Torque on a particle

Thus, combining (7-20) and (7-21), as allowed by the linearity of the governing equations, we find that the force and torque on a particle that moves with arbitrary velocities U and 17 through an otherwise quiescent fluid is... 

Problem 8-10. Symmetry of the Grand Resistance Tensor. Use the reciprocal theorem to show that the grand resistance tensor is symmetric. The grand resistance tensor relates the hydrodynamic force/torque on a particle to its velocity/angular velocity ... 

The time-averaged torque on a particle is given by Jones [22] ... 

According to Eq. (20), the multipolar DEP force and ROT torque on a particle can be determined as long as the electric field distribution in the system is known. 

Another parameter of interest for comparison with experimental measurements is the couple or torque on a particle arising from rotational motion. The torque T about the center of rotation of a body with angular velocity to can be written in Cartesian tensor form as... 

A review of early works on hydrodynamic interaction between two solid spherical particles is contained in [13]. For the most part, these works focus on derivation of forces and torques acting on particles placed relatively far apart, forces and torques on a particle moving perpendicular or parallel to a flat surface, and forces and torques on a particle moving relative to another particle along, or perpendicular to, the line connecting the particle centers. The more general case of translation and rotation of two rigid particles was considered in [40, 41]. Hydrodynamic interaction of two drops with a proper account of the mobility of their surfaces and the internal circulation is analyzed in [39, 42-50]. 

In brief, electrorotation involves an induced torque on a particle that is subjected to a rotating electric field (Fig. 4a). The electric field induces a dipole in the particle, and as the field rotates, the dipole attempts to rotate with the electric field. As the dipole lags behind the rotating field, it attempts to reorient itself and a torque is induced on the particle. This dipole orientation lag is due to its relaxation time. The general equation for the electrorotational torque induced on a homogenous spherical particle is... 

The fiber is suspended in the liquid, which means that due to small time scales given by the pure viscous nature of the flow, the hydrodynamic force and torque on the particle are approximately zero [26,51]. Numerically, this means that the velocity and traction fields on the particle are unknown, which differs from the previous examples where the velocity field was fixed and the integral equations were reduced to a system of linear equations in which velocities or tractions were unknown, depending on the boundary conditions of the problem. Although computationally expensive, direct integral formulations are an effective way to find the velocity and traction fields for suspended particles using a simple iterative procedure. Here, the initial tractions are assumed and then corrected, until the hydrodynamic force and torque are zero. 

Consider a single, freely suspended axisymmetric particle in a homogeneous shear flow held of an incompressible Newtonian liquid. The free suspension condition implies that the net instantaneous force and torque on the particle vanish. There is, however, a finite net force along the axis that one half of the particle exerts on the other, as shown schematically in Fig. 7.25. 

Until fairly recently, the theories described in Secs. II and III for particle-surface interactions could not be verified by direct measurement, although plate-plate interactions could be studied by using the surface forces apparatus (SFA) [61,62]. However, in the past decade two techniques have been developed that specifically allow one to examine particles near surfaces, those being total internal reflection microscopy (TIRM) and an adapted version of atomic force microscopy (AFM). These two methods are, in a sense, complementary. In TIRM, one measures the position of a force-and torque-free, colloidal particle approximately 7-15 fim in dimension as it interacts with a nearby surface. In the AFM method, a small (3.5-10 jam) sphere is attached to the cantilever tip of an atomic force microscope, and when the tip is placed near a surface, the force measured is exactly the particle-surface interaction force. Hence, in TIRM one measures the position of a force-free particle, while in AFM one measures the force on a particle held at a fixed position. 

Finally, linearity of the creeping-flow equations and boundary conditions allows a great a priori simplification in calculations of the force or torque on a body of fixed shape that moves in a Newtonian fluid. To illustrate this assertion, we consider a solid particle of arbitrary shape moving with translational velocity U(t) and angular velocity il(t) through an unbounded, quiescent viscous fluid in the creeping-flow limit Re 1 and Re/S < 1. The problem of calculating the force or torque on the particle requires a solution of... 

Equations (167) and (168) express the fact that the force and torque on each particle are linear vector functionals of the slip velocities of all the particles. It should be clearly noted that these operators are calculable solely from a knowledge of the intrinsic translational and rotational solutions of Stokes equations resulting from motion of the yth particle when all the other particles, as well as the fluid at inflnity, are at rest. In the case where u = 0, Eqs. (167) and (168) reduce to (148) and (149), respectively. As yet, there exists no system for which these operators are known. 

If the fluid is macroscopically at rest, convective fluxes can arise only from the action of external forces and torques on the particles. The particle velocities can then be obtained from the given forces and torques by solving Eqs. (38) and (39) for Uo and a, bearing in mind that the hydrodynamic and external forces acting on a particle are in equilibrium. Equivalently, these velocities may be obtained by inverting the matrix equation (49),... 

Up to now we were limited to considering the behavior of isolated particles or hydrodynamic interactions of two particles (pair interactions). However, in concentrated suspensions multi-particle interactions are possible, and one cannot neglect them. The problem of determining force and torque acting on a particle in suspension in the presence of other particles is very difficult and so far has not been completely solved. However, in the last years, a significant progress [13] in... 

Electrokinetic Motion of Polarizable Particles, Fig. 2 (a) Mechanism for ICEP torque on a rodlike, polarizable particle in a uniform electric field, which... 

Moreover, the local variation of these quantities along the surface results in a hydrodynamic torque on the particle ... 

This frequency, a>, dependent factor, K(ct)), dynamically reflects the polarizability of a particle (subscript p) in a conductive medium (subscript m). The Clausius-Mosotti factor is a ratio of complex permittivities, e, of the form e = s — ia/co, where co is the frequency, e is the dielectric constant, and a is the electrical conductivity of the medium. As can be seen, the complex K(o)) factor has an imaginary corr5)onent, which is out of phase with the applied electric field while the real component is in phase [2]. The imaginary component manifests itself as a torque on the particle and is detected in electrorotation or impendence measurements. However, the traditional dielectrophoretic force is dependent on the in-phase, or real component of the Clausius-Mosotti factor. In order to examine the special case when no frequency component is involved, DC-DEP can be estimated as the residual of this factor when frequency goes to zero [3] ... 

Consider now a colloid particle of radius a immersed in one of these macroscopic flows. The particle influences the flow locally, which produce a microscopic disturbance. This leads to the appearance of force and torque on the particle, often called the liquid drag or driving... 

An analysis of particle motion parallel to a planar interface in a qmescent fluid is more complicated because the translational motion induces a coupled rotational motion, which produces a hydrodynamic torque on the particle. These effects have been calculated by Goren and O Neill [99] using bispherical coordinates. It was shown that particle velocity can be expressed as... 

Besides the flow-induced forces and torques acting on a particle, F and T, external forces can also be considered in the equation of motion. Under consideration of particle acceleration due to gravity, Eq. (2.13) can be modified in this way ... 

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