By A. J. Chorin, J. E. Marsden (auth.)
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Extra resources for A Mathematical Introduction to Fluid Mechanics
Rather than build the wing itself, it may be faster and more economical to perform the initial tests on a scaled-down version. We design our model so that it has the same geometry as the full-scale wing, and choose values for the undisturbed velocity, coefficient of viscosity, etc. such that the Reynolds number for the flow in our experiment match that of the actual flow. We can then expect the results of our experiment to be relevant to the actual flow over the full-scale wing. We shall be especially interested in cases where We stress that one cannot say that if v R is large.
Grad p. We use this remark to prove existence. Indeed, grad Pl that given let ~ p be defined by the solution to the Neumann problem div w in D , w·n in ClD It is known * that the solution to this problem exists and is unique up to the addition of a constant to define div u ~ = 0, w - grad p. u·n =0 p. Then clearly by construction of With this choice of ~ has the desired properties p. 3-2. natural to introduce the operator operator, which maps preceding theorem, P w P P, It is an orthogonal projection onto its divergence free part is well-defined.
More general fluids. 3-1. S Here the velocity field is parallel to a surface ~ but jumps in magnitude either suddenly or rapidly as we cross If the forces are all normal to S, S. there will be no transfer of Figure 1. 3-1. S will diffuse across S I in S Faster molecules from and impart momentum to the fluid below and likewise, slower molecules from below across B However, if we remember the kinetic theory of matter, we see that this is actually unreasonable. above Band to slow down the fluid above S.