By Command. Duchêne
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Extra info for Flight Without Formulae - Simple Discussions on the Mechanics of the Aeroplane, (2nd Ed.)
Solution. a) For any smooth function f (x, t), ∇f · dx = f (x(1, t), t) − f (x(0, t), t). C(t) The right-hand side represents the diﬀerence of f values at the endpoints of C(t). If f (x, t) is a convected scalar, the endpoint values are timeindependent, so d ∇f · dx = 0. 14-1) is satisﬁed if g is a gradient of a convected scalar. We derive the PDE for g(x, t): Let x(s, t) be a parametric representation of C(t) with s in [0, 1] labeling material points. 14-2) 1 d dt g(x(s, t), t) · τ (s, t) ds 0 1 = 0 Dg · τ + g · ((τ · ∇)u) Dt ds.
5) f (t + t ) = f (t) + f (t ), all t, t > 0. 3). Here is a heuristic treatment of higher moments xn , for n > 2: By evenness of g(x, t) in x, odd moments (with n odd) are zero. 1), so one expects even moments (with n even) to be proportional to (Dt)n/2 . The diﬀusion PDE. Now suppose that at t = 0, the density of x’s is a given nonnegative function c(x, 0). What is the density c(x, t) of x’s at time t > 0? Consider the particles at time t + τ which came from positions between x and x +dx at time t.
14-7), whose initial values at t = 0 are ω(x, 0). Integral curves of v are material curves, and since they coincide with the integral curves of ω at t = 0, they are the same as the integral curves of ω. Let x(s, t) be a parametric representation of one of these integral curves C(t), with s labeling material points. We can choose the labeling s so that v(x(s, 0), 0) = τ (s, 0) := xs (s, 0). 14-8) v(x(s, t), t) = τ (s, t) for all t. 14-9) ω(x(s, t), t) = λ(s, t)v(x(s, t), t). The local coeﬃcient λ(s, t) remains to be determined.