Introduction to Mathematical Fluid Dynamics by Richard E. Meyer

Author:Richard E. Meyer
Language: eng
Format: epub
ISBN: 9780486151403
Publisher: INscribe Digital
Published: 2014-01-27T05:00:00+00:00

The fluid and bodies are therefore capable only of a rigid translation; none of the bodies can even be made to rotate! For a proof, see Appendix 19.

The situation is quite different for a viscous fluid. Since p = p(x), but u = u(r), (19.8) implies

dp/dx = const = µA,

say, and u = – Ar2/4 + c1 log r + c2, and (19.7) and (19.6) show the constants of integration to be c1 = 0 and c2 = Aa2/4, so that Poiseuille’s solution,

is obtained, where dr is the total mass flow rate. There is therefore a unique steady flow independent of x and θ and of inlet conditions, which may be approached with increasing distance from the pipe inlet.

It is worth noting that Poiseuille’s flow is an exact solution of the nonlinear partial differential system (4.5), (19.3), sometimes referred to as Navier-Stokes equations (the full system of those equations will be found in Section 38). The number of known exact solutions is not very large, although the term “exact” is conventionally extended to cover not only explicit solutions, like Poiseuille’s and those of Problems 19.1 to 19.3, but also any solution obtainable by (numerical) integration of a system of ordinary differential equations. The exact solutions are therefore particular solutions distinguished by symmetries of one kind or another; surveys of such solutions are found in [21] and [22].

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