Download Advances in Applied Mechanics, Vol. 25 by Eds. Theodore Y. Wu & John W. Hutchinson PDF

By Eds. Theodore Y. Wu & John W. Hutchinson

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However, serious consideration must be given to the suggestion of Fraenkel (1962) for laminar flow that greater accuracy will result when a basic solution can be found that fits the geometry more closely. The technique for calculating higher approximations on that basis will have to be developed, for it was not apparent to us how to improve systematically even the tangent-wedge approximation for plane potential flow through a symmetric channel. Similarly, a systematic technique must be developed for calculating higher approximations in thin three-dimensional regions.

Laminar Flow Kotorynski treats the corresponding laminar flow by using coordinates that differ slightly from ours in orientation. ) His basic solution is, of course, the Poiseuille flow through a straight circular pipe; with our notation and normalization this gives u=2(1-r2)=2(1-y2-z2). 24) After some analysis Kotorynski finds the first approximation to the transverse elocities; in our notation again (and with a missing factor a restored) these are u=o, w=2m(l-y 2 2 - 2 ) . 25) We notice that this flow field again represents at each axial station simply a parallel stream slightly inclined to the central axis.

We have seen that the cylindrical solution can be found for a wide variety of problems. It has a trivial form for the Laplace or biharmonic equation, and for the NavierStokes equations it is known for many cross sections, including ellipses. By contrast, the wedge and cone solutions needed to fit the slope of a boundary are much more limited. For example, not only does the Jeffery-Hamel solution for laminar flow through a wedge have no axisymmetric counterpart for a cone, but even the solution for the wedge involves elliptic functions, which greatly complicates the applications.

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