4 - Boundary layer equations in fluid dynamics

Abstract

In this article we will derive the basic equations of motion for viscous flows. It is worth noting that there are two main physical principles involved: (i) conservation of mass and (ii) Newton's second law of motion, the latter leading to a system of equations expressing the balance of momentum. In addition, we will utilize Newton's law of viscosity in the guise of what will be termed a “constitute relation” and, of course, all this will be done within the confines of the continuum hypothesis. We should also note that Newton's second law of motion is formally applied to point masses, i.e., discrete particles, its application to fluid flow seems difficult at best. But we will see that because we can define fluid particles (via the continuum hypothesis), the difficulties are not actually so great. We will begin with a brief discussion of the two types of reference frame widely used in the study of fluid motion, and provide a mathematical operator that relates them. We then review some additional mathematical constructs that will be needed in subsequent derivations. Once this groundwork has been laid, we will derive the “continuity equation” which expresses the law of conservation of mass for a moving fluid, and we will consider some of its practical consequences. We then provide a similar analysis leading to the momentum equations, thus arriving at the complete set of equations known as the Navier–Stokes (NS) equations. These equations are believed to represent all the fluid motion within the confines of the continuum hypothesis. There are many ways to derive the equation of motion of a fluid; we can use the approach which requires solving a set of partial differential equations. There is another way, called the control volume approach, and in any case, Newton's second law is applied.