Angular velocity of a sphere in a simple shear at small Reynolds number

Abstract

We analyse the angular velocity of a small neutrally buoyant spheroid log rolling in a simple shear. When the effect of fluid inertia is negligible the angular velocity ω equals half the fluid vorticity. We compute by singular perturbation theory how weak fluid inertia reduces the angular velocity in an unbounded shear, and how this reduction depends upon the shape of the spheroid (on its aspect ratio). In addition we determine the angular velocity by direct numerical simulations. The results are in excellent agreement with the theory at small but not too small values of the shear Reynolds number, for all aspect ratios considered. For the special case of a sphere we find ω/s = −1/2 + 0.0540 Re 3/2 s where s is the shear rate and Res is the shear Reynolds number. This result differs from that derived by Lin et al. [J. Fluid Mech. 44 (1970) 1] who obtained a numerical coefficient roughly three times larger.

Angular Dynamics of a Small Particle in Turbulence

“…flows such as a simple shear [24,25], and for planar flows because vorticity is orthogonal to the flow plane.…”

Hydrodynamic force on a small squirmer moving with a time-dependent velocity at small Reynolds numbers

“…However, we found that the two expressions are numerically equivalent for all centre-of-mass motions that we examined: sudden start (Sano 1981), linearly increasing velocity ẋ = v 0 t/t 0 , and sinusoidally varying centre-of-mass velocity ẋ = v 0 sin(ω 0 t), with coefficients v 0 , t 0 and ω 0 . It is quite common that the two methods, reciprocal theorem and asymptotic matching, yield expressions for the hydrodynamic force and torque that look different but are equivalent (Meibohm et al 2016).…”

“…Following Meibohm et al. (2016), we denote the first term on the right-hand side of (3.5 a ) by . This is the well-known Stokeslet describing the far-field disturbance flow produced by an active particle in the Stokes limit.…”

“…It is quite common that the two methods, reciprocal theorem and asymptotic matching, yield expressions for the hydrodynamic force and torque that look different but are equivalent (Meibohm et al. 2016).…”

“…From (A3), we are led to Transforming back to configuration space gives The term is singular in -space (Meibohm et al. 2016), proportional to . The singular term in the expansion (A6) is determined by evaluating the limit (Childress 1964; Saffman 1965; Meibohm et al.…”