Boundary integral equations of fluid flow and electrical and heat conduction in a medium composed of blocks

Abstract

For many applications in the theory of flow through porous media, diffusion, convective transport, and electrical and heat conduction it is important to consider problems involving systems of blocks separated by boundaries (in particular, cracks or conductors), the transport process taking place both through the blocks themselves and along the separating boundaries. Such problems can conveniently be solved by the method of boundary integral equations [1–4]. The object of this study is to propose a form for these equations and methods of solving them specially designed to take into account the particularities of the problems in question. Firstly, the boundary integral equation is given in a form that contains only the net inflow (of fluid, heat or electricity) to a unit area of the boundary and not the individual inflows from each of the blocks separated from that boundary. This almost halves the number of unknowns subject to determination. Secondly, the principal aspects associated with the specifics of the flow along the boundaries and their intersections and discontinuities are discussed. Thirdly, a numerical experiment to realize the proposed form of the equations on a computer using algorithms with different structures leads to quite general conclusions which may be useful for further developing the method of boundary integral equations in relation to the class of applied problems considered.