In this post I am going to go through the solution to the moving wall Couette flow problem.
An illustration of the problem is given below:
In this problem, the fluid between two parallel plates is being driven by the motion of the top plate. Here, we assume that the flow is axial (v=w=0), incompressible (ρ=constant), fully developed (u only a function of y). We also assume that gravity can be neglected and that no pressure gradient is present (dp/dx=0).
First we start with the Navier-Stokes momentum equation in the x-direction:
By applying our assumptions listed above, you should be able to see that the equation simply becomes:
To solve the problem we must integrate this equation and solve using the boundary conditions defined by the problem. The integration results in:
Our boundary conditions come from the problem and our super smart knowledge of the no-slip condition ;). Ie.
After subbing in we get two equations, with two unknowns (the integration constants)
Therefore we can see that B=0 and A=V/h. This leads to the final solution of this plane Couette flow problem:
Now we have shown that the velocity profile in this case is the linear profile above. We can also calculate the shear stress:
Any thoughts or questions please respond in the comments!