Happel Brenner

To understand the forces acting on small particles in solution, let us first review some low Reynolds number flow results in which inertial forces are negligible. The basic text on this topic is that of Happel Brenner (1983) with more recent advances to be found in Kim Karrila (1991), who also include computational and variational considerations, and in Leal (1992). 

An important result from low Reynolds number theory is that for a body of arbitrary shape in translational motion with a velocity U the resultant force F exerted by the body depends on its orientation and may be written in Cartesian tensor notation as (Happel Brenner 1983, Batchelor 1967)... 

When Brownian motion and its attendant randomizing effect is important, an ellipsoid, or colloidal particles of arbitrary shape, will have various orientations. As we have seen, there is a translation coefficient for each orientation. It can be shown that the mean translation coefficient, mean friction coefficient, and mean mobility are given, respectively, by the formulas (Perrin 1936, Happel Brenner 1983)... 

Cell models akin to those discussed in Section 8.5 have also been applied to the determination of the properties of concentrated suspensions (Happel Brenner 1983, van de Ven 1989). Although it is another method which has been used to obtaining approximate expressions for the high shear relative viscosity, we choose not to expand upon it here, instead referring the reader to the references cited. One of the difficulties is that the determination of the boundary conditions at the cell surface is somewhat arbitrary. Furthermore, expressions obtained by this approach indicate that the cell model is inappropriate for highly concentrated suspensions and is most satisfactory only at low to moderate concentrations. 

Other research workers have investigated the relation betwem concentration, particle size and permeability [Happel Brenner, 1965]. 

R. J. Hunter, foundations of Colloid Science, Vol. 2, Clarendon Press, Oxford, UK, 1989, Chapt. 14, p. 992 J. Happel and H. Brenner, Cow Reynolds Humber Hydrodynamics, Martiaus Nijhoff PubHsliers, The Hague, The Netherlands, 1983, pp. 431—473. 

J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Prentice Hall, Englewood Cliffs, NJ, 1965. 

A number of authors from Ladenburg (LI) to Happel and Byrne (H4) have derived such correction factors for the movement of a fluid past a rigid sphere held on the axis of symmetry of the cylindrical container. In a recent article, Brenner (B8) has generalized the usual method of reflections. The Navier-Stokes equations of motion around a rigid sphere, with use of an added reflection flow, gives an approximate solution for the ratio of sphere velocity in an infinite space to that in a tower of diameter Dr ... 

The use of these resistance tensors is developed in detail by Happel and Brenner (H3). While enabling compact formulation of fundamental problems, these tensors have limited application since their components are rarely available even for simple shapes. Here we discuss specific classes of particle shape without recourse to tensor notation, but some conclusions from the general treatment are of interest. Because the translation tensor is symmetric, it follows that every particle possesses at least three mutually perpendicular axes such that, if the particle is translating without rotation parallel to one of these axes, the total... 

H3. Happel, J., and Brenner, H., Low Reynolds Number Hydrodynamics, 2nd ed. Noordhoff, Leyden, Netherlands, 1973. 

For a complete review on low Re motion in bounded fluids, see Happel and Brenner (H3). Some general results are of immediate interest. For a particle moving through an otherwise undisturbed fluid, without rotation and with velocity U parallel to a principal axis both of the body and the container,... 

Surprisingly, intuition fails to predict the behavior of the same solute and solvent in a membrane with a uniform pore size larger than both the solvent and solute. The expectation that such a membrane will provide no rejection of the solute has been refuted repeatedly. Indeed, careful experiments indicate that partial rejection of the solute occurs even when the solute is considerably smaller (say 1/1 Oth as large as the pore size) (Miller, 1992 Deen, 1987 Ho and Sirkar, 1992 Happel and Brenner, 1965). The extent of rejection increases monotonically to the total rejection limit as the solute size approaches the pore size. These effects arise both from entropic suppression of partitioning and from augmented hydrodynamic resistance to transport through the fine pores. Thus, in this case, for a porous membrane, thermodynamic partitioning can play a role in the physical chemical processes of transport. 

For a solute of finite dimensions, a decrease in solute entropy occurs upon partitioning from an unbounded external solution into a confined pore space. The decrease in entropy results in a lower solute concentration in the pore compared to the external solution to allow equalization of chemical potential with the solute in the external fluid. For cylindrical pores, dh = dp, and the partition coefficient, Ki = (Ci)intemai/(Ci)externai, between the internal and external solutions is Ki = (1 - k)2, where k = ds/dp the ratio of solute to pore diameter (Happel and Brenner, 1965). Even for solutes 50% as large as the pore diameter, this factor equals 0.25, yielding a fourfold reduction in concentration within the pore. As k approaches unity, Ki drops tremendously. Related expressions exist for other geometries, and the trends are similar (Happel and Brenner, 1965). 

As a consequence of Lorentz reciprocal theorem (see Happel and Brenner, 1965) the grand resistance matrix R(xJV, e ) possesses many internal symmetries, greatly reducing the number of its independent elements. Another important feature of R is that it depends only on the instantaneous configuration ( N, eN) of the particulate phase. 

In a series of papers, Felderhof has devised various methods to solve anew one- and two-sphere Stokes flow problems. First, the classical method of reflections (Happel and Brenner, 1965) was modified and employed to examine two-sphere interactions with mixed slip-stick boundary conditions (Felderhof, 1977 Renland et al, 1978). A novel feature of the latter approach is the use of superposition of forces rather than of velocities as such, the mobility matrix (rather than its inverse, the grand resistance matrix) was derived. Calculations based thereon proved easier, and convergence was more rapid explicit results through terms of 0(/T7) were derived, where p is the nondimensional center-to-center distance between spheres. In a related work, Schmitz and Felderhof (1978) solved Stokes equations around a sphere by the so-called Cartesian ansatz method, avoiding the use of spherical coordinates. They also devised a second method (Schmitz and Felderhof, 1982a), in which... 

Similar experiments have been performed with settling suspensions, which leads to comparable empirical correlations. Early experiments are reviewed by Happel and Brenner (1965). Reviews by Fitch (1979) and by Davis and Acrivos (1985) provide the state of the art. 

The majority of theories describing the concentration dependence of viscosity of diluted and moderately concentrated disperse systems is based on the hydrodynamic approach developed by Einstein [1]. Those theories were fairly thoroughly analyzed in the reviews written by Frish and Simha [28] and by Happel and Brenner [29], In a fairly large number of works describing the dependence of viscosity on concentration the final formulas are given in the form of a power series of the volume concentration of disperse phase particles — [Pg.111]

According to Happel and Brenner [29], the value of Einstein s coefficient (KE = 2.5) has not been completely proved and therefore it is difficult to obtain theoretically the exact value of the coefficient before the quadratic term. 

Figure 11.3 is a plot of G(p) as a fimction of p for both prolate and oblate ellipsoids. For simplicity and reasonable accuracy over all ranges of the volume fraction, the factor hi ) is given by Happel and Brenner... 

Famularo [35] investigated the problem further and found values of the constant for different assemblies as follows cubic, 1.91 rhombohedral, 1.79 random, 1.30. Burgers [36] considered a random assembly of particles and replaced the second term in the denominator with 6.88c. The numerical constant has been questioned by Hawksley [17 pl31] who suggested that in practice the particles would accelerate to an equilibrium arrangement with a reduced constant of 4.5. The form of expression has also been criticized by Happel and Brenner [7]. 

---