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  • Print publication year: 2007
  • Online publication date: June 2012

4 - An Introduction to Asymptotic Approximations

Summary

Although the full Navier–Stokes equations are nonlinear, we have studied a number of problems in Chap. 3 in which the flow was either unidirectional so that the nonlinear terms u · ∇u were identically equal to zero or else appeared only in an equation for the cross-stream pressure gradient, which was decoupled from the primary linear flow equation, as in the 1D analog of circular Couette flow. This class of flow problems is unusual in the sense that exact solutions could be obtained by use of standard methods of analysis for linear PDEs. In virtually all circumstances besides the special class of flows described in Chap. 3, we must utilize the original, nonlinear Navier–Stokes equations. In such cases, the analytic methods of the preceding chapter do not apply because they rely explicitly on the so-called superposition principle, according to which a sum of solutions of linear equations is still a solution. In fact, no generally applicable analytic method exists for the exact solution of nonlinear PDEs.

The question then is whether methods exist to achieve approximate solutions for such problems. In fluid mechanics and in convective transport problems there are three possible approaches to obtaining approximate results from the nonlinear Navier–Stokes equations and boundary conditions.

First, we may discretize the DEs and boundary conditions, using such formalisms as finite-difference, finite-element, or related approximations, and thus convert them to a large but finite set of nonlinear algebraic equations that is suitable for attack by means of numerical (or computational) methods.

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