By A. J. Chorin, J. E. Marsden (auth.)
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Extra resources for A Mathematical Introduction to Fluid Mechanics
See, and thus any mechanism for starting Thus, intuitively, we might expect rotation Since rotation is intimately related to the for example, J. Marsden and A. Tromba, Vectop CaZcuZus, p. 161. 28 vorticity as we have just shown, we can expect the vorticity to be involved. Let Ct We shall now prove that this is so. C be a simple closed contour in the fluid at be the contour carried along the flow. where ~t = O. 1 (See Figure 1. 2-1). 2-1 The circuZation around Kelvin's Circulation Theorem. Ct is defined to be the line integral For isentropic fZow, in time.
E. related to a disc by a one-to-one invertible differentiable transformation) and that the resulting tube is diffeomorphic to the product of the disc and the real line. ). 36 Helmholtz' Theorem. (aJ If and Cl Assume the fZuid is isentropic. Then are any two curves encircZing the vortex C2 tube, then This common vaZue is caZZed the strength of the vortex tube. (b) The strength of the vortex tube is constant in time, as the tube moves with the fZuid. Proof. 2-6. v = Region Enclosed Figure 1. 2-6 The lateral surface of the vortex tube enclosed between Cl is denoted by and denoted by surface 8 81 8·, i.
R2 ' i. e. :2 satisfy exactly Thus we can conclude that may be obtained from a suitably rescaled solution ~l A be made for each flow and let U2 then the dimensionless velocity fields words, be two flows on ~2; ~l in other are similar. This idea of the similarity of flows is utilized in the design of experimental models. For example, suppose we are contemplating a new design for an aircraft wing and we wish to know the behavior of a fluid flow around it. Rather than build the wing itself, it may be faster and more economical to perform the initial tests on a scaled-down version.
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