Analytical solution of the problem of flow in a porous rectangular bulkhead

The object of study is an unsteady pressureless filtration flow in a porous isotropic medium, in which the region of motion is limited from above by a free surface, on which the fluid pressure is constant and equal to the external atmospheric pressure. Such currents are characteristic of groundwater filtration through hydraulic structures (dams, water drawdowns, drainages, foundations, pits during their drainage). The aim of the study is to solve the problem of non-stationary gravitational flow of a fluid in a scalar porous medium with two-dimensional filtration motion in a vertical plane. To achieve it, the limiting problem of non-stationary filtration theory (Boussinesq) for a scalar porous medium is formulated using dimensionless factorization, which allows solving groups of problems for areas with similar domains of definition. The Boussinesq limit problem is reduced to a typical limit problem for Crocco's ordinary differential equation. Crocco's limiting problem is formulated and solved. An analytical solution of the limit problem for a rectangular bridge is obtained. The solution is used to determine the depth of the filtration flow in the downstream of the cofferdam. The paper proves that the limiting problems of non-stationary filtration in the vertical plane are identical to the limiting problems of the stationary theory of the boundary layer in the von Mises variables - the longitudinal coordinate-current function.