Hydrodynamic effects forces

The methodology discussed previously can be applied to the study of colloidal suspensions where a number of different molecular forces and hydrodynamic effects come into play to determine the dynamics. As an illustration, we briefly describe one example of an MPC simulation of a colloidal suspension of claylike particles where comparisons between simulation and experiment have been made [42, 60]. Experiments were carried out on a suspension of AI2O3 particles. For this system electrostatic repulsive and van der Waals attractive forces are important, as are lubrication and contact forces. All of these forces were included in the simulations. A mapping of the MPC simulation parameters onto the space and time scales of the real system is given in Hecht et al. [42], The calculations were carried out with an imposed shear field. 

For the purpose of this study, particles are classified as Brownian or non-Brownian, where Brownian particles are defined as those for which the diameter is less than five microns and non-Brownian are those with diameter greater than five microns. The major focus of this work is on the second category. The particle release process has been studied both theoretically and experimentally, and it is found that for non-Brownian particles the surface charge and the electrolyte composition of the flowing phase are less significant factors than the hydrodynamic effects. However, Van der Waals forces are found to be important and the distortion of particles by these forces is shown to be crucial. 

This formulation results very insightful according to Equation 8.30, particles move under the action of an effective force — We , i.e., the nonlocal action of the quantum potential here is seen as the effect of a (nonlocal) quantum force. From a computational viewpoint, these formulation results are very interesting in connection to quantum hydrodynamics [21,27]. Thus, Equations 8.27 can be reexpressed in terms of a continuity equation and a generalized Euler equation. As happens with classical fluids, here also two important concepts that come into play the quantum pressure and the quantum vortices [28], which occur at nodal regions where the velocity field is rotational. 

On the contrary, the definition of the collision process, Eq. (429), is such that through a sequence of such events, the perturbation caused by the external force may be propagated at long distances. For instance, in the diagram of Fig. 21, corresponding to a typical term of the iterative solution of Eq. (428), the T operators are not localized around the B-particle. This allows long-range hydrodynamical effects. 

However, the central result of our above analysis has been that two-particle long-range hydrodynamical effects could be represented by a non-Hamiltonian Markoffian force which has the Fourier transform (see Eq. (461)) ... 

In most industrial applications, multistage countercurrent contacting is required. The hydrodynamic driving force necessary to induce countercurrent flow and subsequent phase separation may be derived from the differential effects of either gravity or centrifugal force on the two phases of different densities. Essentially there are two types of design by which effective multistage operation may be obtained ... 

For nondeformable particles, the theories describing the interaction forces are well advanced. So far, most of the surface force measnrements between planar liquid surfaces (TFB) have been conducted under conditions such that the film thickness is always at equilibrium. In the absence of hydrodynamics effects, the forces are correctly accounted considering classical theories valid for planar solid surfaces. When approached at high rate, droplets may deform, which considerably complicates the description it is well known that when the two droplets are sufficiently large, hydrodynamic forces result in the formation of a dimple that flattens prior to film thinning. Along with the hydrodynamic interactions, the direct... 

It is noted that a cluster as referred to here is a lump of solid particles over which flow properties such as voidage do not vary substantially. It is formed mainly as a result of hydrodynamic effects. The mechanism of particle clustering is different from that of agglomeration, in which particles adhere to one another mainly by surface attraction (e.g., van der Waals force and electrostatic forces), and mechanical or chemical interaction [Horio and Clift, 1992],... 

Fig. 26. Schematic design of field flow fractionation (FFF) analysis. A sample is transported along the flow channels by a carrier stream after injection and focusing into the injector zone. Depending on the type and strength of the perpendicular field, a separation of molecules or particles takes place the field drives the sample components towards the so-called accumulation wall. Diffusive forces counteract this field resulting in discrete layers of analyte components while the parabolic flow profile in the flow channels elutes the various analyte components according to their mean distance from the accumulation wall. This is called normal mode . Particles larger than approximately 1 pm elute in inverse order hydrodynamic lift forces induce steric effects the larger particles cannot get sufficiently close to the accumulation wall and, therefore, elute quicker than smaller ones this is called steric mode . In asymmetrical-flow FFF, the accumulation wall is a mechanically supported frit or filter which lets the solvent pass the carrier stream separates asymmetrically into the eluting flow and the permeate flow which creates the (asymmetrical) flow field...

The hierarchy of equations thereby obtained can be closed by truncating the system at some arbitrary level of approximation. The results eventually obtained by various authors depend on the implicit or explicit hypotheses made in effecting this closure—a clearly unsatisfactory state of affairs. Most contributions in this context aim at calculating the permeability (or, equivalently, the drag) of a porous medium composed of a random array of spheres. The earliest contribution here is due to Brinkman (1947), who empirically added a Darcy term to the Stokes equation in an attempt to represent the hydrodynamic effects of the porous medium. The so-called Brinkman equation thereby obtained was used to calculate the drag exerted on one sphere of the array, as if it were embedded in the porous medium continuum. Tam (1969) considered the same problem, treating the particles as point forces he further assumed, in essence, that the RHS of Eq. (5.2a) was proportional to the average velocity and hence was of the explicit form... 

S-FFF has been compared with analytical ultracentrifugation (AUC) with respect to the fractionation of a 10-component latex standard mixture with narrow particle size distribution, known diameters (67-1220 nm) and concentration [ 127]. With an analytical ultracentrifuge, the particle sizes as well as their quantities could be accurately determined in a single experiment whereas in S-FFF deviations from the ideal retention behavior were found for particles >500 nm resulting in smaller particle size determination in the normal as well as in the programmed operation. It was concluded that, without a modified retention equation which accounts for hydrodynamic lift forces and steric exclusion effects, S-FFF cannot successfully be used for the size characterization of samples in that size range. 

Most dielectrophoretic separations of cells to date have used steric-DEP-FFF. The cells are usually effectively immobilized in potential energy minima [282] near the electrodes by a combination of gravity and electrical field forces. Afterwards, the applied hydrodynamic flow forces transport those particles that are held less strongly at the electrodes. 

On the other hand, the mechanism of steric-FFF is complicated by a number of hydrodynamic phenomena [79]. The most important are the hydrodynamic lift forces that drive particles away from the wall and thus counteract the physical field [289-291]. This effect can even lead to an increase in the separation... 

Recent systematic studies used steric-S-FFF with well-characterized latex beads of diameters 2-50 pm where the sedimentation force was adjusted to exactly counterbalance the lift force FL [79,301] to provide a measure of FL. This has led to a more subtle view of the hydrodynamic lift forces than expressed by Eq. (85). There are indications that the hydrodynamic lift force is presumably composed of two different contributions (a) the lift force due to the fluid inertial effect ly., which may be described by the theory of hydrodynamic lift forces [289-291], and (b) the hydrodynamic lift force by a near-wall effect ly [61,62,79,301,302]. The latter was experimentally found to be a function of particle diameter dH, the distance of the particle bottom from the wall 8, the fluid shear rate s0, and the fluid viscosity T by ... 

The behavior of particles under the simultaneous effect of field forces and lift forces can vary with the nature of different applied primary field forces [298]. The force acting on the particles is proportional to the third power of the particle diameter in S-FFF, but only to the first power of the particle diameter in Fl-FFF thus indicating that S-FFF is probably best suited for a fine balance between the external field and hydrodynamic lift forces. 

There are three basic distinct types of phenomena that may be responsible for intrinsic instabilities of premixed flames with one-step chemistry body-force effects, hydrodynamic effects and diffusive-thermal effects. Cellular flames—flames that spontaneously take on a nonplanar shape—often have structures affected most strongly by diffusive-thermal... 

Lift Forces Combined with the Field Action. The hydrodynamic lift forces that appear at high flow rates of the carrier liquid combined with the primary field are able to concentrate the hard suspended particles into the focused layers. The retention behavior of the particles under the simultaneous effect of the primary field and lift forces generated by the high longitudinal flow rate can vary with the nature of various applied primary field forces. 

Below, in Sections 5.2 and 5.3, we consider effects related to the surface tension of surfactant solution and capillarity. In Section 5.4 we present a review of the surface forces due to intermo-lecular interactions. In Section 5.5 we describe the hydrodynamic interparticle forces originating from the effects of bulk and surface viscosity and related to surfactant diffusion. Section 5.6 is devoted to the kinetics of coagulation in dispersions. Section 5.7 regards foams containing oil drops and solid particulates in relation to the antifoaming mechanisms and the exhaustion of antifoams. Finally, Sections 5.8 and 5.9 address the electrokinetic and optical properties of dispersions. 

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