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. The rotating disk acts as a fluid pump and 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. Flow description. Consider a planar disk of infinite  radius rotating at a constant angular velocity formula_1 in fluid which is initially at rest everywhere. Near to the surface, the fluid is being turned by the disk, due to friction, which then causes centrifugal forces which move the fluid outwards. This outward radial motion of the fluid near the disk 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 formula_2 and formula_3 are functions of formula_4 only, where formula_5 are the velocity components in  cylindrical formula_6 coordinate with formula_7 being the axis of rotation and formula_8 represents the plane disk. Due to symmetry, pressure of the fluid can depend only on radial and axial coordinate formula_9. Then the continuity equation and the incompressible Navier–Stokes equations reduce to where formula_11 is the kinematic viscosity. No rotation at infinity. Since there is no rotation at large formula_12, formula_13 becomes independent of formula_14 resulting in formula_15. Hence formula_16 and formula_17. Here the boundary conditions for the fluid formula_18 are Self-similar solution is obtained by introducing following transformation, where formula_22 is the fluid density. The self -similar equations are with boundary conditions for the fluid formula_24 are The coupled ordinary differential equations need to be solved numerically and an accurate solution is given by Cochran(1934). The inflow axial velocity at infinity obtained from the numerical integration is formula_27, so the total outflowing volume flux across a cylindrical surface of radius formula_14 is formula_29. The tangential stress on the disk is formula_30. Neglecting edge effects, the torque exerted by the fluid on the disk with large (formula_31) but finite radius formula_32 is The factor formula_34 is added to account for both sides of the disk. From numerical  solution, torque is given by formula_35. The torque predicted by the theory is in excellent agreement with the experiment on large disks up to the Reynolds number of about formula_36, the flow becomes turbulent at high Reynolds number. Rigid body rotation at infinity. This problem was addressed by George Keith Batchelor(1951). Let formula_37 be the angular velocity at infinity. Now the pressure at formula_38 is formula_39. Hence formula_40 and formula_41. Then the boundary conditions for the fluid formula_18 are Self-similar solution is obtained by introducing following transformation, The self-similar equations are with boundary conditions for the fluid formula_24  is The solution is easy to obtain only for formula_50 i.e., the fluid at infinity rotates in the same sense as the plate. For formula_51, the solution is more complex, in the sense that many-solution branches occur. Evans(1969) obtained solution for the range formula_52. Zandbergen and Dijkstra showed that the solution exhibits a square root singularity as formula_53 and found a second-solution branch merging with the solution found for formula_54. The solution of the second branch is continued till formula_55, at which point, a third-solution branch is found to emerge. They also discovered an infinity of solution  branches around the point formula_56. Bodoyni(1975) calculated solutions for large negative formula_57, showed that the solution breaks down at formula_58. If the rotating plate is allowed to have uniform suction velocity at the plate, then meaningful solution can be obtained for formula_59. For formula_60 (formula_61 represents solid body rotation, the whole fluid rotates at the same speed) the solution reaches the solid body rotation at infinity in an oscillating manner from the plate. The axial velocity is negative formula_62 for formula_63 and positive formula_64 for formula_65. There is an explicit solution when formula_66. Nearly rotating at the same speed, formula_67 . Since both boundary conditions for formula_68 are almost equal to one, one would expect the solution for formula_68 to slightly deviate from unity. The corresponding scales for formula_70 and formula_71 can be derived from the self-similar equations. Therefore, To the first order approximation(neglecting formula_73), the self-similar equation becomes with exact solutions These solution are similar to an Ekman layer solution. Non-Axisymmetric solutions. The flow accepts a non-axisymmetric solution with axisymmetric boundary conditions discovered by Hewitt, Duck and Foster. Defining formula_76 and the governing equations are with boundary conditions The solution is found to exist from numerical  integration for formula_79. Bödewadt flow. Bödewadt flow describes the flow when a stationary disk is placed in a rotating fluid. Two rotating coaxial disks. This problem was addressed by George Keith Batchelor(1951), Keith Stewartson(1952) and many other researchers. Here the solution is not simple, because of the additional length scale imposed in the problem i.e., the distance formula_80 between the two disks. In addition, the uniqueness and existence of a steady solution are also depend on the corresponding Reynolds number formula_81. Then the boundary conditions for the fluid formula_82 are In terms of formula_85, the upper wall location is simply  formula_86. Thus, instead of the scalings used before, it is convenient to introduce following transformation, so that the governing equations become with six boundary conditions and the pressure is given by Here boundary conditions are six because pressure is not known either at the top or bottom wall; formula_93 is to be obtained as part of solution. For large Reynolds number formula_94, Batchelor argued that the fluid in the core would rotate at a constant velocity, flanked by two boundary layers at each disk for formula_95 and there would be two uniform counter-rotating flow of thickness formula_96 for formula_97 . However, Stewartson predicted that for formula_98 the fluid in the core would not rotate at formula_94, but just left with two boundary layers at each disk. It turns out, Stewartson predictions were correct. There is also an exact solution if the two disks are rotating about different axes but for formula_61. Applications. Von Kármán swirling flow finds its applications in wide range of fields, which includes rotating machines, filtering systems, computer storage devices, heat transfer and mass transfer applications, combustion-related problems, planetary formations, geophysical applications etc.