Motion shear flow

Once particles are present in a volume of gas, they collide and agglomerate by different processes. The coagulation process leads to substantial changes in particle size distribution with time. Coagulation may be induced by any mechanism that involves a relative velocity between particles. Such processes include Brownian motion, shearing flow of fluid, turbulent motion, and differential particle motion associated with external force fields. The theory of particle collisions is quite complicated even if each of these mechanisms is isolated and treated separately. 

Sundararajan et al. [131] in 1999 calculated the slurry film thickness and hydrodynamic pressure in CMP by solving the Re5molds equation. The abrasive particles undergo rotational and linear motion in the shear flow. This motion of the abrasive particles enhances the dissolution rate of the surface by facilitating the liquid phase convective mass transfer of the dissolved copper species away from the wafer surface. It is proposed that the enhancement in the polish rate is directly proportional to the product of abrasive concentration and the shear stress on the wafer surface. Hence, the ratio of the polish rate with abrasive to the polish rate without abrasive can be written as... 

Rumscheidt, F. D., and Mason, S. G., Particle motions in sheared suspensions. XII. Deformation and burst of fluid drops in shear and hyperbolic flow. J. ColloidScL 16,238-261 (1961). Rwei, S. P., Manas-Zloczower, I., and Feke, D. L., Observation of carbon black agglomerate dispersion in simple shear flows. Polym. Eng. ScL 30, 701-706 (1990). 

With the gel equation, we can conveniently compute the consequences of the self-similar spectrum and later compare to experimental observations. The material behaves somehow in between a liquid and a solid. It does not qualify as solid since it cannot sustain a constant stress in the absence of motion. However, it is not acceptable as a liquid either, since it cannot reach a constant stress in shear flow at constant rate. We will examine the properties of the gel equation by modeling two selected shear flow examples. In shear flow, the Finger strain tensor reduces to a simple matrix with a shear component... 

Studies on orthokinetic flocculation (shear flow dominating over Brownian motion) show a more ambiguous picture. Both rate increases (9,10) and decreases (11,12) compared with orthokinetic coagulation have been observed. Gregory (12) treated polymer adsorption as a collision process and used Smoluchowski theory to predict that the adsorption step may become rate limiting in orthokinetic flocculation. Qualitative evidence to this effect was found for flocculation of polystyrene latex, particle diameter 1.68 pm, in laminar tube flow. Furthermore, pretreatment of half of the latex with polymer resulted in collision efficiencies that were more than twice as high as for coagulation. 

This characterizes the time taken for the restoration of the equilibrium microstructure after a disturbance caused, for example, by convective motion, i.e., this is the relaxation time of the microstructure. The time scale of shear flow is given by the reciprocal of the shear rate, 7. The dimensionless group formed by the ratio (tD ff/tShear) is the Peclet number... 

Couette and Poiseuille flows are in a class of flows called parallel flow, which means that only one velocity component is nonzero. That velocity component, however, can have spatial variation. Couette flow is a simple shearing flow, usually set up by one flat plate moving parallel to another fixed plate. For infinitely long plates, there is only one velocity component, which is in the direction of the plate motion. In steady state, assuming constant viscosity, the velocity is found to vary linearly between the plates, with no-slip boundary conditions requiring that the fluid velocity equals the plate velocity at each plate. There... 

Fig. 2 compares collision kernels calculated for a 250 nm particle as function of the collision partner size for Brownian motion, laminar and turbulent shear flows as well as sedimentation at 25 °C in water based on the equations given... 

Hence, when solving a non-isothermal problem the question arises -is this a problem where the equations of motion and energy are coupled To address this question we can go back to Example 6.1, a simple shear flow system was analyzed to decide whether it can be addressed as an isothermal problem or not. In a simple shear flow, the maximum temperature will occur at the center of the melt. By substituting y = h/2 into eqn. (6.5), we get an equation that will help us estimate the temperature rise... 

Fiber motion — Jeffery orbits. The motion of ellipsoids in uniform, viscous shear flow of a Newtonian fluid was analyzed by Jeffery [32, 33] in 1922. For a prolate spheroid of aspect ratio a (defined as the ratio between the major axis and the minor axis) in simple shear flow, u°° = (zj), the angular motion of the spheroid is described... 

Applications of optical methods to study dilute colloidal dispersions subject to flow were pioneered by Mason and coworkers. These authors used simple turbidity measurements to follow the orientation dynamics of ellipsoidal particles during transient shear flow experiments [175,176], In addition, the superposition of shear and electric fields were studied. The goal of this work was to verify the predictions of theories predicting the orientation distributions of prolate and oblate particles, such as that discussed in section 7.2.I.2. This simple technique clearly demonstrated the phenomena of particle rotations within Jeffery orbits, as well as the effects of Brownian motion and particle size distributions. The method employed a parallel plate flow cell with the light sent down the velocity gradient axis. 

For simple shear flow (as between parallel plates in relative motion) Eq. 2.7-9 reduces to ... 

Simple shear flow is the characteristic of Couette flow, and no pressure gradient exists in the direction of motion. Since the pressure gradient term is also zero, the governing equation reduces to... 

The vast majority of work on particle-surface electrostatic interactions has neglected any effects due to particle motion. However, both theoretical [31,32] and experimental work [33-35] have been done on the problem of a charged particle interacting with a charged wall in a linear shear flow. In the theoretical treatment, it is assumed that the double layer thickness is small compared to both the particle diameter and the surface-to-particle gap. Hence, changes in the pressure and potential profiles in the gap caused by motion can be written as small perturbations to their equilibrium profiles. In the region outside the small double layers, the fluid velocity v and perturbation pressure dp are governed by Stokes equations... 

Now that we have settled on a model, one needs to choose the appropriate algorithm. Three methods have been used to study polymers in the continuum Monte Carlo, molecular dynamics, and Brownian dynamics. Because the distance between beads is not fixed in the bead-spring model, one can use a very simple set of moves in a Monte Carlo simulation, namely choose a monomer at random and attempt to displace it a random amount in a random direction. The move is then accepted or rejected based on a Boltzmann weight. Although this method works very well for static and dynamic properties in equilibrium, it is not appropriate for studying polymers in a shear flow. This is because the method is purely stochastic and the velocity of a mer is undefined. In a molecular dynamics simulation one can follow the dynamics of each mer since one simply solves Newton s equations of motion for mer i,... 

Here, the sphere center is instantaneously situated at point 0 the sphere center translates with velocity U, while it rotates with angular velocity (a r is measured relative to 0 its magnitude r is denoted by r. Moreover, f = r/r is a unit radial vector. The latter solution is derivable in a variety of ways e.g., from Lamb s (1932) general solution (Brenner, 1970). [Equation (2.12) represents a superposition (Brenner, 1958) of three physically distinct solutions, corresponding, respectively, to (i) translation of a sphere through a fluid at rest at infinity (ii) rotation of a sphere in a fluid at rest at infinity (iii) motion of a neutrally buoyant sphere suspended in a linear shear flow. The latter was first obtained by Einstein (1906, 1911 cf. Einstein, 1956) in connection with his classic calculation of the viscosity of a dilute suspension of spheres, which formed part of his 1905 Ph.D. thesis.]... 

The most interesting case arises for simple shear flows (A = 0). It can be proven mathematically that time-periodic motion is possible only when the direction of the flow is parallel to either a lattice plane or a lattice line. These are respectively termed slide and tube flows the origin of such terminology is evident from Fig. 4. Very similar configurations have been experimentally observed by Ackerson and Clark (1983). 

Similar methods were employed by Schonberg et al. (1986) to investigate the multiparticle motions of a finite collection of neutrally buoyant spheres suspended in a Poiseuille flow. They were also used by Ansell and Dickinson (1986) to simulate the fragmentation of a large colloidal floe in a simple shear flow. 

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