigensolution are given. As an example, the influence of inlet radius on Stokes flow in a circular tube is studied, and some laws are derived. The spatial flow driven by two end lids is also the example discussed. At the end of the dissertation, non-steady low Reynolds number flow problem is studied preliminarily.The research results show that the eigensolutions of the canonical equations have their definite physical meanings: zero eigenvalue solutions are the fundamental flows; and the nonzero eigenvalue solutions reveal the local effects, they can describe end effect and its decay process. The symplectic system method is simple, direct and efficient. It also provides a path for solving other problems.