Hydrodynamic interaction boundary conditions

If the Brownian particles were macroscopic in size, the solvent could be treated as a viscous continuum, and the particles would couple to the continuum solvent through appropriate boundary conditions. Then the two-particle friction may be calculated by solving the Navier-Stokes equations in the presence of the two fixed particles. The simplest approximation for hydrodynamic interactions is through the Oseen tensor [54],... 

The Smoluchowski-Levich approach discounts the effect of the hydrodynamic interactions and the London-van der Waals forces. This was done under the pretense that the increase in hydrodynamic drag when a particle approaches a surface, is exactly balanced by the attractive dispersion forces. Smoluchowski also assumed that particles are irreversibly captured when they approach the collector sufficiently close (the primary minimum distance 5m). This assumption leads to the perfect sink boundary condition at the collector surface i.e. cp 0 at h Sm. In the perfect sink model, the surface immobilizing reaction is assumed infinitely fast, and the primary minimum potential well is infinitely deep. 

Today it is generally understood that the no-slip condition is valid because the interaction between a fluid particle and a wall is similar to that between neighboring fluid particles [23]. This is clearly true for ordinary fluids made of small or short chain molecules. Thus in formulation of hydrodynamics for non-polymeric liquids it is not necessary to consider any measurable violation of the no-slip boundary condition on macroscopic length scales. 

Mazur (1982) and Mazur and van Saarloos (1982) developed the so-called method of induced forces in order to examine hydrodynamic interactions among many spheres. These forces are expanded in irreducible induced-force multipoles and in a hierarchy of equations obtained for these multipoles when the boundary conditions on each sphere were employed. Mobilities are subsequently derived as a power series-expansion in p 1. In principle, calculations may be performed to any order, having been carried out by the above authors through terms of 0(p 7) for a suspension in a quiescent fluid. To that order, hydrodynamic interactions between two, three, and four spheres all contribute to the final result. This work is reviewed by Mazur (1987). 

Koelman and Hoogerbrugge (1993) have developed a particle-based method that combines features from molecular dynamics (MD) and lattice-gas automata (LGA) to simulate the dynamics of hard sphere suspensions. A similar approach has been followed by Ge and Li (1996) who used a pseudo-particle approach to study the hydrodynamics of gas-solid two-phase flow. In both studies, instead of the Navier-Stokes equations, fictitious gas particles were used to represent and model the flow behavior of the interstial fluid while collisional particle-particle interactions were also accounted for. The power of these approaches is given by the fact that both particle-particle interactions (i.e., collisions) and hydrodynamic interactions in the particle assembly are taken into account. Moreover, these modeling approaches do not require the specification of closure laws for the interphase momentum transfer between the particles and the interstitial fluid. Although these types of models cannot yet be applied to macroscopic systems of interest to the chemical engineer they can provide detailed information which can subsequently be used in (continuum) models which are suited for simulation of macroscopic systems. In this context improved rheological models and boundary condition descriptions can be mentioned as examples. 

The effect of the interparticle interactions on the electrophoretic mobihty in concentrated dispersions was theoretically studied by Levine and Neale. They used a cell model with two alternative boundary conditions at the cell boundary to describe the hydrodynamic flow the free surface model of HappeF and the zero vorticity model of Kuwabara. The results suggested that the zero vorticity model is more appropriate, because it represents in a more correct way the... 

The components of represent stochastic displacements and are obtained using the multivariate Gaussian random number generator GGNSM from the IMSL subroutine library (30). p ° is the initial hydrodynamic interaction tensor between subunits iJand j. Although the exact form of D. is generally unknown, it is approximated here using the Oseen tensor with slip boundary conditions. This representation has been shown to provide a reasonable and simple point force description of the relative diffusion of finite spheres at small separations (31). In this case, one has... 

In recent years a lot of attention has been devoted to the application of electroacoustics for the characterization of concentrated disperse systems. As pointed out by Dukhin [26,27], equation (V-51) is not valid in such systems because it does not account for hydrodynamic and electrostatic interactions between particles. These interactions can typically be accounted for by the introduction of the so-called cell model, which represents an approach used to model concentrated disperse systems. According to the cell model concept, each particle in the disperse system is inclosed in the spherical cell of surrounding liquid associated only with that individual particle. The particle-particle interactions are then accounted for by proper boundary conditions imposed on the outer boundary of the cell. The cell model provides a relationship between the macroscopic (experimentally measured) and local (i.e. within a cell) hydrodynamic and electric properties of the system. By employing a cell model it is also possible to account for polydispersity. Different cell models were described in the literature [26,27]. In each case different expressions for the CVP were obtained. It was argued that some models were more successful than the others for characterization of concentrated disperse systems. Nowadays further development of the theoretical description of electroacoustic phenomena is a rapidly growing area. 

The presence of particles in the liquid complicates the hydrodynamic problem very considerably, because in order to determine the velocity field, it is necessary to solve the boundary value problem with the additional boundary conditions of sticking at the surface of each particle (i.e., zero relative velocity of liquid at the particle s surface). Physically, the problem reduces to the determination of disturbances that arise in the velocity field of a pure liquid due to the presence of particles in the liquid. The assumption of low volume concentration of particles allows us to begin with finding the disturbance provoked by one particle. Then, because in the considered approximation, particles do not interact with each other, the total disturbance will be the superposition of disturbances, assuming a uniform particle distribution in the liquid volume. The solution, which was obtained in [33], is presented below. 

Consider the coalescence of drops with fiilly retarded (delayed) surfaces (which means they behave as rigid particles) in a developed turbulent flow of a lowconcentrated emulsion. We make the assumption that the size of drops is much smaller than the inner scale of turbulence R Ao), and that drops are non-deformed, and thus incapable of breakage. Under these conditions, and taking into account the hydrodynamic interaction of drops, the factor of mutual diffusion of drops is given by the expression (11.70). To determine the collision frequency of drops with radii Ri and Ri (Ri < Ri), it is necessary to solve the diffusion equation (11.36) with boundary conditions (11.39). Place the origin of a spherical system of coordinates (r, 0,0) into the center of the larger particle of radius i i. If interaction forces between drops are spherically symmetrical, Eq. (11.36) with boundary conditions (11.39) assumes the form... 

The differences between Pitts (P) and Fuoss-Onsager (F-O) are first, the above mentioned omission by F-O of the effect of asymmetric potential on the local velocities of the solvent near the ions second, the use of the more usual boundary conditions 5.2.28b by F-O compared to the P assumption that perturbations cease to be important at r = a. Pitts, Tabor and Daly, who have analysed in detail both treatments, concluded that the discrepancy due to the different boundary conditions is small but has the effect of reducing ionic interactions in the P treatment with respect to the F-O. This is confirmed by the analysis of data with both theories. Usually P requires a smaller value of the a parameter than F-O. The third discrepancy between the theoretical treatments is in the expression of Vj, in eqn. 5.2.5, for which F-O add a term which involves the effect of the asymmetry of the ionic atmosphere upon the central ion surrounded by such atmosphere. The last difference lies in the hydrodynamic approaches and the corresponding boundary conditions. P imposes the condition that the velocity of the smoothed... 

The Yamakawa-Fujii theory [2, 3] was developed by using the Kirkwood-Riseman formalism with the effect of chain thickness approximately taken into account. The following remarks may be in order. The Oseen interaction tensor was preaveraged. Force points were distributed along the centroid of the wormlike cylinder (not over the entire domain occupied by the cylinder). The no-slip hydrodynamic condition was approximated by equating the mean solvent velocity over each cross-section of the cylinder to the velocity of the cylinder at that cross-section (Burgers approximate boundary condition). 

The dimensionless coefficient, accounts for the change in the hydrodynamic friction between the fluid and the particles (created by the hydrodynamic interactions between the particles). The dimensionless surface mobility coefficient, takes into account the variation of the friction of a molecule in the adsorption layer. The diffusion problem, Eqs. (4) and (5), is connected with the hydrodynamic problem, Eqs. (1) and (2), through the boundary conditions at the material interface. 

The last term in Eq. 19 is the non-dissipative part of the stress tensor, representing the hydrodynamic interactions. The parameter C is estimated as jly- flaMr], where the interfacial tension y and aM is the diffusivity. For a fluid with high viscosity, one has C -C 1, which implies that the hydrodynamic interactions can be neglected. The parameter IT (nondimensional pressure) mathematically acts like a Lagrange multiplier, which generates the incompressibility condition V V = 0. The boundary conditions on p and / are as follows (where n represents a normal direction to the boundary surface) ... 

The surface boundary conditions are critical to modeling electrokinetic phenomena in nanofluidics. For the hydrodynamic boundary condition, we use the nonslip model at the silica surfaces. Although the slip boundaries have been adopted and have shown significant effects to improve the energy-conversirai efficiency, a careful molecular study showed that the hydro-dynamic boundary conditirm, slip or not, depended on the molecular interactions between fluid and solid and the channel size. For the dilute solution in silica nanochannels considered in this work(/x 2 nm), the nonslip boundary condition is still valid very well. 

The role of the Maxwell pressure residting from a normal gas phase interfacial electric field that scales as /R in elongating the Uquid meniscus into a cylindrical microjet stracture can also be verified through a dynamic simulation in which the equations governing the coupled interactions between the hydrodynamics (Eqs. 1-3) and electrodynamics (Eq. 9) are solved simultaneously for a constant potential liquid meniscus in the longwave limit in axisymmetric polar coordinates (r,0,z), subject to the boundary conditions... 

Pulsatile flow in an elastic vessel is very complex, since the tube is able to undergo local deformations in both longitudinal and circumferential directions. The unsteady component of the pulsatile flow is assumed to be induced by propagation of small waves in a pressurized elastic tube. The mathematical approach is based on the classical model for the fluid-structure interaction problem, which describes the dynamic equilibrium between the fluid and the tube thin wall (Womersley, 1955b Atabek and Lew, 1966). The dynamic equilibrium is expressed by the hydrodynamic equations (Navier-Stokes) for the incompressible fluid flow and the equations of motion for the wall of an elastic tube, which are coupled together by the boundary conditions at the fluid-wall interface. The motion of the liquid is described in a fixed laboratory coordinate system (f , 6, f), and the dynamic... 

Equation 3.84 is solved for a single collector with the same boundary conditions as before. The flux to a single collector can thus be calculated. In some cases it is possible to solve Equation 3.84 using the bulk value of D given by Equation 3.72 and formulate a suitable boundary condition that includes particle-collector interactions and hydrodynamic effects in the immediate vicinity of the wall (Ruckenstein and Prieve, 1973). 

The friction depends on the microscopic details of the substrate and its interaction with the polymer on the atomistic scale," - but some universal aspects of the slip of polymer over substrates have been considered. " In order to incorporate microscopic friction in a continuum description, for example, a hydrodynamic description in terms of a Navier-Stokes equation, one routinely employs a boundary condition devised by Navier in 1823 ... 

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