# Stokes Problem

## Stokes’ Problem

The problem of determining the external gravitation field of a planet from the planet’s external equipoten-tial surface S, the mass within S, and the angular velocity of rotation about some axis is sometimes referred to as Stokes’ problem. G. G. Stokes showed that the problem can be solved and provided an approximate solution for an oblate spheroid with a relative error of the order of the square of the spheroid’s flattening. He treated the problem as a first boundary value problem. An exact solution of Stokes’ problem for an ellipsoid was put forth by the Italian mathematician P. Pizzetti and by M. S. Molodenskii.

To an arbitrary shape of S there corresponds the boundary condition

and the equation for φ

under the condition

∫ φ ds = 0

Here, ζ is the height of S above the reference ellipsoid S0 containing the given mass, T is the disturbing potential defined by the equation

φ is the density of the simple layer at S, W0 is the gravitational potential at the point where ζ = 0 on the intersection of S and S0, U0 is the corresponding potential on S0, γ is the gravity in the field of the ellipsoid, r is the distance between the element ds and the point on S with height ζ, and r0 is the distance between ds and the point where ζ = 0. The axes of rotation of S and S0 coincide. The equation for φ can be replaced by a system of linear algebraic equations. The determination of φ solves the problem known as Stokes’ problem.

The solution presented above is also applicable to the case where S is a nonequipotential surface and ζ is the height of the quasi-geoid (seeGEOID).

### REFERENCES

Molodenskii, M. S., V. F. Eremeev, and M. I. Iurkina. Metody izucheniia vneshnego gravitatsionnogo polia i figury Zemli. (Tr. Tsentr. n.-i. in-ta geodezii, aeros’emki i kartografii, fasc. 131.) Moscow, 1960.
Stokes, G. G. “On Attractions and on Clairaut’s Theorem.” Cambridge and Dublin Mathematical Journal, 1849, vol. 4.

M. I. IURKINA

References in periodicals archive ?
CHABARD, Some fast 3D finite element solvers for the generalized Stokes problem, Internat.
2] of a covolume method for the generalized Stokes problem are obtained under higher regularity assumption
SOUNDALGEKAR, Free convection effects on stokes problem for an infinite vertical plate, Journal of Heat transfer, 99 (1977), 449-501.
The coextrusion problem can be written as a Stokes problem (Eqs.
On a Parallel Implementation of the BDDC Method and Its Application to the Stokes Problem
We consider the Stokes problem stemming from the simulation of a steady horizontal flow in a channel driven by a pressure difference between the two ends [14, Example 5.
BOCHEV, A stabilized finite element method for the Stokes problem based on polynomial pressure projections, Internat.
With these results in mind, one can easily prove the unique solvability of the Stokes problem, as for example done in [7].
2 the results for MINRES preconditioned with the norm in the case of the Stokes problem.
In this paper, we focus on two types of saddle point problems, namely the symmetric Stokes problem discretized with a stabilized finite element method and the more general, nonsymmetric Jacobian system arising in the Newton-Krylov-Schwarz (NKS) algorithm for solving nonlinear incompressible Navier-Stokes equations.
MADAY, Generalized inf-sup conditions for Chebyshev spectral approximation of the Stokes problem, SIAM J.
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