Method Of Regularized Sources For Stokes Flow Problems

The solution of Stokes flow problems with Dirichlet and Neumann boundary con-ditions is performed by a non-singular Method of Fundamental Solutions which does not require an artificial boundary, i.e. source points of fundamental solution (MFS) coincide with the collocation points on the boundary. Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are smaller compared with viscous forces. This fluid typically occurs in situations where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. The present research is initiated by the simulation needs in microfluidics, specifically gas focused micro-jets. Generally the following method is used to seek for the solution of Stokes flow. They are stream function, Green’s function:the Stokeslet, Papkovich - Neuber solution and boundary element method. Method of regularized source is used to get the solution of the Stokes flow in the thesis. Compared with the traditional method of fundamental solution, the numerical solution of Stokes flow is presented by a non-singular function which approximates to fundamental solution. Recently, MFS is used to solve partial differential equations due to its high accuracy and easy to be used. However, how to choose an artificial boundary is still an open problem which restricts MFS to be used in the practical engineering. Instead of the Dirac delta function, a non-singular function is employed. As a result the artificial boundary is no longer needed, i.e, source points are fixed on the boundary in MRS.The fundamental solution of the Stokes pressure and velocity is obtained from the analytical solution due to the action of the Dirac delta type of force. Instead of the Dirac delta force, a rational and exponential function with a free parameter epsilon is employed, which approximates to Dirac delta function when epsilon tends to zero. The analytical expressions for related Stokes flow pressure and velocity around such regularized sources are derived for rational and exponential blobs in an ordered way. The solution of the problem is sought as a linear combination of the fields due to the regularized sources that coincide with the boundary and with their intensities chosen in such a way that the solution complies with the boundary conditions. A two dimensional driven cavity numerical example and a flow between parallel plates are chosen to assess the properties of the method. The results of the posed Method of Regularized Sources (MRS) are compared with the results obtained by the fine-grid second-order classical Finite Difference Method and analytical solution. The results converge with finer discretisation, however they depend on the value of epsilon. The method gives reasonably accurate results for the range of epsilon between 0.1 and 0.3 of the typical nodal distance on the boundary. A correction of the method is proposed which can be used to properly assess also the derivatives at the boundary. A robust and efficient strategy to find the optimal value of epsilon is needed in the perspective.