Steady Flow over a Rotating Disk: von Kármán Swirling Flow

The steady flow induced by a infinite disk that rotates in its own plane at is a classical problem in fluid mechanics. It is one of the few examples of a viscous flow that involves all three components of velocity and admits an exact solution to the Navier–Stokes equations. The velocity field for the swirling flow is given by: . The Navier–Stokes equations reduce to:
The boundary conditions at the disk surface are no slip and impenetrability conditions:
where is the rotational speed of the disk. Far away from the surface of the disk, viscous effects are negligible, such that
However, the viscous pumping action of the disk is balanced by a uniform axial inflow at infinity: .
von Kármán was able to show that the above equations admit a self-similar solution defined by: , , , and , where , and the functions , , , and are determined by:
and ,
subject to the boundary conditions:
and .
It is evident from the structure of the equations that the equation for the pressure field is decoupled from the equations that define the velocity field given by the functions , , and . The boundary value problem for the velocity field can be solved by a shooting method using Mathematica. Also of interest are the fluid particle trajectories in the swirling flow; they can be computed from the velocity field by solving:
, , .
It is noteworthy that the boundary layer thickness in the above swirling flow is constant over the disk surface; this feature of the flow is exploited in electrochemistry to study mass transfer.
This Demonstration shows the velocity components of the swirling flow as a function of the angular velocity of the disk and the kinematic viscosity of the fluid. The thickness of the boundary layer is also displayed. Fluid particle trajectories are also shown for different starting positions in the flow.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


[1] F. M. White, Viscous Fluid Flow, New York: McGraw–Hill, 1974.
[2] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge: Cambridge University Press, 1967.
[3] H. Schlichting, Boundary-Layer Theory, 6th ed., New York: McGraw–Hill, 1968.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+