Particle of an Arbitrary Shape

Asymptotic formulas for the Sherwood number and diffusion flux. For low 

Peclet numbers, the problem of mass exchange between a particle of arbitrary shape and a uniform translational flow were studied by the method of matched asymptotic expansions in [62]. The following expression was obtained for the mean Sherwood number up to first-order infinitesimals with respect to Pe  

Formula (4.4.21) is quite general and holds for solid particles, drops, and bubbles of arbitrary shape in a uniform translational flow at any Re as Pe —F 0. 

It gives a good approximation of the Sherwood number ratio for PeM 5. In the special case of a spherical particle, (4.4.21) coincides with (4.4.19). For nonspherical particles, in (4.4.21) one must use the values of II from Table 4.2. 

For a particle of arbitrary shape in a translational flow, the first three terms of the asymptotic expansion of the dimensionless total diffusion flux as Pe — 0 have the form [62] 

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The above construction and accordingly the conclusion, obviously hold for an arbitrary electrolyte in 1Zn. 772 and symmetry of the electrolyte, resulting in the sinh p nonlinearity instead of a general sum of exponents, are not essential for the derivation above and were assumed for the sake of brevity of presentation only. The appropriate result—force saturation—is most likely true for particles of an arbitrary shape, although the corresponding generalization still has to be carried out. Another relevant task would be an actual computation of the limiting repulsion force as a function of, say, particle separation. 

Proof of boundedness of the force of interaction between two charged particles of an arbitrary shape in H3, held at a given distance from each other in an electrolyte solution, upon an infinite increase of the particle s charge. (It was shown in 2.2 that the repulsion force between parallel symmetrically charged cylinders saturates upon an infinite increase of the particle s charge. This is also true for infinite parallel charged plane interaction [9]. The appropriate result is expected to be true for particles of an arbitrary shape.)... 

A particle of an arbitrary shape moving in an infinite fluid that is at rest at infinity is subject to the action of the hydrodynamic force and angular momentum due to its translational motion and rotation, respectively [179] ... 

The steady-state settling rate of a particle of an arbitrary shape (for Newtonian regime of a flow at high Reynolds numbers) can be obtained by the formula [94]... 

The following general statement was proved in [63] for the case of a uniform translational Stokes flow (Re -4 0) or a potential flow past a particle of an arbitrary shape the mean Sherwood number remains the same if the flow direction is changed to the opposite. 

For smooth particles of an arbitrary shape in ideal fluid (this model is used, say, to describe heat exchange between particles and liquid metals at Pr -C 1 and Re 3> 1) and in the absence of regions with closed streamlines, the mean Nusselt number can be calculated by the formula... 

In the case a = b, the value of R for the sphere ensues from (8.10). The Brownian motion of an ellipsoidal particle or a particle of an arbitrary shape has a stochastic character. Accordingly, the orientation of the partide in space is also stochastic. Therefore, for such a motion, the concepts of average coefficients of resistance, friction, and mobility are introduced ... 

For a particle of an arbitrary shape having a scattering density difference of Ap(r), the pair-distance distribution function (PDDF) is given by... 

Fig. 3.7. Systems described by interacting beads, (a) Polymer (the freely jointed model), (b) Solid particle of an arbitrary shape. In s model, it is convenient to assume that the inside of the particle is filled with the same fluid as that outside,...

Formula (5.6.4) is valid for an arbitrary laminar flow without closed streamlines for particles and drops of an arbitrary shape. The quantity Sh(l,Pe) corresponds to the asymptotic solution of the linear problem (5.6.1) at Pe > 1. For spherical particles, drops, and bubbles in a translational or linear straining shear flow, the values of Sh(l, Pe) are shown in the fourth column in Table 4.7. 

Induced-Charge Electrokinetic Motion of Particle in a MicroChannel, Fig. 1 Charging process of an arbitrary shape conducting particle under a unifram applied electric field (a) initial electric field passing through the... 

It can be shown that equation (3) is not only valid for the Gaussian chain but also for particles with an arbitrary shape as long as q Rg) 1. Considering the interparticle interference between the scattered lights, Debye (10) showed in 1947 that the concentration dependence can be virial expanded as a power series in the concentration, the combination of which with equations (2) and (3) leads to... 

The Transition Probability. Suppose we have a Brownian particle located at an initial instant of time at the point xo, which corresponds to initial delta-shaped probability distribution. It is necessary to find the probability Qc,d(t,xo) = Q(t,xo) of transition of the Brownian particle from the point c 0 Q(t,xo) = W(x, t) dx + Jrf+ X W(x, t) dx. The considered transition probability Q(t,xo) is different from the well-known probability to pass an absorbing boundary. Here we suppose that c and d are arbitrary chosen points of an arbitrary potential profile (x), and boundary conditions at these points may be arbitrary W(c, t) > 0, W(d, t) > 0. 

The sum rule (4.81) for extinction was first obtained by Purcell (1969) in a paper which we belive has not received the attention it deserves. Our path to this sum rule is different from that of Purcell s but we obtain essentially the same results. Purcell did not restrict himself to spherical particles but considered the more general case of spheroids. Regardless of the shape of the particle, however, it is plausible on physical grounds that integrated extinction should be proportional to the volume of an arbitrary particle, where the proportionality factor depends on its shape and static dielectric function. 

This means that if the half time is calculated based on the unit of volume to external surface area, the non-dimensional times defined are closer for all three shapes of particle. The small difference is attributed to the curvature effect during the course of adsorption, which can not be accounted for by the simple argument of volume to surface area. Eq. (9.2-37) is useful to calculate the half time of solid having an arbitrary shape, that is by simply measuring volume and external surface area of the solid object the half time can be estimated from eq. (9.2-37). 

For a spherical particle, the diameter is taken as the size. However, the size of an irregularly shaped particle is a rather uncertain quantity. We therefore need to define what the particle size represents. One simple definition of the size of an irregularly shaped particle is the diameter of the sphere having the same volume as the particle. This is not much help because in many cases the volume of the particle is ill-defined or difficult to measure. Usually, the particle size is defined in a fairly arbitrary manner in terms of a number generated by one of the measuring techniques described later. A particle size measured by one technique may therefore be quite different from that measured by another technique, even when the measuring instruments are operating properly. 

The solid particle can have an arbitrary shape, but attention for now will focus on the case of a sphere. The radius of the sphere is denoted by a and its complex dielectric function by s((o). The dielectric function will be assumed to be local in this discussion so there is no dependence on the wave-vector of the photon. The dielectric function for the solvent is denoted by the local function Ss( ) particle size is assumed to be sufficiently small compared to the wave length of the relevant photons that the electrostatic approximation to electrodynamics is warranted. Thus retardation effects will be neglected here. 

In this expression the particle is considered to be exposed to a homogeneous electric field of strength . The shape of the particle is of no importance just as the electro-osmosis equation (11) is valid for an arbitrary shape of the porous plug. Restrictions on the applicability of expression (26) are similar to the case of electro-osmosis y. the double-layer must be thin compared with the dimensions of the particle, p. the particle must be insulating and the surface conductance at the interface must be so small that the distribution of the external field is practically uninfluenced by it. 

Particles subject to Brownian motion tend to adopt random orientations, and hence do not follow these rules. A particle without these symmetry properties may follow a spiral trajectory, and may also rotate or wobble. In general, the drag and torque on an arbitrary particle translating and rotating in an unbounded quiescent fluid are determined by three second-order tensors which depend on the shape of the body ... 

For shapes whose boundaries are not simply described in a single coordinate system, numerical solution of Eq. (4-54) is required. However, it is possible to provide upper and lower bounds for the conductance (P8) in much the same way as for the drag. A lower bound for an arbitrary particle is the conductance of the sphere of the same volume, i.e.. 

A discussion of some theoretical approaches to scattering by randomly inhomogeneous particles is followed in the final section by an outline of recent progress in constructing solutions to problems of scattering by nonspherical particles, including those of arbitrary shape. 

The classical method of solving scattering problems, separation of variables, has been applied previously in this book to a homogeneous sphere, a coated sphere (a simple example of an inhomogeneous particle), and an infinite right circular cylinder. It is applicable to particles with boundaries coinciding with coordinate surfaces of coordinate systems in which the wave equation is separable. By this method Asano and Yamamoto (1975) obtained an exact solution to the problem of scattering by an arbitrary spheroid (prolate or oblate) and numerical results have been obtained for spheroids of various shape, orientation, and refractive index (Asano, 1979 Asano and Sato, 1980). 

Analytical expression for the electrophoretic velocity of a sphere can be obtained for a thin but distorted double layer. Dukhin [6] first examined the effect of distortion of thin ion cloud on the electrophoresis of a sphere in a symmetric two-species electrolyte. Dukhin s approach was later simplified and extended by O Brien [7] for the case of a general electrolyte and a particle of arbitrary shape. Since 0(k 1) double layer thickness is much smaller than the characteristic particle size L, the ion cloud can be approximated as a structure composed of a charged plane interface and an adjacent diffuse cloud of ions. Within the double layer, the length scales for variation of quantities along the normal and tangential directions are k ] and L, respectively. 

In a systematic study of particle-particle interactions that also has often been overlooked, Cox and Brenner (1971) developed a comprehensive general theory for calculating the rheological properties of a suspension of particles of arbitrary shape to 0(02), including wall effects. They did not, however, attempt an explicit numerical calculation of ku and it is perhaps for this reason that their work has not received the thoughtful attention it deserves. 

Here B is an optical constant, or is the total polarizability of the particle, and n is the number of components in each particle. The indexes i and j refer to components of the same particle. If the assumption of independent particles was not made, then the indexes could refer to components of any two particles, and the autocorrelation expression could not be written as a simple sum of contributions from individual particles. The spatial vector r(r) refers to the center of mass of the particle. R(r). In the case of a nonspherical particle (arbitrary shape), Eq. (I0) would describe the coupled motion of the center of mass and the relative arrangement of the components of the particle. For spherical particles, translational and rotational motion arc uncoupled and we have a simplified expression for the electric field time correlation function ... 

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... 

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