Introduction

The examples we have studied thus far have had rather simple kinematics: flow parallel or nearly parallel to a wall and ideal or nearly ideal extension. Thus, we have been able to obtain exact solutions for the flow or to obtain approximate solutions based on the small difference between the actual flow and an ideal case for which an exact solution is available. Even for the case of fiber spinning, where an analytical solution to the thin filament equations cannot be obtained under conditions relevant to industrial practice, we simply need to obtain a numerical solution to a pair of ordinary differential equations, which is a task that can be accomplished using elementary and readily available commercial software.

The flow in many real processing geometries is too complex for us to apply the analytical methods utilized in the preceding chapters. Indeed, even when the flow field is a simple one, the coupled heat transfer problem may not be amenable to a simple treatment; the elementary extruder in Chapter 3 is an example of a case in which we are unable to obtain an exact or even approximate solution for the spatial development of the two-dimensional temperature field.

Complex coupled flow and heat transfer problems can be solved using numerical techniques in which the partial differential equations are converted to a large set of coupled algebraic equations, and the algebraic equations are then solved using conventional methods developed specifically to be efficient on digital computers.