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Circulation, vorticity, and vortices

Fluids will invariably swirl around, and it is often useful to characterize the amount of swirling. This is done by introducing the circulation  tex2html_wrap_inline1074 , defined as


with the line integral taken is around a closed circuit C in a counter-clockwise sense. By using Stokes' theorem we may write this as a surface integral,


where the vorticity  tex2html_wrap_inline1078 of the fluid is defined as


If the vorticity of the fluid is zero, we say that the fluid is irrotational :


We've already mentioned that there are some useful analogies between magnetostatics and incompressible fluid flow. I've collected these together in Table 2.1 below. To transform the magnetostatics expressions into those for fluid flow, we make the substitutions tex2html_wrap_inline1080 , tex2html_wrap_inline1082 , and tex2html_wrap_inline1084 (where here tex2html_wrap_inline1039 is the mass density of the fluid). If a fluid is both incompressible and irrotational, then tex2html_wrap_inline1088 and tex2html_wrap_inline1090 , which are the same as the equations of electrostatics and magnetostatics in free space (i.e., in the absence of charges and currents).

Magnetostatics Incompressible fluid flow
tex2html_wrap_inline1043 tex2html_wrap_inline1090
tex2html_wrap_inline1096 (Ampére's Law) tex2html_wrap_inline1098 (circulation)
tex2html_wrap_inline1100 (energy density of the magnetic field) tex2html_wrap_inline1102 (kinetic energy density of a fluid)
Table 2.1: Analogies between incompressible fluid flow and magnetostatics.


As an example of fluid flow with vorticity, we can consider a single vortex , which would be produced when draining a bathtub, and which is illustrated in Fig. 2.5.


Figure 2.5: Velocity field of a vortex. Flow lines are drawn for equal increments of the speed, as shown by arrows. Click here to see animations.

In cylindrical coordinates tex2html_wrap_inline1104 there is only one component of the velocity field, tex2html_wrap_inline1106 . In calculating the circulation, the line element tex2html_wrap_inline1108 , so that tex2html_wrap_inline1110 . If the circulation is independent of the integration path, then we must have tex2html_wrap_inline1112 , with C a constant. The circulation is then


so that tex2html_wrap_inline1116 . Therefore, the velocity field of a vortex is


By making the substitutions suggested above, we can use this result to find the magnetic field produced by a wire carrying a current I:


next up previous
Next: Steady flows of incompressible Up: Fluid Mechanics: Preliminaries Previous: Streamlines and streamtubes

Vittorio Celli
Wed Sep 10 01:02:02 EDT 1997