Interim Summary

In this section, we have studied incompressible Couette flow. Although it is a some­what academic flow, it has all the trappings of many practical viscous flow problems, with the added advantage of lending itself to a simple, straightforward solution. We have taken this advantage, and have discussed incompressible Couette flow in great detail. Our major purpose in this discussion is to make the reader familiar with many concepts used in general in the analysis of viscous flows without clouding the picture with more fluid dynamic complexities. In the context of our study of Couette flow, we have one additional question to address, namely, What is the effect of compressibility? This question is addressed in the next section.

Consider the geometry sketched in Figure 16.2. The velocity of the upper plate is 200 ft/s, and the two plates are separated by a distance of 0.01 in. The fluid between the plates is air. Assume incompressible flow. The temperature of both plates is the standard sea level value of 519° R.

(a) Calculate the velocity in the middle of the flow.

(,b) Calculate the shear stress.

(c) Calculate the maximum temperature in the flow.

(d) Calculate the heat transfer to either wall.

(e) If the lower wall is suddenly made adiabatic, calculate its temperature.

Solution

Assume that д is constant throughout the flow, and that it is equal to its value of 3.7373 x 10~7 slug/ft/s at the standard sea level temperature of 519°R.

(a) From Equation (16.6),

difference between compressible and incompressible flows. With all this in mind, Equation (16.3), repeated below

can be written as

The temperature variation of ц is accurately given by Sutherland’s law, Equation

(15.3) , for the temperature range of interest in this book. Hence, from Equation (15.3) and recalling that it is written in the International System of Units, we have

The solution for compressible Couette flow requires a numerical solution of Equation (16.62). Note that, with p. and к as variables, Equation (16.62) is a nonlinear differential equation, and for the conditions stated, it does not have a neat, closed – form, analytic solution. Recognizing the need for a numerical solution, let us write Equation (16.62) in terms of the ordinary differential equation that it really is. (Recall that we have been using the partial differential notation only as a carry-over from the Navier-Stokes equations and to make the equations for our study of Couette flow look more familiar when treating the two-dimensional and three-dimensional viscous flows discussed in Chapters 17 to 20—just a pedagogical ploy on our part):

Equation (16.64) must be solved between у = 0, where T = 7„ , and у = D, where T = Te. Note that the boundary conditions must be satisfied at two different locations in the flow, namely, at у = 0 and у = D; this is called a two-point boundary value problem. We present two approaches to the numerical solution of this problem. Both approaches are used for the solutions of more complex viscous flows to be discussed in Chapters 17 through 20, and that is why they are presented here in the context of Couette flow—simply to “break the ice” for our subsequent discussions.