By A. J. Chorin, J. E. Marsden (auth.)
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Extra resources for A Mathematical Introduction to Fluid Mechanics
A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow", Gordon and Breach (19·69). See also R. Temam , "The Navier-Stokes Equations," North Holland (1977). 48 the boundary one sees that the velocity approaches zero at the boundary to a high degree of precision. The no slip condition is also reason- able if one contemplates the physical mechanism responsible for the viscous terms, namely molecular motion. Our opening example indicates that molecular interaction between the solid wall with zero tangential velocity (or zero average velocity on the molecular level) should impart the same condition to immediately adjacent fluid.
A curve ~) which is tangent to the vorticity vector ~ at each of its points. Theorem. If a surface (or curve) moves with the flow of an isentropic fZuid and is a vortex sheet (or line) at for aZl time. 2-3). u vortex line Figure 1. 2-3 Proof. ~ = 0 n be the unit normal to by hypothesis. e. identically on S, so S remains a vortex sheet. One can show that (or take it as part of the definition) if ~(~) ~ Q, ~ then locally, a vortex line is the intersection of two 32 vortex s'heets. * Thus the result for vortex lines follows from that for vortex sheets.
We now use Stokes' theorem which will bring in the vorticity. 2-2. dA = ndA Thus, as a corollary of the circulation theorem, we can conclude that the flux of vorticity across a surface moving with the fluid is constant in time. By definition a vortex sheet (resp. line) is a surface S 31 (resp. a curve ~) which is tangent to the vorticity vector ~ at each of its points. Theorem. If a surface (or curve) moves with the flow of an isentropic fZuid and is a vortex sheet (or line) at for aZl time. 2-3).
A Mathematical Introduction to Fluid Mechanics by A. J. Chorin, J. E. Marsden (auth.)