Theory And Application Of Natural Boundary Integral Method For STOKES Flow

The Stokes flow problem is the classical task of computational fluid dynamics, and as one of the active branch of fluid mechanics, it is widely used in mineral dressing, grouting, chemical industry and environmental engineering. Due to the fact that the disturbance attenuation of the Stokes flow is slow, the pure numerical method needs a prodigious computational domain as to the boundless flow. Accordingly, the semi-analytical and semi-numerical method becomes the main technique of Stokes problems. As a new valid method, the natural boundary integral method is a kind of semi-analytical and semi-numerical method with many advantages. In this paper, the natural boundary integral method and the coupled method of the natural boundary integral method and the finite difference method are adopted to study Stokes flows systematically. The main research results and new ideas are as follows:(1) With regard to the axial symmetric Stokes flow in half space, new analytical formulae of velocity and pressure solutions for the Stokes equations in cylindrical coordinate system are obtained. The obtained formulae are much more convenient and intuitionistic to study axial symmetric Stokes flow from orifices in a plane wall.(2) By finiteness and single value conditions of the velocity and stress in multi-connectivity region, the analytical functions for Stokes problems of exterior circular region were deduced. Synchronously, velocity and pressure boundary integral formulae are deduced with arbitrary velocity conditions for the multi-connectivity region and infinite region of the Stokes equations. Accordingly, the bounded flows with arbitrary velocity conditions and the unbounded flows under zero velocity magnitude are studied. The natural boundary integral method is used to analyze the velocity-pressure mixed problem for Stokes problems of exterior circular region when pressure magnitude of inner boundary is zero, and the above results are compared with the numerical results finally.(3) The coupled method of the natural boundary integral method and the finite difference method is studied. Stokes problems of irregular domain which can not be calculated by the natural boundary integral method are resolved. Stokes problems of interior domain are discussed synchronously, and comparied with pure numerical method. It is shown that the coupled method can save the time and decrease amount of calculation.(4) According to Fourier series technique and correlative property of singular function, and based on the theory of the stokes flow in porous medium, the pressure boundary integral formula for Darcy flow equation of interior circular region with Dirichlet problem and the seepage pressure field of exterior vertical shaft is deduced. Based on above formulae and the experimentation measure, the serous pressure distribution and spreading status of multi-hole grouting problems with arbitrary forms of hole-sitting are calculated. Then relationships between consolidating parameters and subsidiary pressure of shaft wall are discussed.(5) Based on the D-N alternative method, the coupled method of the natural boundary integral method and the finite difference method is adopted to calculate coupling Darcy-Stokes problem on the viscous flow past a circular cavity. Finally, the groundwater seepage velocity distributions of different permeability coefficients with the influence of bore well are studied.