Vorticity and the equations of motion
The momentum equation is
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(11.9) |
The convective term may be written as
[ = 0, by continuity] |
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Invoking (11.8) into the above one obtains |
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(11.10) |
The viscous term may be written as
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[ = 0, by continuity] |
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Invoking (11.8) into the above one obtains |
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(11.11) |
The momentum equation thus reads as
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(11.12) |
In the above equation and in this section of the text, we shall use p=p/ρ. This equation shows two inferences
- for irrotational flow,
for steady flows
or
= constant (Bernoulli's equation!) |
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- is analogous to the coriolis force
if the coordinate system were rotating with an angular velocity . The vorticity is twice the angular velocity . The vorticity term is also related to lift generated by a vortex line exposed to a velocity . In the equation for , this term is . A graphical interpolation is shown below.
Figure 11.2 |