On Exact Solutions to Problems of Steady Flow of Incompressible Viscous Fluid in a Channel and in a Circular Pipe

Abstract

Turning to the well-known Couette flow, we again, but now in a new formulation, consider the steady motion of an incompressible viscous fluid in a channel with two parallel flat walls, one of which is stationary while the other moves at a constant speed in its plane. In the classical formulation of this problem on a layered (laminar) fluid flow in a channel, the wall velocity and the constant pressure drop along the channel are independent quantities and are considered predetermined. However, when a solid body moves along the fluid surface, the fluid particles are set in motion. This occurs due to the arising pressure generated in the fluid caused by the movement of the solid body. Therefore, here the pressure or pressure drop is not predetermined but is determined as a result of solving the corresponding two-dimensional boundary value problem of a steady flow of a viscous fluid in a channel. For this, the corresponding boundary value problem of a fluid flow in the channel is formulated based on the linearized Navier-Stokes equations obtained from the general nonlinear Navier-Stokes equations for a steady flow of a viscous fluid neglecting the convective terms, which is true for small Reynolds numbers. Using the Fourier integral transform method, an exact (closed) solution to this boundary value problem is constructed; the velocity and pressure components are determined. Then it is shown that the resulting solution, different from the known Couette flow solution, also satisfies the original nonlinear Navier-Stokes equations, the continuity equation, and boundary conditions. A comparative analysis of the new solution with the Couette flow solution is carried out. The well-known Hagen-Poiseuille problem is also discussed from the point of view of establishing the conditions for the laminar axisymmetric and steady flow of an incompressible viscous fluid in a straight circular pipe According to the study's results, to implement laminar flow under the indicated conditions, the pressure drop along the pipe must be constant. The study demonstrates that this condition is simultaneously sufficient, and therefore, the necessary and sufficient condition is established for a laminar flow in a straight circular pipe.