The kinematics of a viscous fluid flow in a channel with a valve

In this paper, results of mathematical modeling of a steady flow of a Newtonian fluid in a flat channel are presented. At some distance from the inlet and outlet of the channel, a valve having a specified diameter and capable to change the width of the flow cross section depending on the position of the locking element is situated. On the walls, the adhesion conditions are performed; at the input, the profile corresponding to the steady flow of a predetermined constant flow is assigned, and the Neumann boundary conditions are used at the output. The mathematical statement of a problem is formulated using a dimensionless stream function and vorticity variables. The numerical solution of the problem is implemented with the application of an original finite-difference method based on the scheme of alternating directions. The physical flow area with a curved boundary is converted into a rectangular domain by introducing new coordinates, and the stationary solution of the problem is obtained using the relaxation method. The calculations demonstrate that in the vicinity of the inlet and outlet boundaries, the onedimensional flow areas corresponding to a steady fluid flow in an infinite channel occur, and in the vicinity of the shutter, a two-dimensional flow with recirculations is observed. The effect of the Reynolds number on the distribution of the kinematic characteristics is investigated. The results of the parametric studies of the flow pattern in relation to the opening position and diameter of the locking element are presented.