Estimation of the hate of flow in porous media

In a number of problems of the theory of flow in porous media it is particularly important to find the integral characteristics of the flow for regions that constitute extended stream tubes. (Such a region can be imagined as the result of the continuous deformation of a cylinder whose lateral surfaces are impermeable while the bases are surfaces of constant pressure, the inlet and outlet of the flow. In the plane case the cylinder becomes a rectangle.) Usually, the flow rate Q is to be found from the difference of head H. In some cases it possible to obtain upper and lower bounds for the flow rate by varying the flow region, the flow resistance field or the form of the flow law 1–4]. The aim of the present study is to find the shape of the region of fixed volume (in the plane case area) which for given constraints realizes an extremum of the steady-state flow rate. It is shown that the extremality requirement leads to an additional local condition on the unknown part of the boundary. A class of plane problems for which the resulting boundary-value problem with an unknown boundary is effectively solved is identified. Examples are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 80–87, March–April, 1986.