Blasius expansions

Since for large Schmidt numbers, the depth of penetration by diffusion is very small, only the region near the wall affects the mass or heat transfer rate. Consequently, in such cases, one can retain only the first term in the Blasius expansions for the velocity components, to obtain... 

The problem of start-up flow for a circular cylinder has received a great deal of attention over the years because of its role in understanding the inception and development of boundary-layer separation. An insightful paper with a comprehensive reference list of both analytical and numerical studies is S. I. Cowley, Computer extension and analytic continuation of Blasius expansion for impulsive flow past a circular cylinder, J. Fluid Mech. 135, 389-405 (1983). 

Turbulent mass transfer near a wall can be represented by various physical models. In one such model the turbulent flow is assumed to be composed of a succession of short, steady, laminar motions along a plate. The length scale of the laminar path is denoted by x0 and the velocity of the liquid element just arrived at the wall by u0. Along each path of length x0, the motion is approximated by the quasi-steady laminar flow of a semiinfinite fluid along a plate. This implies that the hydrodynamic and diffusion boundary layers which develop in each of the paths are assumed to be smaller than the thickness of the fluid elements brought to the wall by turbulent fluctuations. Since the diffusion coefficient is small in liquids, the depth of penetration by diffusion in the liquid element is also small. Therefore one can use the first terms in the Taylor expansion of the Blasius expressions for the velocity components. The rate of mass transfer in the laminar microstructure can be obtained by solving the equation... 

Show that an approximate solution can be obtained for 2(x, Y) in the form of a series expansion in x with coefficients that are functions of the Blasius similarity variable rj. Determine the governing DEs and boundary conditions for the first tow terms in this expansion. It can be shown that... 

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