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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.

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You set the velocity equal to zero at the outer cylinder and equal to the imposed velocity at the inner cylinder. This determines A and B.

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