Solution of linear Navier-Stokes equations in a cylindrical coordinates

Question

Can you advise me please how to solve analytically the linear Navier-Stokes equations in a cylindrical coordinate system. Or share the link please if this solution exists somewhere.

Also there is сontinuity equation

Comments

This set of differential equation lacks the initial and boundary conditions for its variables $p$ and $\bf{v}$. As there is no time derivative, this set describes a steady flow within some "geometry" - in a pipe, cone, etc., not necessarily with cylindrical symmetry.

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In this task a solid ball rotates around Oy with a constant angular velocity and it is streamlined with velocity U on the Ox by flow of liquid. In cylindric coordinates z is Oy.

Maybe there is some symmetry which reduces equations.

The boundary conditions on this photo:

Answer 1

I do not know what is the analytical solution is in this case, but it may exist in textbooks. What I know is that there is the pressure difference between co-current and counter-current sides. This pressure difference creates a force acting perpendicularly the body velocity and it is known as a Magnus effect. Football players and tennis players use it. Also, cylindrical tubes (rotors) were installed on some ships. I personally can throw a round stone in water and this stone will first enter the water and then reappear above the surface again (jump out of water).

Maybe you can pick up (guess) a solution from qualitative behaviour and dimensional reasoning.