Couette Flow: General Discussion
Consider the flow model shown in Figure 16.2. Here we see a viscous fluid contained between two parallel plates separated by a distance D. The upper plate is moving to the right at velocity ue. Due to the no-slip condition, there can be no relative motion between the plate and the fluid; hence, at у = D the flow velocity is и = ue and is directed toward the right. Similarly, the flow velocity at у = 0, which is the surface of the stationary lower plate, is и = 0. In addition, the two plates may be at different temperatures; the upper plate is at temperature Te and the lower plate is at temperature Tw. Again, due to the no-slip condition as discussed in Section 15.7, the fluid temperature at у = D is T = Te and that at у = 0 is T = Tw.
Clearly, there is a flow field between the two plates; the driving force for this flow is the motion of the upper plate, dragging the flow along with it through the mechanism of friction. The upper plate is exerting a shear stress, re, acting toward the right on the fluid at у — D, thus causing the fluid to move toward the right. By an equal and opposite reaction, the fluid is exerting a shear stress ze on the upper plate acting toward the left, tending to retard its motion. We assume that the upper plate
is being driven by some external force which is sufficient to overcome the retarding shear stress and to allow the plate to move at the constant velocity ue. Similarly, the lower plate is exerting a shear stress rw acting toward the left on the fluid at у = 0. By an equal and opposite reaction, the fluid is exerting a shear stress tw acting toward the right on the lower plate. (In all subsequent diagrams dealing with viscous flow, the only shear stresses shown will be those due to the fluid acting on the surface, unless otherwise noted.)
In addition to the velocity field induced by the relative motion of the two plates, there will also be a temperature field induced by the following two mechanisms:
1. The plates in general will be at different temperatures, thus causing temperature gradients in the flow.
2. The kinetic energy of the flow will be partially dissipated by the influence of friction and will be transformed into internal energy within the fluid. These changes in internal energy will be reflected by changes in temperature. This phenomenon is called viscous dissipation.
Consequently, temperature gradients will exist within the flow; in turn, these temperature gradients result in the transfer of heat through the fluid. Of particular interest is the heat transfer at the upper and lower surfaces, denoted by qe and qw, respectively.
These heat transfers are shown in Figure 16.2; the directions for qe and qw show heat being transferred from the fluid to the wall in both cases. When heat flows from the fluid to the wall, this is called a cold wall case, such as sketched in Figure 16.2. When heat flows from the wall into the fluid, this is called a hot wall case. Keep in mind that the heat flux through the fluid at any point is given by the Fourier law expressed by Equation (15.2); that is, the heat flux in the у direction is expressed as
where the minus sign accounts for the fact that heat is transferred from a region of high temperature to a region of lower temperature; that is, qy is in the opposite direction of the temperature gradient.
Let us examine the geometry of Couette flow as illustrated in Figure 16.2. An x-y cartesian coordinate system is oriented with the x axis in the direction of the flow and the у axis perpendicular to the flow. Since the two plates are parallel, the only possible flow pattern consistent with this picture is that of straight, parallel streamlines. Moreover, since the plates are infinitely long (i. e., stretching to plus and minus infinity in the x direction), then the flow properties cannot change with x. (If the properties did change with x, then the flow-field properties would become infinitely large or infinitesimally small at large values of x—a physical inconsistency.) Thus, all partial derivatives with respect to x are zero. The only changes in the flow – field variables take place in the у direction. Moreover, the flow is steady, so that all time derivatives are zero. With this geometry in mind, return to the governing Navier-Stokes equations given by Equations (15.19a to c) and Equation (15.26). In these equations, for Couette flow,
ди ЗT dp
dx dx dx
Hence, from Equations (15.19a to c) and Equation (15.26), we have
„ 3 /, Э7Л 3 / du n
Energy equation: — 3y / + dy “ dy ) = °
Equations (16.1) to (16.3) are the governing equations for Couette flow. Note that these equations are exact forms of the Navier-Stokes equations applied to the geometry of Couette flow—no approximations have been made. Also, note from Equation (16.2) that the variation of pressure in the у direction is zero; this in combination with the earlier result that dp/dx = 0 implies that the pressure is constant throughout the entire flow field. Couette flow is a constant pressure flow. It is interesting to note that all the previous flow problems discussed in Parts 2 and 3, being inviscid flows, were established and maintained by the existence of pressure gradients in the flow.
In these problems, the pressure gradient was nature’s mechanism of grabbing hold of the flow and making it move. However, in the problem we are discussing now—being a viscous flow—shear stress is another mechanism by which nature can exert a force on a flow. For Couette flow, the shear stress exerted by the moving plate on the fluid is the exclusive driving mechanism that maintains the flow; clearly, no pressure gradient is present, nor is it needed.
This section has presented the general nature of Couette flow. Note that we have made no distinction between incompressible and compressible flow; all aspects discussed here apply to both cases. Also, we note that, although Couette flow appears to be a rather academic problem, the following sections illustrate, in a simple fashion, many of the important characteristics of practical viscous flows in real engineering applications.
The next two sections will treat the separate cases of incompressible and compressible Couette flow. Incompressible flow will be discussed first because of its relative simplicity; this is the subject of Section 16.3. Then compressible Couette flow, and how it differs from the incompressible case, will be examined in Section 16.4.
As a final note in this section, it is obvious from our general discussion of Couette flow that the flow-field properties vary only in the у direction; all derivatives in the x direction are zero. Therefore, as a matter of mathematical preciseness, all the partial derivatives in Equations (16.1) to (16.3) can be written as ordinary derivatives. For example, Equation (16.1) can be written as
However, our discussion of Couette flow is intended to serve as a straightforward example of a viscous flow problem, “breaking the ice” so-to-speak for the more practical but more complex problems to come—problems which involve changes in both the x and у directions, and which are described by partial differential equations. Therefore, on pedagogical grounds, we choose to continue the partial differential notation here, simply to make the reader feel more comfortable when we extend these concepts to the boundary layer and full Navier-Stokes solutions in Chapters 17 and 20, respectively.