Analytical solution of the Navier-Stoks equations reduced to the third-order equation for the problem of fluid motion in a round pipe

Abstract

Abstract The paper defines a rheological law that takes into account the molecular and molar transfer of fluid particles between the layers of the flow; the equation of fluid motion, taking into account two mechanisms of molecular and molar exchange of momentum in the flow; the form of the obtained new equations in the form of an equation of the boundary layer, neglecting the terms whose order is much lower than the order held in the equations; statement of the problem of stationary fluid flow in cylindrical coordinates with the corresponding boundary conditions using the transition to new dimensionless variables; a technique for solving the Navier-Stokes equation reduced to a third-order differential equation for studying the motion of a fluid in a round pipe; analytical solution of the formulated problem; the role of the newly introduced molar transfer coefficient in describing the flow pattern. An analytical solution of the problem of fluid motion in a cylindrical pipe is obtained, taking into account these two transfer mechanisms where third-order terms are formed in the Navier-Stokes equations. For small Reynolds numbers, the influence of the newly introduced term on the flow pattern is a shortening of the length of the initial segment of motion. A decrease in the value of the new number is associated with an increase in this region.