New general solutions and MFS methodology of Stokes equations

Abstract

Three-dimensional (3D) Stokes equations are reformulated to be the third-order partial differential equations, with four specific solutions being derived in Theorems 1-4. Then a third-order method of fundamental solutions (MFS) to solve the Stokes flow problems is developed. The Papkovich-Neuber solution is proven with an easier manner, which needs four 3D harmonic functions. The new solution with one 3D harmonic function and three 2D in-plane harmonic functions is more saving. Three important methods in Theorems 5-7 are proven to seek new solutions of the Stokes equations by means of a biharmonic potential function or a biharmonic vector; they are complete through a lengthy proof. An effort is made in Theorem 8 for providing a complete general solution, and the Slobodyanskii general solution is re-derived via a simple way; both of them are presented in terms of three harmonic functions in the Cartesian coordinates. The new solutions under a point force are employed to generate the Stokeslet as an application. Five fresh numerical methods are developed which automatically satisfy the incompressibility condition. A reduced MFS together with the Papkovich-Neuber solution are merged into an unsymmetric Stokeslet method. Two MFS with dipole source are also derived. To explore the efficiency and accuracy of the proposed numerical methods, some examples including a benchmark problem are tested.