So far we have been able to cover a lot of ground with a minimum of material on fluid flow. At this point I need to present to you some more topics in fluid dynamics—inviscid fluid flow, the Bernoulli equation, turbulence, boundary layers, and flow separation—before returning to flow past spheres. This material also provides much of the necessary background for discussion of many of the topics on sediment movement to be covered in Part II. But first we will make a start on the nature of flow of a viscous fluid past a sphere.
The Navier-Stokes Equation
The idea of an equation of motion for a viscous fluid was introduced in the Chapter 2. It is worthwhile to pursue the nature of this equation a little further at this point. Such an equation, when the forces acting in or on the fluid are those of viscosity, gravity, and pressure, is called the Navier–Stokes equation, after two of the great applied mathematicians of the nineteenth century who independently derived it.
It does not serve our purposes to write out the Navier–Stokes equation in full detail. Suffice it to say that it is a vector partial differential equation. (By that I mean that the force and acceleration terms are vectors, not scalars, and the various terms involve partial derivatives, which are easy to understand if you already know about differentiation.) The single vector equation can just as well be written as three scalar equations, one for each of the three coordinate directions; this just corresponds to the fact that a force, like any vector, can be described by its scalar components in the three coordinate directions.
The Navier–Stokes equation is notoriously difficult to solve in a given flow problem to obtain spatial distributions of velocities and pressures and shear stresses. Basically the reasons are that the acceleration term is nonlinear, meaning that it involves products of partial derivatives, and the viscous-force term contains second derivatives, that is, derivatives of derivatives. Only in certain special situations, in which one or both of these terms can be simplified or neglected, can the Navier–Stokes equation be solved analytically. But numerical solutions of the full Navier–Stokes equation are feasible for a much wider range of flow problems, now that computers are so powerful.