Numerical investigation of a viscous fluid flow through the gap between two parallel plates

The process of unsteady flow of a viscous incompressible fluid through the gap between two parallel plates is considered. To describe the process, a one-dimensional model of the plane-parallel flow of a viscous fluid is proposed. Within the framework of this model, the problem with nonlocal additional condition for obtaining the pressure drop versus the time for a given volumetric flow rate of the fluid through the gap is posed. This problem belongs to the class of inverse problems associated with the reconstruction of right-hand sides of parabolic equations as functions of time. Using the time sampling, the posed problem is converted to a semidiscrete problem. In order to solve this problem, a special formulation is suggested. As a result, the solution of the original problem at each time step is reduced to solving two boundary value problems with local boundary conditions and the linear equation with respect to the approximate value of the pressure drop.