Exact solutions of problems of the three-dimensional flow of ideal viscous incompressible fluids near cylindrical surfaces

Abstract

Three-dimensional flows of an incompressible fluid, the parameters of which depend on two coordinates and time, are considered. The stream surfaces of such flows are cylindrical. The equations of continuity and the Navier-Stokes equations can be transformed to relations, one of which is the equation for the stream function the other is the integral of the equations relating the pressure and the stream function, and the third is a linear equation for the projection of the velocity vector onto the axis parallel to the generatrix of the cylindrical surfaces. The problems of modelling the flows are considered on the basis of the exact solutions of the Navier-Stokes equations and Euler's equations using examples. Relations for the distribution of the flow parameters in the channel created by hyperbolical cylinders are derived for the case of unsteady inviscid flow. The streamlines of these flows are situated on the side surfaces of the hyperbolical cylinders and intercept the generatrices of the cylinders at certain indirect angles. The flow around a circular cylinder and the flow of fluid inside an elliptic cylinder are considered in the case of steady inviscid flow. The streamlines on the circular cylinder are arranged transverse to the cylinder (the projection of the velocity vector onto the coordinate axis, parallel to the generatrix of the cylinder, is equal to zero). Far from the cylinder the streamlines are also situated on a cylindrical surfaces, but not transverse to the cylinder, making certain indirect angles with the generatrix. Viscous three-dimensional flows, possessing a certain symmetry, are considered. In the case of radial symmetry the streamlines are helical lines. The non-planar Couette flow between parallel moving planes is characterized by the fact that the velocity vectors, being situated in the same plane, are collinear, while the velocity vectors in parallel planes are not collinear. Relations for viscous steady three-dimensional flows, using well-known relations, obtained for the stream function of two-dimensional flows, are given.