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  Solution of linear Navier-Stokes equations in a cylindrical coordinates

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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

asked Dec 20, 2016 in Theoretical Physics by avenior (0 points) [ revision history ]

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.

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:

1 Answer

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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.

answered Dec 21, 2016 by Vladimir Kalitvianski (102 points) [ revision history ]
edited Dec 21, 2016 by Vladimir Kalitvianski

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