In another section, analytical solutions were obtained for flow between fixed parallel plates. In this section, the parallel plates are no longer fixed, and can be moved at a given velocity. This type of problem is generally referred as Couette flow.

Couette Flow

Consider flow between parallel plates, as shown in the figure where the top plate moves at a velocity U. The general velocity profile can be obtained from the Navier-Stokes equations for flow between fixed parallel plates as discussed in another section. That is,

     

Recall that this result is obtained based on the assumption that the flow is steady, incompressible and laminar.

Applying the boundary conditions (u = 0 at y = 0 and u = U at y = h) to obtain the coefficients as follows:

     

Simple Shear Flow

Velocity Profiles

The velocity profile now becomes   
   
     

For simple shear flow, there is no pressure gradient in the direction of the flow. The fluid motion is simply created by the moving top plate, and the velocity profile is linear (u = Uy/h).

The above equation can be recast into a dimensionless form as follows:    

     

where P is the dimensionless pressure gradient, and is given by

     

The velocity profiles for various P values are plotted in the figure. It can be seen that when P = 0 (i.e., no pressure gradient), the velocity profile is linear, as mentioned previously. Note that for P < -1, the fluid motion created by the top plate is not strong enough to overcome the adverse pressure gradient, hence backflow (i.e., u/U is negative) occurs at the lower-half region.