Rubinow-Keller equation

Most experimental data obtained to date have been interpreted on the basis of the Rubinow-Keller equation (216) for the lift force on a spinning, translating sphere in an unbounded fluid at rest (or in uniform flow) at infinity. The spin is assumed to be given by Eq. (253), whereas the velocity U appearing in Eq. (216) has usually been interpreted as the axial slip velocity. This gives rise to a lift force... 

As observed repeatedly, most investigators concerned with the radial migration problem have attempted to analyze their data on the basis of a rather broad interpretation of the Rubinow-Keller equation. In the neutrally buoyant case, i.e., UaMV - Q, this interpretation predicts [cf. Eqs. (256) and (247)] a radial velocity... 

The solution of the Rubinow-Keller problem had previously been attempted by Garstang (Gla) on the basis of the Oseen equations. His result for the lift force is larger than (216) by a factor of 4/3. But as Garstang himself pointed out, his result was not unequivocal. Rather, different results were obtained according as the integration of the momentum flux was carried out at the surface of the sphere or at infinity. Garstang s paradox is clearly due to the fact that the term U-Vv does not represent a uniformly valid approximation of the inertial term v Vv throughout all portions of the fluid, at least not to the first order in R. 

Spin-orbit coupling problems are of a genuine quantum nature since a priori spin is a quantity that only occurs in quantum mechanics. However, already Thomas (Thomas, 1927) had introduced a classical model for spin precession. Later, Rubinow and Keller (Rubinow and Keller, 1963) derived the Thomas precession from a WKB-like approach to the Dirac equation. They found that although the spin motion only occurs in the first semiclassical correction to the relativistic classical electron motion, it can be expressed in merely classical terms. 

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

Rizk MA, Elghobashi SE (1989) A Two-Equation Turbulence Model for Dispersed Dilute Confined Two-Phase Flows. Int J multiphase Flow 15(1) 119-133 Rubinow SI, Keller JB (1961) The transverse force on a spinning sphere moving in a viscous fluid. J Fluid Mech 11 447-459... 

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