By Peter L Duren; Richard Askey; Uta C Merzbach; Harold M Edwards

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4). Assume that u2 < u1 . Imagine that a particle comes down from u1 at y = −∞. Without friction it will pass 38 CHAPTER 2. ANALYTICAL METHODS through u2 at some point and go further down. This is prevented by the the presence of the term λU0′ which is a friction term. The value of λ gives the just right amount of friction so that the particle reaches u2 as y → +∞. 47) At this time scale (which is sometimes called the convective time scale, in contrast to the diffusive time scale discussed below), the interface moves to the region of lower free energy, with a constant normal velocity that depends on the difference in the free energy of the two phases u1 and u2 .

J. Durlofsky, “Numerical calculation of equivalent grid block permeability tensors for heterogeneous poros-media,” Water. Resour. , vol. 27, pp. 699–708, 1991. [24] W. E and B. Engquist, “The heterogeneous multi-scale methods,” Comm. Math. , vol. 1, pp. 87–133, 2003. [25] W. E and B. Engquist, “Multiscale modeling and computation,” Notices of the American Math. , vol. 50, no. 9, pp. 1062–1070, 2003. [26] B. , Springer-Verlag, vol. 1270, pp. 10–22, 1987. [27] C. Garcia-Cervera, W. Ren, J. Lu and W.

24) where the pressure P = P (x, t) is assumed to be given. Prandtl’s boundary layer theory is derived under the assumption that the flow field can be decomposed into an outer region where viscous effects can be neglected and an inner region that lies close to the wall. This assumption is valid for laminar flows. However, it is well-known that at high Reynolds number (in our setting, small ε), fluid flow generally becomes turbulent. In that case, the assumption stated above no longer holds and Prandtl’s theory ceases to be valid.