By A. J. Chorin, J. E. Marsden (auth.)
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Additional info for A Mathematical Introduction to Fluid Mechanics
Thus, the stretching of vortex tubes can increase vorticity, but cannot create it. A vortex tube with non-zero strength cannot "end" in the interior of the fluid; it either forms a ring (such as the smoke from a cigarette), extends to infinity or is attached to a solid boundary. The usual argument supporting this statement goes like this: suppose the tube ended at a certain cross-section fluid. Cl . S, Since the tube cannot be extended, we must have inside the ~ =0 on Thus the strength is zero, a contradiction.
P These equations must be supplemented by boundary conditions. e. fluid does not cross the boundary, but may move tangentially to the boundary. For the Navier-Stokes equations, the extra term number of derivatives of ~ vLlu involved from one to two. raises the For both experimental and mathematical reasons, this is accompanied by an increase in the number of boundary conditions. e. that a unique solution exists and depends continuously on the initial data. In three dimensions, it is known that smooth solutions to the incompressible equations exist for short time and depend continuously on the initial data.
3-4). 3-4 The presence of viscosity allows the pressure forces to be balanced by the term state. lu"(Y) R and allows the fluid to achieve a stationary We saw that this was not possible for ideal flow.
A Mathematical Introduction to Fluid Mechanics by A. J. Chorin, J. E. Marsden (auth.)