By A. J. Chorin, J. E. Marsden (auth.)
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Additional resources for A Mathematical Introduction to Fluid Mechanics
P, u. By the Notice that by construction is a linear operator and that ~~ See Courant and Hilbert, Methods of MathematicaZ Physics, Wiley (1953) The equation ~p = f Clp/Cln = g has a solution uniq"ue up to a constant if and only if fD f dV = faD ensures this condition in our case. g dA. 4). Since u true of Since If we apply the operator P to both sides, we obtain is divergence-free and vanishes on the boundary, the same is at~ (if P(grad p) ~ = 0, is smooth enough). 7), pat~ = at~. 8) Although is divergence free, it need not be parallel to the bound- 6~ ary and so we cannot simply write P~~ = ~~.
5) be a parametrization of the loop Then a parametrization of Ct is ~(~(s),t), C, 0 ~ s ~ 1. Thus, by definition of the line integral and the material derivative, ~tfc t uods d dt + Since d~ = u at -' r o~ ( ¢( ~ ( s) , t) , t f: f: Du d TIt( ¢ (~( s ) , t ) °as ~(~(~(s) lO o (since Proof of the Circulation Theorem. Dt -'Vw ~(~(s),t)ds ,t),t)°a-t d as ~ (~( s ) , t ) ds . ~(
Marsden and A. Tromba, Vectop CaZcuZus, p. 161. 28 vorticity as we have just shown, we can expect the vorticity to be involved. Let Ct We shall now prove that this is so. C be a simple closed contour in the fluid at be the contour carried along the flow. where ~t = O. 1 (See Figure 1. 2-1). 2-1 The circuZation around Kelvin's Circulation Theorem. Ct is defined to be the line integral For isentropic fZow, in time. is constant rC t For example we note that if the fluid moves in such a way that Ct shrinks in size, then the angular velocity around Ct increases.