By A. J. Chorin, J. E. Marsden (auth.)
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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~~ = ~~.
1) is a matrix', about which some assumptions will have to be (Jon need not be parallel to n. 1) is somewhat ambiguous since component parallel to n. (Jon may contain a This issue will be resolved below when we give a more definite functional form to g. l). Thus we see that of momentum across the boundary of it approximates in a reasonable way Wt • ~ modifies the transport We will choose (J = so that the transport of momentum by molecular motion. 1) acting on should be a linear function of n. S In fact if one merely assumes it is an arbitrary continuous function of ~, then using balance of 45 momentum, one can actually prove it is linear in is called Cauahy's Theorem.
67. 33 Proof. One has the vector identity Substitution into the equations of motion yields nay + .! u) 2 - - - ~ x curl ~ = Vw Taking the curl gives (see a table of vector identities), ~~ curl(u x ~) =0 Le. Le. 9) by the continuity equation. y)~. On the other hand, (chain rule) Thus F equation. and G satisfy the same linear, first order differential Since they coincide at t = 0, they are equal. 10) proved in § 1. 1. 10). For two-dimensional flow, where component ~t ~ = (O,O,s). 7). e . sip is propagated as a scalar by the flow.