Exact Solutions to Flow Problems of an Incompressible Viscous Fluid

Abstract

We now present exact solutions to some flow problems obtained by solving the governing differential equations. Though exact solutions are known to only a small number of problems due to the complexity of the governing equations, these solutions are very valuable not only for the insight that they offer, but also as a means of testing the accuracy of numerical schemes such as the one presented in [49]. All the solutions that are presented in this chapter have been obtained by guessing (fully, or in terms of some function to be determined) the velocity field. If one guesses the stress field, then of course, one would have to check the compatibility conditions presented in Vol. I; the same compatibility conditions derived for linear elasticity hold, with the rate of deformation tensor and velocity playing the roles of the linearized strain tensor and displacements, respectively. In contrast to the control volume approach, the solution for each unknown field, such as the velocity or pressure field, is obtained at each point of the flow domain as a function of the position vector and time, from which other quantities of interest (e.g., the drag force) can be derived. Needless to say, this detailed information is obtained only at the expense of the increased complexity of having to solve the governing differential equations. Since the fluid is assumed to be incompressible, the pressure field is not constitutively related to the density and pressure, and is either prescribed, or has to be determined from the governing equations and boundary conditions. Often, we shall be interested in finding the temperature field. Since, in this chapter, we assume the fluid to be incompressible, and the viscosity of the fluid to be constant, the equations of momentum and energy are decoupled. Hence, we can first find the velocity and pressure fields using the continuity and momentum equations, before proceeding to find the temperature using the energy equation. We consider only laminar, fully developed flows in this chapter.