The splitting of Navier–Stokes equations for a class of axisymmetric flows

In the framework of the Navier–Stokes equations, unsteady axisymmetric flows of a homogeneous viscous incompressible fluid, in which the axial and circumferential velocities depend only on radius and time are considered, and the radial velocity is zero. It is shown that the velocity of such flows is the sum of the velocities of two flows of a viscous incompressible fluid: axial flow (radial and circumferential velocities are zero) and circumferential flow (radial and axial velocities are zero). Axial and circumferential movements occur independently, without exerting any mutual influence. This allows us to split the boundary value problems for the type of flows under consideration containing three unknown functions (pressure, circumferential and axial velocities) into two problems, each of which contains two unknown functions (pressure and one of the velocity components). In this case, the sum of pressures of the axial and circumferential flow will be the pressure of the initial flow. The discovered possibility of splitting allows using known solutions to replenish the “reserves” of axial and circumferential exact solutions. These solutions, in its turn, can be summed in various combinations and, as a result, give the velocities and pressures of new exact solutions of the Navier–Stokes equations.