Particle shear flow field

Solid particles dispersed in a liquid have a two-particle collision frequency due to Brownian movement or to a shear flow field in the suspension after stirring, mixing, pumping or otherwise. 

J. A. Schonberg and E. J. Hinch, Inertial migration of a sphere in Poiseuille flow, J. Fluid Mech. 203, 517-524 (1989) E. S. Asmolov, The inertial lift on a small particle in a weak-shear parabolic flow, Phys. of Fluids 14, 15-28 (2002) P. Cherukat, J. B. McLaughlin, and D. S. Dandy, A computational study of the inertial lift on a sphere in a linear shear flow field, Int. J. of Multiphase Flow, 25, 15-33 (1999). 

The Newtonian behavior of suspensions in Newtonian liquids is limited to low concentrations. An exception seems to be the exten-sional flow of anisometric particles (irrotational flow field) where the rate of strain independent region extends to concentrations where strong non-Newtonian behavior would be expected in shear. These rate of deformation dependent phenomena will be summarized below. 

A flow field is laminar, when it is governed by the viscous properties. That applies to very slow fluid velocities or very high viscosities, which are typical for polymer composites or highly concentrated suspensions (e.g. paints). There are three ideal types of laminar flow uniform flow (e.g. experienced by settling particles), shear flow (e. g in rheometers or pipes), and elongational flow (e.g. in nozzles and diffusers). Only the latter two are relevant for deagglomeration. 

Figure 5.5). This last result clearly shows good dispersibility in OI of pure Na-MMT type particles, whereas appropriate surface treatment of Clay B particles is required to homogeneously disperse the clay in the polymer volume, even without applying a strong shear flow field.

Ounis, H. and Ahmadi, G, (1991), Motions of Small Particles in Simple Shear Flow Field under Microgravity Condition, Phys, Fluids A, Vol. 3, pp. 2559-2570. 

The dielectric constant of poly(lithium methacrylate) dispersed in Cereclor oil at a volume fraction of 0,3 vs. the frequency of electric field under different shear rates is shown in Figure 7. The dielectric constant reaches a maximum value when Eq. (5) is satisfied. This phenomenon is called the FMP resonance between a shear field and an electric field. The flow field definitely has an influence on the particle polarization and hence on the dielectric constant. The dielectric constant thus becomes a function of both the shear rate and the frequency of the applied electric field. Especially when the flow field is rotational and strong enough to such an extent that the particle is able to spin, it may compete with the applied electric field for particle polarization. In other words, the particles or particle clusters can be orientated not only under an electric field, but also under a shear flow field. FMP was also observed in rigid or flexible polymer solutions [25-271. 

The elongational flow field was found to be two to three times more effective for dispersing than the shear flow field. A figure of 6 times has been given for experiments using cohesionless particle clusters [6, 7]. 

Lift force on a particle in shear flow When a particle is flowing in a shear flow field, it experiences a lift force normal to the fluid flow direction. The magnitude of... 

Oscillatory shear experiments are the preferred method to study the rheological behavior due to particle interactions because they directly probe these interactions without the influence of the external flow field as encountered in steady shear experiments. However, phenomena that arise due to the external flow, such as shear thickening, can only be investigated in steady shear experiments. Additionally, the analysis is complicated by the different response of the material to shear and extensional flow. For example, very strong deviations from Trouton s ratio (extensional viscosity is three times the shear viscosity) were found for suspensions [113]. 

This additional Eq. (18) was discretized at the same resolution as the flow equations, typical grids comprising 1203 and 1803 nodes. At every time step, the local particle concentration is transported within the resolved flow field. Furthermore, the local flow conditions yield an effective 3-D shear rate that can be used for estimating the local agglomeration rate constant /10. Fig. 10 (from Hollander et al., 2003) presents both instantaneous and time-averaged spatial distributions of /i0 in vessels agitated by two different impellers color versions of these plots can be found in Hollander (2002) and in Hollander et al. (2003). 

Figure 3.10 The dilation of the flow field around a spherical particle. The shear field has a vorticity equal to y/2 and the particle rotates with this constant angular velocity...

The application of a shear rate to a linear viscoelastic liquid will cause the material to flow. The same will happen to a pseudoplastic material and to a plastic material once the yield stress has been exceeded. The stress that would result from the application of the shear rate would not necessarily be achieved instantaneously. The molecules or particles will undergo spatial rearrangements in an attempt to follow the applied flow field. 

As the shear rate increases so the particles begin to align with the flow field and pack more efficiently. This is a system dominated by the hydrodynamic forces overcoming the Brownian motion. Goddard, and... 

As with spherical particles the Peclet number is of great importance in describing the transitions in rheological behaviour. In order for the applied flow field to overcome the diffusive motion and shear thinning to be observed a Peclet number exceeding unity is required. However, we can define both rotational and translational Peclet numbers, depending upon which of the diffusive modes we consider most important to the flow we initiate. The most rapid diffusion is the rotational component and it is this that must be overcome in order to initiate flow. We can define this in terms of a diffusive timescale relative to the applied shear rate. The characteristic Maxwell time for rotary diffusion is... 

For solid particles a sufficient set of boundary conditions is provided by the no slip condition, the requirement of no flow across the particle surface, and the flow field remote from the particle. For fluid particles, additional boundary conditions are required since Eqs. (1-1) and (1-9) apply simultaneously to both phases. Two additional boundary conditions are provided by Newton s third law which requires that normal and shearing stresses be balanced at the interface separating the two fluids. 

Theoretical predictions relating to the orientation and deformation of fluid particles in shear and hyperbolic flow fields are restricted to low Reynolds numbers and small deformations (B7, C8, T3, TIO). The fluid particle may be considered initially spherical with radius ciq. If the surrounding fluid is initially at rest, but at time t = 0, the fluid is impulsively given a constant velocity gradient G, the particle undergoes damped shape oscillations, finally deforming into an ellipsoid (C8, TIO) with axes in the ratio where... 

Whilst the physical and chemical nature of the filler will determine its effectiveness in a functional role, the presence of sohd additives in thermoplastics melts inevitably influence their processability. The extent to which this occurs depends on many factors including the amount of filler present, possible interactive effects between the filler and polymer, or between the filler particles themselves, together with the conditions experienced during melt processing, in particular the shear and/or elongational flow fields developed. 

NMR imaging techniques were applied to the measurements of velocity field in opaque systems such as tomato juice and paper pulp suspensions [58-60]. In both cases, the particle concentrations are sufficiently high that widely applied techniques such as hot film and laser Doppler anemometry could not be used. The velocity profile for a 6 % tomato juice slurry clearly showed a power-law behavior [58, 59]. Row NMR images for a 0.5 % wood pulp suspension provided direct visual of three basic types of shear flow plug flow, mixed flow and turbulent flow as mean flow rate was increased. Detailed analysis of flow NMR image is able to reveal the complex interaction between the microstructure of suspensions and the flow [60]. 

If the mixing device generates a simple shear flow, as shown in Fig. 3.23, the maximum separation forces that act on the particles as they travel on their streamline occur when they are oriented in a 45° position as they continuously rotate during flow. However, if the flow field generated by the mixing device is a pure elongational flow, such as shown in Fig. 3.24, the particles will always be oriented at 0° the position of maximum force. 

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. 

Figure E7.10b shows the SDF and compares it to that of circular tube flow of a Newtonian fluid. The SDF is broad with about 75% of the flow rate experiencing a strain below the mean strain. A better insight into the meaning of the SDF is obtained by following simultaneously the reduction of the striation thickness and the flow rates contributed by the various locations between the plates (Fig. E7.10c). The distance between the plates is divided into 10 layers. We assume for the schematic representation of the SDF that the strain is uniform within each layer. Let us consider in each alternate layer two cubical minor particles separated by a certain distance, such that the initial striation thickness is Tq. By following the deformation of the particles with time, we note that although the shear rate is uniform, since the residence time is different, the total strain experienced by the particle is minimal at the moving plate and increases as we approach the stationary plate. But the quality of the product of such a mixer will not be completely determined by the range of strains or striations across the flow field the flow rate of the various layers also plays a role, as Fig. E7.10c indicates. A sample collected at the exit will consist, for example, of 17% of a poorly mixed layer B and only 1%...

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. 

Furthermore, the friction forces acting in the flow field can induce phase segregation at the mould surface [189]. As pointed out by Cakmak and Cronin [191], in PP/EP blends with a high content of EP particles even shear amplification phenomena may occur due to the presence of the small rubber particles. The shear amplification results from considerable shear fields occurring in small gaps between rubber particles which in turn are subjected to the macroscopic shear field extended over the whole width of the sample. 

In the present rheological context, lattice deformation may be regarded as arising from the transport of neutrally buoyant lattice points suspended within a macroscopically homogeneous linear shear flow. The local vector velocity field v at a general (interstitial or particle interior) point R of such a spatially periodic suspension can be shown to be of the form... 

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