How do I find a stream function given a volumetric flow rate? How do I find a stream function given a volumetric flow rate?
The flow only occurs in one direction, between 2 plates,  and I have no knowledge of velocity.
I know that volumetric flow rate = change in stream function between two points but I have no idea how to apply this.
 A: If it is pure shear flow, with the top plate moving with velocity V and the lower plate stationary, the stream function is given by $$\psi=Q\left(\frac{y}{h}\right)^2$$where Q is the total volumetric flow rate, equal to Vh/2, h is the distance between the plates, and y is the distance measured upward from the bottom plate.
A: If you have two stationary plates h apart and know Q. You know that the fluid is not flowing at the boundary of the plates (right up against the plates, this is called the no-slip boundary condition), so at y= 0 and y= h, the velocity is zero.
You also know that the flow is symmetrical about the height y= h/2 because the plates are not moving, and the max velocity and max stream function is at y= h/2. The region from y= 0 to y= h/2 has Q/2 flowing through it (as does the region from y= h/2 to y= h) due to the symmetry.
The difference between the stream function at y= 0 and y= h/2 is equal to the flow through that region, which is Q/2, because Q/2 also flows through the other symmetrical region. We also know stream function at y= 0 is zero. These two facts mean the stream function is Q/2 at y= h/2, which is the max value of the stream function. It goes from 0 at y=0 to Q/2 at y=h/2 back to 0 at y=h. That’s all we know for sure.
If it is pure sheer, then it is linear. Stream = yQ/h from y= 0 to h/2 and Stream = Q - yQ/h from y= h/2 to h.
