# Modified Taylor-Couette flow

I have a question regarding the case of two nested cylinders (both infinitely long and massive) with a fluid between them, but rather than being fixed the inner cylinder is free to rotate while the outer cylinder has a constant torque applied to it.

I assume that the problem has axisymmetric flow, such that the fluid velocity is only in the angular direction and depends only on the radial distance $$r$$.

Now, assuming Navier-Stokes, and the solution to the fluid velocity will be of the form $$A(t) r + B(t)/r$$, what are our expected boundary conditions for the problem?

On the outside, we have constant torque, which means at the outer cylinder we will have a $$A(t)r + B(t)/r = ctr$$, where $$c$$ is determined by the constant torque and $$t$$ is time. However, on the inner cylinder, I am having some trouble. Since we are starting from rest but we are free to move, is there something I can say about the system evolving with time, without having to solve numerically?

My most simple solution would be to use Torque = Force * Lever Arm, where the lever arm would be calculated based on the distance from the two cylinders, but I do not think we can assume this.

Any help would be appreciated.