Papkovich-Neuber solution

The Papkovich–Neuber solution is a technique for generating analytic solutions to the Newtonian incompressible Stokes equations, though it was originally developed to solve the equations of linear elasticity.

It can be shown that any Stokes flow with body force $mathbf\{f\}=0$ can be written in the form:

:$mathbf\{u\}\; =\; \{1over\{2\; mu\; abla\; (\; mathbf\{x\}\; cdot\; mathbf\{Phi\}\; +\; chi)\; -\; 2\; mathbf\{Phi\}$:$p\; =\; abla\; cdot\; mathbf\{Phi\}$

where $mathbf\{Phi\}$ is a harmonic vector potential and $chi$ is a harmonic scalar potential. The properties and ease of construction of harmonic functions makes the Papkovich-Neuber solution a powerful technique for solving the Stokes Equations in a variety of domains.

References

* Citation
last = Neuber
first = H.
year = 1934
title = Ein neuer Ansatz zur Lösung räumblicher Probleme der Elastizitätstheorie
periodical = Z. Angew. Math. Mech.
volume = 14
pages = 203–212.

* Citation
last = Papkovish
first = P. F.
year = 1932
title = Solution Générale des équations differentielles fondamentales d'élasticité exprimée par trois fonctions harmoniques
periodical = Compt. Rend. Acad. Sci. Paris
volume = 195
pages = 513–515.