Rubinow-Keller

Saffmann [125] (p 394) compared the Magnus force developed by Rubinow Keller [123] with the shear force given above and showed that unless the rotational speed is much larger than the rate of shear k = dv /dy, and for a freely rotating particle 17 = the lift force due to shear is an order... 

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. 

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... 

Denson provides a quantitative theoretical analysis which adequately accounts for these phenomena. Agreement is quite good, especially in the range Rep < 40. No adjustable parameters appear in the treatment. The unsteady-state analysis depends critically upon the applicability of the Rubinow-Keller theory to the instantaneous particle motion, and the observed agreement is construed by Denson as evidence of its applicability for RCp < 40. (See, however, the remarks in the last paragraph of Section III,E,3.)... 

Upon comparing Eqs. (265) and (267) with Eqs. (254)-(255) using the experimentally measured angular and axial slip velocities, Jeffrey concludes that the radial velocities predicted by the Rubinow-Keller theory are too small by an order of magnitude, except at the larger values of l> where their theory yields results too small by a factor of only about 1.5-3. 

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... 

This differs radically from Eq. (275) although it does predict the correct direction of the lateral motion in the two possible cases, %p>0 and < 0. Even apart from this formal difference is the fact that, to terms of lowest order, the lift force predicted by Saffman is independent of the angular velocity of the particle that is, the lift force would be the same even if the particle were prevented from rotating. On the other hand, the Rubinow-Keller lift force depends critically upon the particle rotation. 

When the conditions required by the inequality (274) are met, the slip-shear Saffman lift force is larger, by an order of magnitude, than the slip-spin Rubinow-Keller lift force for from Eqs. (275) and (276) we find... 

This ratio clearly becomes infinite in the limit Re. - 0. Saffman demonstrates that terms of the Rubinow-Keller type, among others, will appear in the next higher-order approximation to Eq. (273). 

As Rep 0 this ratio becomes infinite, showing that the restriction required by Eq. (274) is indeed met. Thus, at least in the neutrally buoyant case, the Rubinow-Keller theory is wholly inapplicable to Poiseuille flows. It should not be surmised, however, that Saffman s results themselves have any direct application to such flows for Saffman s calculations take no account of either the nonconstancy of the local shear rate or of the presence of boundaries constraining the flow. And either of these effects may result in the appearance of contributions to the lift force more dominant than Saffman s, at least for some ranges of the many independent variables. 

As mentioned previously, Repetti and Leonard (R4a) attempted an explanation of this phenomenon on the basis of a questionable modification of the Rubinow-Keller theory. It is now clear, however, that their proposal cannot be correct. 

With regard to the interpretation of available experimental data, the only outstanding point seemingly in need of further discussion is the observation by Denson (D4b) that the Rubinow-Keller theory agrees well with his... 

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. 

Theoretical attempts to explain lift have concentrated on flow at small but nonzero Re, using matched asymptotic expansions in the manner of Proudman and Pearson for a nonrotating sphere (see Chapter 3). In the absence of shear, Rubinow and Keller (R6) showed that the drag is unchanged by rotation. With... 

Rubinow, S. I. and Keller, J. B. (1961). The Transverse Force on a Spinning Sphere Moving in a Viscous Fluid. J. Fluid Mech., 11,447. 

Rubinow and Keller [123] calculated the flow around a rotating sphere moving in a viscous fluid for small Reynolds numbers. They determined the drag, torque, and lift force (Magnus) on the sphere to O(Rep). The results were ... 

The measured lift force on a solid particle compared well with the model of Saffman (5.64) when the shear Reynolds number was small, and with the model of Rubinow and Keller (5.60) when the shear Reynolds number was large. The shear Reynolds number was defined as ... 

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... 

G. K. Batchelor, Slender-body theory for particles of arbitrary cross-section in Stokes flow, J. Fluid Mech. 44, 419-40 (1970) R. G. Cox, The motion of long slender bodies in a viscous fluid, Part 1, General theory, J. Fluid Mech. 44, 791-810, (1970) Part 2, Shear Flow, J. Fluid Mech. 45, 625-657 (1971) J. B. Keller and S. T. Rubinow, Slender-body theory for slow viscous flow, J. Fluid Mech. 75, 705-14 (1976) R. E. Johnson, An improved slender-body theory for Stokes flow, J. Fluid Mech. 99, 411-31 (1980) A. Sellier, Stokes flow past a slender particle, Proc. R. Soc. London Ser. A 455, 2975-3002 (1999). 

See, for example, the paradox uncovered by Garstang (Gla) and subsequently resolved by Rubinow and Keller (R7). 

Using these asymptotic matching techniques, Rubinow and Keller (R7) calculated, to 0(R), the force and torque on a translating sphere simultaneously spinning about any axis through its center in a fluid at rest at infinity. 

Rubinow and Keller empirically suggest multiplying the above by the factor - (P - to bring it at least into qualitative agreement with experiment, but this suggestion has little rational basis to commend it. 

In a sense, the case where the flow is bounded is conceptually easier to treat than is the comparable unbounded flow for when the flow is bounded the disturbance created by the sphere dies away exponentially with axial distance (Tla, S16) rather than inversely with some power of the distance, as is true for the unbounded case. For this reason, the outer boundary conditions satisfied by the lower-order terms of the inner expansion can be determined without having to apply the matching principle except in a trivial sense that is, the outer boundary conditions satisfied by the lower-order terms of the inner expansion are identical to the physical outer boundary conditions. Since the force and torque on the sphere can be determined from the inner expansion alone, it is therefore not necessary to compute any terms in the outer expansion in order to obtain the leading terms in the force and torque. Thus, the analysis differs greatly from those of Rubinow and Keller (R7) and Saffman (Sla). 

Flaheity, J. E., Keller, J. B., and Rubinow, S. I., Post buckling behavior of elastic tubes and rings with opposite sides in contact, SIAM J. Appl. Math., 23 446—455,1972. 

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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