We will first cite the following problem; a liquid is contained in a vessel which it completely fills; various moving solid bodies are immersed in this liquid; we know the movements of these bodies and we suppose that there is a function of the speeds; we ask what is the movement of the liquid.
This is the problem of the pulsating spheres of M. Bjerknes (hydrodynamic imitation of electrical phenomena).
From the analytical point of view, it is a question of finding a function V which satisfies the Laplace equation inside a certain space and such that on the surface which limits this space the derivative dV/dn has given values.
I recall what is the meaning of this notation dV/dn which will be made a frequent use in the sequel. Consider any surface element; and α, β, γ the three direction cosines of the normal to this element; we have:
dV/dn = α dV/dx + β dV/dy + γ dV/dz
Thus in the hydrodynamic problem, we find the same differential equation as in the thermal and electrical problems; only the boundary conditions differ. It will still be the same in the problem of magnetic induction.
Source: H. Poincaré, Sur les Equations aux Dérivées Partielles de la Physique Mathématique, American Journal of Mathematics, Mar., 1890, Vol. 12, No. 3 (Mar., 1890), pp. 211-294. Translation by Nicolae Sfetcu. © 2022 Nicolae Sfetcu