Seeking closed form solution of Navier-Stokes for a fluid in an annular space.

[MudEngineer]

I have a pressure flow problem where I'm trying to understand the velocity profile of a fluid in an annular space between a stationary exterior cylinder and a rotating, longitudinally advancing cylinder at its center.

So the boundary conditions a zero velocity at the exterior surface and a constant angular and longitudinal velocity at the interior surface.

I begin by simplifying the usual form of Navier-Stokes in cylindrical coordinates to the following three equations, knowing that acceleration and velocity in the radial direction are zero:

[1] -ρ(u_θ^2)/r=μ*(2/r^2)((∂u_θ)/∂θ)-(∂u_θ)/∂θ+ρ*g_r

[2] ρ((∂u_θ)/∂t+(u_θ/r)((∂u_θ)/∂θ)+u_z*((∂u_θ)/∂z))=μ[(∂^2*u_θ)/(∂r^2 )+(1/r^2)*((∂^2 u_θ)/(∂θ^2))+(∂^2 u_θ)/(∂z^2 )]-(1/r)(∂p/∂θ)+ρ*g_θ

[3] ρ((∂u_z)/∂t+(u_θ/r)(∂u_z)/∂θ+u_z*(∂u_z)/∂z)=μ[(∂^2*u_z)/(∂r^2)+(1/r^2)(∂u_z)/(∂θ^2)+(∂^2*u_z)/(∂z^2)]-∂p/∂z+ρ*g_z

How do I solve for u_θ and u_z??

 

[arildno]

Why should the solution be "closed form"?
Most answers to the questions of life, universe and everything are not closed form solutions.

 

[MudEngineer]

I've been considering different ways to find a numerical solution, but for the sake of repeatability I'd like to find a solution with a couple of neat equations that I can just plug and chug down the road.

Most of the numerical modeling methods involve several different steps, including generation of a mesh, iterative solving using some form of programming, and then finding a way to make that data usable for later calculations. Unfortunately, generating the velocity profile is only the first step of the problem.

 

[HallsofIvy]

I'm not going to do the entire calculation, but, if I remember correctly, there is an exact solution to fluid motion through a pipe (cylinder) and the fronts (motion of what was initially a cross section of the pipe) are paraboloids.

 

[MudEngineer]

That's the standard solution for steady-state, pressure driven flow in a stationary pipe. In this case we have two boundary layers. In the attached image, I'm trying to solve for the velocity profile between the two boundary layers, where the outer (brown) layer is stationary, and the inner (blue pipe) layer is moving to the right and rotating.

 

[bigfooted]

I think what you are describing is Taylor-Couette flow. For very low rotational speeds it has a known solution (like V=Ar+B/r, where A and B depend on radius and rotational speed), but it quickly becomes difficult to get analytic solutions.