Hydrodynamic force, suspensions

The electroviscous effect present with solid particles suspended in ionic liquids, to increase the viscosity over that of the bulk liquid. The primary effect caused by the shear field distorting the electrical double layer surrounding the solid particles in suspension. The secondary effect results from the overlap of the electrical double layers of neighboring particles. The tertiary effect arises from changes in size and shape of the particles caused by the shear field. The primary electroviscous effect has been the subject of much study and has been shown to depend on (a) the size of the Debye length of the electrical double layer compared to the size of the suspended particle (b) the potential at the slipping plane between the particle and the bulk fluid (c) the Peclet number, i.e., diffusive to hydrodynamic forces (d) the Hartmarm number, i.e. electrical to hydrodynamic forces and (e) variations in the Stern layer around the particle (Garcia-Salinas et al. 2000). 

The hydrodynamic forces acting on the suspended colloids determine the rate of cake buildup and therefore the fluid loss rate. A simple model has been proposed in literature [907] that predicts a power law relationship between the filtration rate and the shear stress at the cake surface. The model shows that the cake formed will be inhomogeneous with smaller and smaller particles being deposited as the filtration proceeds. An equilibrium cake thickness is achieved when no particles small enough to be deposited are available in the suspension. The cake thickness as a function of time can be computed from the model. 

The prime difficulty of modeling two-phase gas-solid flow is the interphase coupling, which deals with the effects of gas flow on the motion of solids and vice versa. Elgobashi (1991) proposed a classification for gas-solid suspensions based on the solid volume fraction es, which is shown in Fig. 2. When the solid volume fraction is very low, say es< 10-6, the presence of particles has a negligible effect on the gas flow, but their motion is influenced by the gas flow for sufficiently small inertia. This is called one-way coupling. In this case, the gas flow is treated as a pure fluid and the motion of particle phase is mainly controlled by the hydrodynamical forces (e.g., drag force, buoyancy force, and so... 

Shear Stress. Because mammalian cells lack a cell wall and are larger than bacteria, they are more susceptible to hydrodynamic forces, or shear stress. Several studies have investigated the effects of shear stress on mammalian cells.45 8 Many indicate that the action of the impeller alone does not decrease the viability of suspension-adapted mammalian cells.46,48,49 Some bioprocess engineers in industry have seen a few cell lines that appear to be less robust, and anecdotally might have been damaged by the impeller. However, bubble rupture does cause sufficient hydrodynamic force to kill all the cells attached to the bubble.48 The effects of bubble... 

The rheology of suspensions generally differs from fluids as a result of the hydrodynamic forces acting on the particles. The following figure illustrates this behavior in a print head. Since flows in print heads are in the range of low Reynolds numbers (Re 1-10 during drop formation in an ink channel with 350 m diameter), the velocity profile within (circular) capillaries is parabolic. This is indicated in Fig. 1. 

At very low shear rates (i.e., flow velocities), particles in a chemically stable suspension approximately follow the layers of constant velocities, as indicated in Fig. 2. But at higher shear rates hydro-dynamic forces drive particles out of layers of constant velocity. The competition between hydrodynamic forces that distort the microstructure of the suspension and drive particles together, and the Brownian motion and repulsive interparticle forces keeping particles apart, leads to a shear dependency of the viscosity of suspensions. These effects depend on the effective volume fraction of... 

For monodisperse or unimodal dispersion systems (emulsions or suspensions), some literature (28-30) indicates that the relative viscosity is independent of the particle size. These results are applicable as long as the hydrodynamic forces are dominant. In other words, forces due to the presence of an electrical double layer or a steric barrier (due to the adsorption of macromolecules onto the surface of the particles) are negligible. In general the hydrodynamic forces are dominant (hard-sphere interaction) when the solid particles are relatively large (diameter >10 (xm). For particles with diameters less than 1 (xm, the colloidal surface forces and Brownian motion can be dominant, and the viscosity of a unimodal dispersion is no longer a unique function of the solids volume fraction (30). 

For a very dilute suspension of rigid noninteracting particles, the rate of sedimentation Vq can be calculated by the application of Stokes law, whereby the hydrodynamic force is balanced by the gravitational force. 

Tenneti, S., Garg, R., Hrenya, C. M., Fox, R. O. Subramaniam, S. 2010 Direct numerical simulation of gas-solid suspensions at moderate Reynolds number quantifying the coupling between hydrodynamic forces and particle velocity fluctuations. Powder Technology 203, 57-69. 

The prediction (7-197) is compared with experimental data for a suspension of monodis-perse spherical particles in Fig. 7-14. The result (7-197) is actually independent of whether the suspension is monodisperse or polydisperse as the contribution of each particle is independent of the presence of other particles. However, we see that the result (7-197) holds only for extremely small volume fractions and then the data begin to increase more rapidly with volume fraction. When this happens, particle interactions begin to be important, both hydrodynamic and also those that are due to any interaction forces that may exist between particles. Then the size distribution, the nature of hydrodynamic forces, and other factors such as the importance of Brownian motion begins to play a role. 

To ensure equilibrium, Einstein assumed that the applied hydrodynamic force must be balanced by a steady thermodynamic force acting on each particle. This force may be identified with the change in Gibbs free energy G of the suspension due to the addition of the particle. It follows from the expression for chemical potential (Eq. 3.3.8), equal to the Gibbs free energy per mole, that... 

Of the particular value is the case of concentrated suspensions for which the volume concentration of disperse phase is not small. The microstmcture of such suspensions depends on relations between hydrodynamic forces of particle interactions and thermodynamic forces causing Brownian motion. In the last years the research of dynamics of concentrated suspensions (Stokes s dynamics [30]) was based on use of the Langevin equation for ensemble of N particles... 

The surrounding fluid exerts a force on each particle that includes components of different physical origin. The main contribution to this hydrodynamic force is usually made by components associated with hydrodynamic drag and buoyancy. If when expressing hydrodynamic drag we use the well-known semi-empirical two-term law, then for the force per particle in a suspension without fluctuations we obtain... 

Stokesian dynamics is a numerical technique for simulating the dynamic hehaviour of colloidal suspensions (sedimentation, rheology), where the motions of the individual particles is driven hy Brownian and volume forces (including particle interactions) and coupled by hydrodynamic interaction. In a more general approach than in Eq. (4.69), the hydrodynamic forces are traced back to the generalised particle velocities Vp and the velocity gradients E ... 

The sedimentation velocity Vq of a very dilute suspension of rigid non-interacting particles vith radius a can be determined by equating the gravitational force with the opposing hydrodynamic force as given by Stokes law, i.e. 

In order to address these issues, researchers have incorporated convection into the process of cell seeding, suppressing some of the mass transfer limitations encountered in the static procedure. Spinner flask bioreactors (Figure 44.2b) have been implemented to create convection and, thereby, hydrodynamic forces that could help increase mass transport. Poly(glycolic acid) (PGA) scaffolds were threaded onto needles and chondrocytes suspensions with a total number of cells between 2 x 10 and 10 x 10 were used. A yield of 60% was obtained after 2 h of seeding. A more uniform distribution of the cells in the scaffold was seen (compared to the static seeding) nonetheless, the concentration of cells in the outer layer of the construct was 60 to 70% higher than that in the bulk [24]. This behavior maybe due to the poor... 

The sedimentation velocity v of a very dilute suspension of rigid noninteracting particles with radius a can be determined by equating the gravitational force with the opposing hydrodynamic force eis given by Stokes law in equation (3.46). Equation (3.46) predicts a sedimentation rate for particles with radius 1 pm in a medium with a density difference of 0.2gcm and a viscosity of ImPas (i.e. water at 20 °C) of 4.4 X 10 ms . Such particles will sediment to the bottom of a 0.1m container in about 60 hours. For 10 pm particles, the sedimentation velocity is 4.4 x 10 ms and such particles will sediment to the bottom of a 0.1 m container in about 40 minutes. 

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