By L. I. Sedov, J. R. M. Radok
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Extra resources for A course in continuum mechanics Vol. I Basic equations and analytical techniques
Extending the above ideas to higher order approximations we replace the Stokes expansion by an intermediate expansion th q*~ E«/(Re)u, (21) We require the n partial sum to be valid to order € n(Re) in some convex domain ord Re ^ ord / < ord /„ where the limit corresponding to /„(Re) is sufficiently slow so that the domain just described overlaps with the corresponding domain of validity of the Oseen expansion. Thus, for η = 1, lim, (g» + «*•> ~ ("° + = ο (22) uniformly in some convex overlap domain.
V , 1954 MECHANICS A N D SINGULAR PERTURBATIONS T h e R o l e of C o o r d i n a t e S y s t e m s i n B o u n d a r y - L a y e r T h e o r y 131 and the optimal ^-coordinates may be taken as : η = IM ζ , (41b) where f - Vz. This gives a conformai set of coordinates, which may be called streamline-parabolic. The optimal coordinates in Examples 1 and 2 are special 2 cases of this system. We have χζ = ξ , ~γζ= 2ξη, and so the optimal boundary-layer approxi mation is: (42) ψ =ϋξί (2η), ζ α where / ais the same function as that in equation (39).
Independent variables. Main correlation theorem. Consider now two boundarylayer approximations to the same exact solution, one based on the system of coordinates ζ = (ξ, η) and the other one on another system χ = (ρ, cr). Here η and a respectively are assumed to be zero at the wall. The two flow fields may be represented by the stream functions ψζ and ψχrespectively. Their relation is then described by the following correlation theorem : THEOREM 1. *• (12b) ( 1 2 c > The theorem states that the result of applying 1πτιχto ψ is identical with that of first applying lim^ and then l i m r The evaluations of the limits (12 b) and (12 c) are easily shown to be correct by considering ξ and η as functions of σ for a fixed ρ, and using the fact that η and a vanish at the wall.