Hard spheres shear flow

The intrinsic viscosity can be related to the overlap concentration, c, by assuming that each coil in the dilute solution contributes to the zero-shear viscosity as would a hard sphere of radius equal to the radius of gyration of the coil. This rough approximation is reasonable as a scaling law because of the effects of hydrodynamic interactions which suppress the flow of the solvent through the coil, as we shall see in Section 3.6.1.2. The Einstein formula for the contribution of suspended spheres to the viscosity is... 

Hard sphere systems are characterized by viscous flow and for low solids loading (less than 5%) they can be described as Newtonian fluid. At higher loadings, cluster formation takes place and the fluid cau acquire shear thinning or thickening behavior. The viscosity and solids loading are correlated with the... 

We have already discussed confinement effects in the channel flow of colloidal glasses. Such effects are also seen in hard-sphere colloidal crystals sheared between parallel plates. Cohen et al. [103] found that when the plate separation was smaller than 11 particle diameters, commensurability effects became dominant, with the emergence of new crystalline orderings. In particular, the colloids organise into z-buckled" layers which show up in xy slices as one, two or three particle strips separated by fluid bands see Fig. 15. By comparing osmotic pressure and viscous stresses in the different particle configurations, tlie cross-over from buckled to non-buckled states could be accurately predicted. 

The constitutive equation, (2-60), for the stress, on the other hand, will be modified for all fluids in the presence of a mean motion in which the velocity gradient Vu is nonzero. To see that this must be true, we can again consider the simplest possible model system of a hard-sphere or billiard-ball gas, which we may assume to be undergoing a simple shear flow,... 

The characteristic flow time in a steady shear flow is simply given by the reciprocal of the shear rate. Comparing the relaxation time to the flow time, as with hard spheres, the ratio is given by the Peclet number (Eq. 5.3.25)... 

Dembo et al. [1988] developed a model based on the ideas of Evans [1985] and Bell [1978]. In this model, a piece of membrane is attached to the wall, and a pulling force is exerted on one end while the other end is held fixed. The cell membrane is modeled as a thin inextensible membrane. The model of Dembo et al. [1988] was subsequentlyextended via a probabilistic approach for the formation of bonds by Coezens-Roberts et al. [1990]. Other authors used the probabilistic approach and Monte Carlo simulation to study the adhesion process as reviewed by Zhu [ 2000]. Dembo s model has also been extended to account for the distribution of microvilli on the surface of the cell and to simulate the rolling and the adhesion of a cell on a surface under shear flow. Hammer and Apte [1992] modeled the cell as a microvilli-coated hard sphere covered with adhesive springs. The binding and breakage of bonds and the distribution of the receptors on the tips of the microvilli are computed using a probabilistic approach. 

Particles elevate the viscosity of the medium (water) through viscous interaction with the water. Thermal or Brownian motion of the particles contributes to this at low rates of shear, but this contribution diminishes with increasing shear rate. At very high rates of shear and with high particle volume fraction, instabilities in the tendency of particles to align in layers with the flow field can result in dilatency. The rheology of hard sphere dispersions has become quite well understood and quantified by theory and experiment, especially in the last decade. 

Computation of shear viscosity of hard spheres has been attempted using NEMD [11], Modified non-equilibrium molecular dynamics methods have also been developed for study of fluid flows with energy conservation [12], NEMD simulations have also been recently performed to compare and contrast the Poiseuille and Electro-osmotic flow situations. Viscosity profiles obtained from the two types of flows are found to be in good mutual agreement at all locations. The simulation results show that both type of flows conform to continuum transport theories except in the first monolayer of the fluid at the pore wall. The simulations further confirm the existence of enhanced transport rates in the first layer of the fluid in both the cases [13, 14]. 

The rheological properties of the suspension are strongly influenced by the spatial distributiOTi of the particles. The relationship between microstructure and rheology of suspensions has been smdied extensively (Brader 2010 Morris 2009 Vermant and Solomon 2005). Most of earlier smdies dealt with the simplest form of suspensions, in which dilute hard-sphere suspensions are subjected only to hydro-dynamic and thermal forces near the equilibrium state (i.e., Peclet number << 1) (Bergenholtz et al. 2002 Brady 1993 Brady and Vicic 1995). In shear flows of such suspensions, the structure is governed only by the particle volume fraction and the ratio of hydrodynamic to thermal forces, as given by the Peclet number. 

The primary effect of a grafted layer of thickness L is to increase the effective hydrodynamic and thermodynamic sizes of a colloidal particle, assuming good solvent conditions. Consequently, dispersions of these particles behave as non-Newtonian fluids with low and high shear limiting relative viscosities (rjo and tjoo), and a dimensionless critical stress a oJkJ) that depend on the effective volume fraction. As for hard spheres the viscosities diverge at volume fractions and oo, respectively, with < 00, for 0 > dispersions yield and flow as pseudoplastic solids. 

---