1.7 Rates of Deformation

In the equations developed so far, we have not identified the nature of the material we are studying. In fact, our equations are so general at this stage that they apply to solid materials as well as to fluids. To narrow the subject to fluids, it is necessary to show how the fluid behaves under applied stresses. The important geometric quantity that describes the fluids behavior under stress is the rate of deformation.

To define the deformation of a fluid under acting stresses, first consider motion in the two-dimensional xy plane. Choose three neighboring points ABC (Figure 1.7.1) selected to make up a right angle at an initial time t. A is a distance ? x along the x-axis from B, and C is a distance ? y above B along the y-axis.

Figure 1.7.1: Rates of deformation two dimensions

As the flow evolves through a short time interval ? t, the angle ABC will change, as will the distance between the three points. At this later time t + ? t, these points will have moved to A ?, B ?, and C ?, as shown. Using Taylor series expansions to the first order, the fluid that initially was at point A will have x and y velocity components given by and . The fluid initially at point A will move a distance to the right of A, and...