OVERVIEW

Having established the equations that govern the motion of an incompressible Newtonian fluid and the associated boundary conditions, we proceed to discuss their solutions. Not surprisingly, we find that computing analytical solutions in closed or transcendental form is hindered by the presence of the nonlinear term associated with the point-particle acceleration. This term renders the governing system of equations nonlinear with respect to the velocity in a quadratic way. An inevitable consequence is that analytical solutions can be found only for a limited class of flows in the majority of which the nonlinear term either happens to vanish or is assumed to make an insignificant contribution. Under more general circumstances, the solution must be found using approximate, asymptotic, and numerical methods suitable for solving ordinary and partial differential equations. Fortunately, with the availability of an extensive arsenal of computational methods, to be discussed in the subsequent chapters, we are in a position to tackle a broad range of problems pertinent to a wide range of physical conditions.

In the present chapter we discuss a family of flows whose solution may be found either analytically, in closed or transcendental form, or by numerically solving ordinary and simple one-dimensional partial differential equations. We shall begin by considering unidirectional flows in channels and tubes for which the nonlinear term in the equation of motion vanishes due to the fact that the fluid particles travel along straight paths and the magnitude of the velocity is constant along the streamlines. Swirling flow...