Flow development region in the boundary layer: two-component molecular viscosity and partial slip

Abstract

The subject of this study is the flow development region of laminar incompressible fluid flow in the boundary layer. This flow is an example where a direct application of the Navier-Stokes equations of gradient-free laminar incompressible fluid flow, in which the molecular viscosity is assumed to be a constant value independent of spatial coordinates, leads to a redefinition of the mathematical model. It is about the fluid boundary layer in the region of flow establishment in the motion problem of a semi-infinite plane, where the pressure gradient is zero. There is a situation when the number of equations is equal to three (two equations of momentum conservation and the equation of continuity), and the number of unknowns is equal to two - the number of the speed component. As a logical solution to the obtained inconsistency, it is proposed, as was already done for the problem of stationary motion of a plane and the problem of acceleration of a plane, to depart from the false statement about the constancy of molecular viscosity in the gradient-free boundary layer of an incompressible flow and consider molecular viscosity as a function of spatial coordinates. The need to consider the variable nature of molecular viscosity led to the discovery of another flaw in the Navier-Stokes theory. This non-trivial flaw was discovered during the application of the original numerical analytical method for solving the flow development region problem. The Navier-Stokes equations are supplemented by boundary conditions. The most important condition is the condition of fluid non-slipping on the surface of a solid body, which, by the way, does not follow any physical law. As a result, on the surface of a half-plane (or a moving body), the component of the velocity, which coincides with the direction of motion, has a constant value equal to the velocity of the body. It immediately follows from the continuity equation that the normal derivative of the normal component of the velocity must be equal to zero along the surface of the plane (body), since the longitudinal derivative of the velocity becomes zero. However, it is quite obvious that the velocity component normal to the surface of the plane (body) changes across the boundary layer in the region of current development, which indicates the presence of a normal gradient (both components) of the velocity. The conflict or contradiction is overcome by moving away from the generally accepted condition of non-slipping to the condition of partial non-slipping, or essentially the presence of sliding. Even with the sudden braking of any vehicle, the complete stop does not occur instantly, but after some finite time and distance, so in the case of the motion of a body in a stationary fluid, there is not an instant sticking, but a gradual one - from complete sliding, when a particle of liquid has just met a moving plane (body), to complete non-slipping at the end (and further) of the flow development region. Research methods. This work uses purely theoretical methods based on the use of calculus of variations, laws of physics, and ideas from everyday life. Conclusions. An improved model of a viscous Newtonian fluid in the area of flow development in the boundary layer was derived. On the basis of assumptions about the variable nature of the molecular viscosity, which already has two components, and the departure from the non-slipping condition, analytical solutions for both components of the velocity and both components of the molecular viscosity were obtained. A comparison of the obtained results with the results of other studies is presented.