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Flow Between Parallel Plates
Consider steady, twodimensional, viscous flow between two parallel plates that are situated a perpendicular distance apart. Let
be a longitudinal coordinate measuring distance along
the plates, and let be a transverse coordinate such that the plates are located at and . See Figure 78.
Figure 78:
Viscous flow between parallel plates.

Suppose that there is a uniform effective pressure gradient in the direction, so that

(963) 
where is a constant.
Here, the quantity could represent a gradient in actual fluid pressure, a gradient in gravitational potential energy (due to an inclination of the plates to
the horizontal), or
some combination of the twoit actually makes no difference to the final result. Suppose that the fluid velocity profile between the plates takes the form

(964) 
From Section 2.18, this profile automatically satisfies the incompressibility constraint
, and is also such that
.
Hence, Equation (961) reduces to

(965) 
or. taking the component,

(966) 
If the two plates are stationary then the solution that satisfies the no slip constraint,
, at each plate is

(967) 
Thus, steady, twodimensional, viscous flow between two stationary parallel plates is associated with a parabolic velocity profile
that is symmetric about the midplane, .
The net volume flux (per unit width in the direction) of fluid between the plates is

(968) 
Note that this flux is directly proportional to the effective pressure gradient, inversely proportional to the fluid viscosity, and
increases as the cube of the distance between the plates.
Suppose that the upper plate is stationary, but that the lower plate is moving in the direction at the constant speed . In this
case, the no slip boundary condition at the lower plate becomes , and the modified solution to Equation (966) is

(969) 
Hence, the modified velocity profile is a combination of parabolic
and linear profiles. This type of flow is known as Couette flow. The net volume flux (per unit width) of fluid between the plates becomes

(970) 
Next: Flow Down an Inclined
Up: Incompressible Viscous Flow
Previous: Introduction
Richard Fitzpatrick
20120427