A Method Of Hamiltonian System For Stokes Flow In The Planar Curved Tube

There are many problems,which can be reduced to Stokes flow,in biomechanics, environment engineering,chemical industry etc.The Stokes flow in a tube is a typical model and has being attracted widely attention and study.So far,there are many methods for solving the problem,however,these methods are limited in some problem.Therefore,it is important to investigate the problem in science and engineering application.In this paper,the angular coordinate is taken in analogy to the time,and Hamiltonian system is introduced to the Stokes flow problem under polar coordinates,which based on the energy dissipation.In Hamiltonian system,the fundamental problem is reduced to the problem of eigenvalues and eigensolutions.Because of the completeness of eigensolution space and symplectic relationships of adjoint ortho-normalization for eigensolutions,the solution can be expanded by a linear combination of eigensolutions.For some simple boundary conditions,the analytical solutions can be obtained directly.And for some complicated boundary conditions,the semi-analytical solution of the problem can be expressed. The close method of the symplectic system is presented.In Hamiltonian system,problems of the lid-driven flow in the sectorial cavity and entrance flow in the channel are investigated.Based on symplectic relationships of adjoint ortho-normalization for eigensolutions,the problem of the governing equations and the boundary conditions is replaced by algebra equations.At the same time,a symplectic numerical method is presented.Numerical results show the streamlines,the velocity vectors, the velocity contours of absolute velocity and the counters of vorticity.The mechanism, characteristics and the end effects of flow are revealed by numerical analysis.The research results show that zero eigenvalue solutions describe the fundamental flow and non-zero eigenvalue solutions reveal the local effect of the flow.The results proved that the method of Hamiltonian System is effective for Stokes flow in the planar curved tube.