By A. J. Chorin, J. E. Marsden
The objective of this article is to give a few of the easy principles of fluid mechanics in a mathematically appealing demeanour, to provide the actual historical past and motivation for a few buildings which were utilized in contemporary mathematical and numerical paintings at the Navier-Stokes equations and on hyperbolic structures and to curiosity a number of the scholars during this appealing and tough topic. The 3rd variation has included a couple of updates and revisions, however the spirit and scope of the unique e-book are unaltered.
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Extra resources for A mathematical introduction to fluid mechanics, Second Edition
1. The circulations about C and C are equal if the ﬂow is potential in Σ. Let the velocity of ∂D be speciﬁed as V, so u · n = V · n. Thus, ϕ solves the Neumann problem: ∆ϕ = 0, ∂ϕ = V · n. 4) where p = −ρ u 2 /2. 1). 4) (with ϕ determined only up to an additive constant) on simply connected regions. This observation leads to the following. Theorem Let D be a simply connected, bounded region with prescribed velocity V on ∂D. 4)) in D if and only if ∂D V · n dA = 0; this ﬂow is the minimizer of the kinetic energy function Ekinetic = 1 2 ρ u 2 dV, D among all divergence-free vector ﬁelds u on D satisfying u ·n = V·n.
One can show (using the implicit function theorem) that if ξ(x) = 0, then, locally, a vortex line is the intersection of two vortex sheets. 4). This fact can also be used to give another proof of the preceding theorem. We assume we are in three dimensions; the two-dimensional case will be discussed later. 4. The vorticity is transported by the Jacobian matrix of the ﬂow map. 1) and ∇ϕt is its Jacobian matrix. Proof Start with the following vector identity (see the table of vector identities at the back of the book) 1 2 ∇(u · u) = u × curl u + (u · ∇)u.
17) that is, u is tangent to the circle of radius r with magnitude |∂r ψ| and oriented clockwise if ψr > 0 and counterclockwise if ψr < 0. Next, observe that 1 ∂ ∂ψ ξ = −∆ψ = − r , r ∂r ∂r a function of r alone. Because ψr = 0, r is a function of ψ so ξ is also a function of ψ. Thus, J(ξ, ψ) = 0. Hence, motion in concentric circles with u deﬁned as above is a solution of the two-dimensional stationary incompressible equations of ideal ﬂow. 18) ∆A = −ξ, div A = 0, u = ∇ × A. Here we used ∇·u = 0 to write u = ∇×A, where div A = 0.