## Flow Between Parallel Plates

**Introduction: **Laminar and Turbulent Flow: Fluid flow can be classified as laminar or turbulent, which can be best described using the classic Reynolds experiment. Consider water flowing through a transparent pipe from a reservoir. A dye is injected at the inlet of the pipe. For a small flow rate, the dye will remain intact and form a single filament as it moves downstream. This is laminar flow since the dye flows in an ordered fashion. For a large flow rate, the dye will fluctuate immediately in a random fashion due to microscopic variations in the velocity field, and rapid dispersion of the dye is observed throughout the pipe. This is turbulent flow since the dye stream is unstable and fluctuating throughout the flow field. At an intermediate flow rate, the flow is in transition. The dye is intact at the beginning, and it slowly becomes fluctuating as it moves downstream. The fluctuations, however, are not as intense as in fully turbulent flow.

Recall that the Reynolds number (Re) is determined by ** Re = ρVL/μ**

And it is used to classify fluid flow as laminar or turbulent. The nature of the flow (laminar vs. turbulent) not only depends on its velocity but also its density, viscosity and length scale. For flow between parallel plates, the flow is laminar when Re < 1,400; whereas for pipe flow, the flow is laminar when Re < 2,100

**Flow between Fixed Parallel Plates:** Consider steady, incompressible, laminar flow between two infinite parallel horizontal plates as shown in the figure. The flow is in the x- direction, hence there is no velocity component in either the y- or z- direction (i.e., v = 0 and w = 0). The steady-state continuity equation becomes

........... (1)

From Eqn. 1, it can be concluded that the velocity u is a function of both y and z only. Since the plates are infinitely wide, it can be argued that the velocity u should not be a function of z, i.e., it must be a function of y only, u = u(y). Applying the Navier-Stokes equations, along with the assumptions that v = 0, w = 0 and u = u(y), yields

Eqn. 4 indicates that the pressure is a function of x and y. Integrate Eqn. 3 to yield

p = -ρgy g1(x)

Hence it can be concluded that is a function of x only. Now, integrate Eqn. 2 twice with respect to y, and treat as a constant (with respect to y) to give:

Applying the no-slip conditions (i.e., the fluid is "stuck" to the plates or u = 0 at y = ±h) to determine the coefficients as follows:

The velocity profile becomes

This is a parabola. The total volumetric flow per linear depth can be obtained integrating the velocity to give

**Note: **q is per linear depth, which is different than Q which is the total volumetric flow rate. Also note that the flow is negative, i.e. to the left, for a positive pressure gradient, dp/dx. This is due to the gradient definition where decreasing pressure to the right is negative.