Reconstruction of the inlet viscous fluid flow by velocity measurements on any observable part of the free moving surface

A method has been developed and an algorithm has been elaborated in order to determine an unknown distribution of inflow velocity of viscous heterogeneous incompressible fluid into its common flow. The main data for solution of this problem are changes in the fluid velocity at some observable section of free surface of this flow. Also, movement of the fluid is considered isothermal and stationary. In order to simplify the model of fluid movement, let us believe that the Reynolds number for the fluid flow under consideration is very small. The number indeed is very small at a high viscosity and (or) slow motion of the fluid. Smallness of the Reynolds number allows discarding the total derivative from the time velocity vector in the Navier–Stokes equation of fluid motion. Thus, in a number of cases, the Stokes equation may be considered as the main one when simulating the motion of viscous fluid. The formulated problem gets formalized as an inverse boundary value problem for the model of motion of viscous fluid. The model includes the Stokes equation, the incompressibility equation, and the corresponding boundary conditions. The problem is ill-posed in regard to perturbation of the velocity under measurement. Therefore, numerical solution of the problem requires development of special sustainable methods. We offer using the variational method. For that, we introduce some assessment function which is a disparity between the observed velocity and the virtual velocity calculated out of a specially-set auxiliary boundary control problem, which is usually called the direct problem. Control is the inflow velocity. Desired solution of the problem is the minimum point of the residual functional which is obtained through the gradient descent method. Implementation of the method leads to a sequential solution of the corresponding correctly-posed boundary control problem. Model example has been calculated.