Here, I am going to go over the solution to fully developed laminar pipe flow. This is a canonical problem and provides an exact solution to the Navier-Stokes equations. It is often referred to as the Hagen-Poiseuille flow problem.
The problem is depicted in the figure below:
In this problem, we are examining laminar flow through a pipe. The problem states that the flow in the pipe is being driven by a constant pressure gradient in the axial direction (dP/dz=constant). We assume that the flow is purely axial (), steady state (), incompressible (ρ=constant), axisymmetric (). We also neglect gravity.
First we start with the axial Navier-Stokes momentum equation in cylindrical coordinates:
By using our assumptions we can reduce this equation to give us:
Because we know that dP/dz is a constant, this function is easily integrated twice. The first integration:
And the second integration:
Now we apply our boundary conditions!
No slip condition at r=R :
Finite axial velocity at r=0:
Since ln(0) is a discontinuity, we know that in order for these equations to be satisfied, A must be equal to zero (ie A=0). Then combining these two equations, we get that:
Thus our final solution is:
And there we have our answer!