Von Kármán swirling flow

Von Kármán swirling flow is a flow created by a uniformly rotating infinitely long plane disk, named after Theodore von Kármán who solved the problem in 1921. This problem is used as a model for centrifugal fans or compressors. This flow is classified under the category of steady flows in which vorticity generated at a solid surface is prevented from diffusing far away by an opposing convection, the other examples being the Blasius boundary layer with suction, stagnation point flow etc.

Consider a plane disk of infinite radius rotating at a constant angular velocity ${\displaystyle \Omega }$ in fluid which is initially rest everywhere. The outward radial motion of the fluid near the disk due to the centrifugal force must be accompanied by an inward axial motion of the fluid towards the disk to conserve mass. Theodore von Kármán noticed that the governing equations and the boundary conditions allow a solution such that ${\displaystyle u/r,v/r}$ and ${\displaystyle w}$ are functions of ${\displaystyle z}$ only, where ${\displaystyle (u,v,w)}$ are the velocity components in cylindrical ${\displaystyle (r,\theta ,z)}$ coordinate with ${\displaystyle r=0}$ being the axis of rotation and ${\displaystyle z=0}$ represents the plane disk. Due to symmetry, pressure of the fluid can depend only on radial and axial coordinate ${\displaystyle p=p(r,z)}$. Then the continuity equation and the incompressible Navier–Stokes equations reduce to

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