By R. M Dudley
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Extra info for Probabilities and metrics: Convergence of laws on metric spaces, with a view to statistical testing
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For plane flows the methods of complex variables are useful tools. 5) ∂x ∂y and is irrotational, that is, ∂u ∂v − = 0. 7) Let which is called the complex velocity. 6) are exactly the Cauchy-Riemann equations for F , and so F is an analytic function on D. Conversely, given any analytic function F , u = Re F and v = −Im F define an incompressible (stationary) potential flow. If F has a primitive, F = dW/dz, then we call W the complex potential . ) Write W = ϕ + iψ. 7) is equivalent to u = ∂x ϕ = ∂y ψ and v = ∂y ϕ = −∂x ψ, that is, u = grad ϕ and ψ is the stream function.
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. 3. Vortex sheets and lines remain so under the flow. 3). Proposition If a surface (or curve) moves with the flow of an isentropic fluid and is a vortex sheet (or line) at t = 0, then it remains so for all time. Proof Let n be the unit normal to S, so that at t = 0, ξ · n = 0 by hypothesis. , ˜t S ξ · n dA = 0. It follows that ξ · n = 0 identically on St , so S remains a vortex sheet.
7), P∂t u = ∂t u. Because P(grad p) = 0, we get ∂t u = P −u · ∇u + 1 ∆u . 8) Although ∆u is divergence free, it need not be parallel to the boundary and so we cannot simply write P∆u = 0. 8) of the Navier– Stokes equations eliminates the pressure and expresses ∂t u in terms of u alone. The pressure can then be recovered as the gradient part of −u · ∇u + 1 ∆u. 11 The pressure in compressible flows is conceptually different than in incompressible flows just as it was in ideal flow. If we think of viscous flow as ideal flow with viscous effects added on, it is not unreasonable to assume that p is still a function of ρ.
Probabilities and metrics: Convergence of laws on metric spaces, with a view to statistical testing by R. M Dudley
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