Analytical Solution of the Steady Navier-Stokes Equation for an Incompressible Fluid Entrained by a Rotating Disk of Finite Radius in the Area of Boundary Layer

Abstract

The flow in the neighborhood of a rotating disk is of great practical importance, particularly in connection with rotary machines. It becomes turbulent at larger Reynolds numbers, [Math Processing Error] , in the same way as the flow about a plate. In this article, we consider a motion of incompressible fluid that is always turbulent in azimuthal direction (Reynolds number based on azimuthal velocity [Math Processing Error] ) and is of both kinds in a radial direction, i.e. laminar (Reynolds number based on radial velocity [Math Processing Error] ) and turbulent ( [Math Processing Error] ). The equations of analyticity of functions of a spatial complex variable (shortly, the equations of tunnel mathematics) afford a possibility to seek the solutions of steady Navier-Stokes equation in view of elementary functions. All vector fields, including those obeying the Navier-Stokes equation, satisfy the equations of tunnel mathematics. The Navier-Stokes equations themselves are afterward applied for verification of obtained solutions and calculation of the pressure. Obtained formulae for pressure allow us to visualize the presence of the boundary layer and estimate its thickness for laminar and turbulent flows. We use Prandtl`s concept of considering fluids with small viscosities, i. e. we suppose that the Reynolds number is enough large and the viscosity has an important effect on the motion of fluid only in a very small region near the disk (boundary layer). We also suppose that the fluid and the disk had at the beginning the same temperatures and the energy dissipation occurs only by means of internal friction.