ABSTRACT
The rotational-diffusion constant may be obtained by means of the angular root-mean-square displacement in a long time period Δt
(10)
Spherical and nonspherical particles isotropically dispersed do not rotate freely according to the average angular velocity, as explained above. They rotate under an external couple caused by the friction between the particles and the fluid. In this case, the angular momentum balance equation and the linear momentum balance equation have to be taken into account. The former is:
(11)
where is the vorticity of the surrounding fluid and ς is the vortex viscosity. It is clear that the friction between the particle and the liquid brings about the relaxation mechanism that leads to the internal mechanical equilibrium of the suspension. The equilibrium is attained when the particle rotation rate follows the vorticity of the fluid flow, that is, when
If the flow velocity is very high, the effects of inertia produce a noninstantaneous coupling between particle internal rotation and local vorticity.[60] Assuming that the relaxation of internal rotation or spin occurs in a time scale shorter than the hydrodynamic one, Eq. 5 decouples from Eq. 11 and the velocity field can be solved independently of the rotation dynamics but according to the boundary conditions. However, the spin still feels the effects of the flow field. Therefore, supposing as constant in the rotation time scale and ω(t=0)=ωav, the spin evolution can be given by the following expression
(12)
where τs is the spin-relaxation time given by
(13)
The identity in Eq. 13 follows from the relations[61] and I=2/5(ρsvd2), where v is the particle volume, ρs is the density of the particle, and d is its characteristic length. τs is of the order of 10 ps for particles of 0.01μm. If the particles are of 1μm, τs increases four orders of magnitude giving the possibility of a coupling between the particle spin and the velocity field dynamics. In this case, the balance of momentum Eq. 5 is modified by the inclusion of a new term arising from the antisymmetric contribution of the viscous stress tensor Qa,[62] namely,
(14)
where
(15)
where is the three subindexes alternating tensor. By substituting Eq. 15 into Eq. 14, the modified linear momentum equation for an incompressible fluid is obtained,
(16)
This result differs from the usual Navier-Stokes equation of hydrodynamics by the presence of the last term, which is related to the coupling of the spin with the flow via the antisymmetric stress tensor. [63]
Spin Velocity Diffusion In the subsection “Rotational Fick’s Diffusion,” the evolution of the angular position of particles was discussed as produced by the mechanism of Brownian motion. Now, the diffusion of the spin velocity is considered. This diffusion is produced by the hydrodynamic interaction via the vorticity induced by the particle rotation.
