Two-Dimensional Poiseuille Flow
Consider the parallel flow between two horizontal plates separated by the distance D as sketched in Figure 16.12. In this case, unlike Couette flow where one of the plates is in motion, we consider both plates to be stationary. Recall that the driving force which established Couette flow was the shear stress between the moving plate and the fluid. In the present case sketched in Figure 16.12, what is the driving force; that is, what makes the fluid move? In Chapter 1, we emphasized that the only way that nature can exert a force on a fluid is by means of shear stress and pressure distributions. In the present problem, since the walls are not moving, there is no shear stress to drive the flow. Hence, the only other possibility is the pressure distribution. Indeed, to establish the flow shown in Figure 16.12, there must be a pressure gradient acting
on the gas. Moreover, in Figure 16.12 the flow extends to infinity in both directions along the x axis. As in the case of Couette flow, this implies that the velocity и is independent of x; that is, и = и (у). Since the streamlines are parallel, v = 0. This flow is called two-dimensional Poiseuille flow, named after the French physician J. L. M. Poiseuille, who studied similar flows in pipes.
Let us examine the Navier-Stokes equations in light of the problem just outlined. For simplicity, we will consider only steady, incompressible flow. First, return to the continuity equation for an incompressible flow, given by Equation (3.39). In cartesian coordinates, this is
du dv dx 9y
Since the flow is parallel, v = 0; hence, dv/dy = 0. From Equation (16.99), then du/dx = 0; this confirms that и is constant with x; that is, и is a function of у only. From the у-momentum equation, Equation (15.19b), we have
— =0 [16.100]
Hence, p varies only in the x direction; p = p(x). From the x-momentum equation, Equation (15.19a), we have
On the left-hand side of Equation (16.102), p is a function of x only. On the right-hand side of Equation (16.102), и is a function of у only. Hence, the left-hand and right- hand sides of Equation (16.102) must be equal to the same constant. This confirms an important aspect of this flow, namely, the pressure gradient is constant along the flow direction. Once again, we emphasize that it is this pressure gradient that drives the flow. The pressure gradient must be provided by an outside mechanism, that is, some source of high pressure toward the left and low pressure toward the right.
The velocity profile across the flow is obtained by solving Equation (16.102). For convenience, and to emphasize that p = p(x), we write the pressure derivative as an ordinary derivative, dp/dx. Integrating Equation (16.102) twice across the flow, we have
^ y2 + ay + b [16.103]
where a and b are constants of integration. Evaluating Equation (16.103) at у = 0, where и = 0, we have
b = 0
Evaluating Equation (16.103) at у = D, where и = 0, we have
Hence, Equation (16.103) becomes
At у = 0, Equation (16.106) yields
/Зи D /dp
Эу/ш 2/л dx
Hence, the wall shear stress is
Note the interesting fact from Equation (16.108) that zw does not depend on the viscosity coefficient (i, but rather only on the separation distance of the walls and on the pressure gradient. Clearly, this flow is a force balance between the pressure gradient acting toward the right on the gas and the shear stress at the walls acting toward the left on the gas.
This flow is sometimes called fully developed flow, for the following reason. Consider an actual flow in the laboratory wherein a uniform flow enters a channel, such as shown in the photograph in Figure 16.13. Here, velocity profiles in water flow are made visible by the hydrogen bubble method, where the bubbles are generated by electrolysis on a fine wire used as a cathode at the entrance of the channel. Near the entrance, the flow is uniform over a large portion of the distance across the channel; the viscous effects are limited to a thin boundary layer at the walls. However, as the flow progresses downstream, the viscous effects are felt over a larger portion of the flow. Finally, after the flow has covered a sufficient distance through the channel, the velocity profile is totally dominated by viscosity; a parabolic velocity profile is achieved, and the real flow becomes essentially the Poiseuille flow studied in this section. When this type of real flow is reached in the channel (at the right of Figure 16.13), it is called “fully developed flow.”
The parallel flows discussed in this chapter illustrate features common to many more complex viscous flows, with the added advantage of lending themselves to a relatively straightforward solution. The purpose of this discussion has been to introduce many of the basic concepts of viscous flows in a fashion unencumbered by fluid dynamic complexities. In particular, we have studied Couette and two-dimensional Poiseuille flows and found the following.