Abstract

The analysis developed in the previous chapter, regarding the vorticity and stream function equations under no-slip conditions, has revealed that an uncoupled formulation of the linear or linearized problem is possible, provided that the vorticity variable is supplemented by conditions of an integral character. These conditions have been found to admit a very simple geometrical interpretation: they impose the orthogonality of the vorticity field with respect to the linear space of the functions which are harmonic in the considered domain, or, at least, fix the orthogonal projection of the vorticity with respect to such a linear space. This geometrical result allows to recast the Navier-Stokes equations for two-dimensional flows in terms of two second-order scalar equations, one parabolic and the other elliptic, the vorticity equation being completely independent in linear situations. It seems therefore worthwhile to investigate whether these abstract geometrical ideas can be generalized to three dimensions.KeywordsVector FieldIntegral ConditionTangential ComponentDivergence TheoremVorticity FieldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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