By Pert, Geoffrey
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Extra resources for Introductory Fluid Mechanics for Physicists and Mathematicians
If this requirement is violated the loop is broken as one part goes along one side of the surface and the other part along the other. Kelvin’s theorem therefore no longer holds. This streamline thus separates the region of zero vorticity from one where the vorticity may be ﬁnite. As a consequence a tangential discontinuity may form along the streamline, across which the tangential component of velocity changes discontinuously, but the pressure and normal velocity component are continuous. The discontinuity itself forms a ‘vortex sheet’ due to the velocity diﬀerence, which is equivalent to an inﬁnitely thin layer of vortices of appropriate strength per unit area.
G. aerofoils. In this case the problem of non-uniqueness is resolved by an additional empirical condition, which frequently allows accurate representations of the ﬂow found in practice. Introductory Fluid Mechanics for Physicists and Mathematicians, First Edition. Geoﬀrey J. Pert. © 2013 John Wiley & Sons, Ltd. Published 2013 by John Wiley & Sons, Ltd. 25 26 Introductory Fluid Mechanics Much of the theory of aerodynamics in consequence has been constructed around inviscid ﬂow models. 2. Waves and instabilities.
41) shows that the inclusion of the additional term ∂φ/∂t yields a form which is time dependent and valid throughout the ﬂow, not just along a streamline, but is only applicable in irrotational ﬂow. 33 Flow of Ideal Fluids Since v · d increases along a streamline, it follows from Stokes’ theorem that, if the potential is single valued (acyclic ﬂow ),4 the streamlines must start and ﬁnish on the surface of a body or at inﬁnity. 12) since ∇φ = v is parallel to the streamline δ , unless the ﬂow velocity is zero and the potential constant.
Introductory Fluid Mechanics for Physicists and Mathematicians by Pert, Geoffrey