Force and torque

As before in Section F.1.2.2 diagonal components of the stress tensor of a dipolar fluid can be obtained from the relation 

Based on the same traasformation to unit-rube coordinates employed before [see Eq. (F.57)j. we realize that the dependence on. s.y is buried in the argument of the functions B and C [see Eqs. (6.28a)) and in the factors fii j vij + n) as far as l/ is concerned. Differentiating these terms with respect to Sy, it is easy to verify that terms of the form 

Tbrning to the Foiuier-space contribution next, we immediately see that 

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The solids forwarding angle q> is related to rate by Eq. 5.7, and the force and torque balances provide the relationship between and the pressure gradient and q> using Eq. 5.23. Here, the equation is written in differential form so that integration can be performed numerically in the z direction using variable physical property data. 

Equations 12 through 14 may be derived from thermodynamic and mechanical expressions for the transverse force and torque acting on a surface which intersects the interfacial region ( ). Moments of an isotropic pressure force and a surface tension acting at R are... 

Fig. 9.11. Deflection of a tube scanner in the unipolar arrangement. (A) A voltage i.s applied to only one quadrant of the outer metal coating. All other quadrants and the inner metal coating are grounded. (B) At equilibrium, a distribution of stress and strain is established such that the total force and torque at each cross section i.s zero. This condition determines the y displacement. (Reproduced from Chen, 1992, with permission.)...

Before discussing other results it is informative to first consider some correlation and memory functions obtained from a few simple models of rotational and translational motion in liquids. One might expect a fluid molecule to behave in some respects like a Brownian particle. That is, its actual motion is very erratic due to the rapidly varying forces and torques that other molecules exert on it. To a first approximation its motion might then be governed by the Langevin equations for a Brownian particle 61... 

The phenomenological Langevin Eqs. (227) and (228) are only applicable to a very restricted class of physical processes. In particular, they are only valid when the stochastic forces and torques have infinitely short correlation times, i.e., their autocorrelation functions are proportional to Dirac delta functions. As was shown in the previous section, these restrictions can be removed by a suitable generalization of these Langevin equations. As we saw in the particular case of the velocity, the modified Langevin equation is... 

The important point to note here is that the 2nd moment of Ky(t) depends on the 2nd and 4th moments of y(t). The 2nd moments of each of the three previously mentioned autocorrelation functions may be calculated from ensemble averages of appropriate functions of the positions, velocities, and accelerations created in the dynamics calculations. Likewise, the 4th moment of the dipolar autocorrelation function may also be calculated in this manner. However the 4th moments of the velocity and angular momentum correlation functions depend on the derivative with respect to time of the force and torque acting on a molecule and, hence, cannot be evaluated directly from the primary dynamics information. Therefore, these moments must be calculated in another manner before Eq. (B.3) may be used. 

Boundary element methods can be used for particulate flows where direct1 formulations can be used. The surface tractions on the solids are integrated to compute the hydrodynamic force and torque on those particles, which for suspended particles must be zero. 

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. 

Next we proceed with the force and torque balances. Since pressure builds up in the down-channel direction, the force and torque balances are made on a differential increment in the down-channel direction this is illustrated in Fig. 9.29, where the various forces acting on the element are also depicted. These forces can be expressed in terms of the coefficients of friction, local geometry, and the differential pressure increment, which compensate for the other forces and torques. For an isotropic stress distribution, these are... 

The formulae given above can be illustrated by numerical estimates of the magnitude of the renormalization of the QM parameters and changes in the DMM forces and torques appearing in the vicinity of an sjA carbon atom located on the intersubsystem frontier. The changes in the QM one-center Hamiltonian parameters due to elongation of one of the MM bonds incident to the frontier carbon atom are ... 

On the other hand the changes in the one- and two-electron densities on the HO centered on the frontier atom lead to the following forces and torques acting on the MM atoms immediately bound to the frontier carbon atom ... 

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. 

T, which are expressed in terms of the translational and angular velocities, U and n. Since the outer surface of the double layer encloses a neutral body (charged interface plus diffuse ions), the particle is force and torque free. Hence, electrophoretic velocities U and Q can be determined. For spherical particles, the electrophoretic velocities are... 

Denote by F( ext and Tf ext the external force and torque (about 0.) acting on particle i. Neglect of particle inertia then leads to the equation... 

This generic formula permits averages to be calculated for any pertinent suspension property /. When time-independent external forces F(e) and torques N(e) act on each of the suspended particles, Eq. (7.14), together with the net force- and torque-free conditions characterizing the neutrally buoyant... 

In the absence of inertia and Brownian motion, the quasistatic dynamics of N rigid suspended particles contained within a unit cell is governed by the composite force and torque balance... 

When the thermostat is applied the momenta and the angular velocities must be peculiar with respect to the linear and angular streaming velocities otherwise the thermostat will exert forces and torques on the system. The streaming angular velocity is sometimes orientation dependent which makes it even more difficult to apply the thermostat correctly [15,16]. In order to avoid these problems one can apply the ordinary Euler equations in angular space and limit the thermostat to the translational degrees of freedom. In this case the square of the curvature becomes... 

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