Displacement theorems for spherical solutions of the linear Navier-Stokes equations. (English) Zbl 0665.76033
The solutions in spherical coordinates of the linear Navier-Stokes equations for steady flow in an incompressible viscous fluid were found long ago by Lamb. For a special choice of coordinates, the desired transformation was found by R. Schmitz and the first author [Physica A 113, 103-116 (1982)]. Here we present the general transformation for an arbitrary direction of the vector connecting the two centers. Regarded as a function of this vector, most of the transformation coefficients satisfy Laplace’s equation. Our derivation is based on addition theorems for spherical wave solutions of the vector Helmholtz equation, as presented by the authors [J. Math. Phys. 28, 836- 839 (1987)]. The derivation is straightforward but lengthy and we present only the final results.
Keywords:
linear Navier-Stokes equations; incompressible viscous fluid; Laplace’s equation; Helmholtz equationReferences:
| [1] | DOI: 10.1016/0378-4371(82)90008-5 · doi:10.1016/0378-4371(82)90008-5 |
| [2] | DOI: 10.1063/1.527572 · Zbl 0663.35071 · doi:10.1063/1.527572 |
| [3] | Cichocki B., PhysicoChem. Hyd. 10 pp 383– (1988) |
| [4] | DOI: 10.1016/0378-4371(86)90043-9 · Zbl 0664.76033 · doi:10.1016/0378-4371(86)90043-9 |
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