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 Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechani ...

. It can be shown that any

Stokes flow Stokes flow (named after George Gabriel Stokes), also named creeping flow or creeping motion,Kim, S. & Karrila, S. J. (2005) ''Microhydrodynamics: Principles and Selected Applications'', Dover. . is a type of fluid flow where advection, advec ...

with body force

\mathbf=0

can be written in the form: :

\mathbf =  \left \nabla ( \mathbf \cdot \mathbf + \chi) - 2 \mathbf \right /math>
: p = \nabla \cdot \mathbf where \mathbf is a

harmonic In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st har ...

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.

Further reading