on to fluid mechanics, Second Edition
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.
A mathematical introduction to fluid mechanics, Second Edition by A. J. Chorin, J. E. Marsden