Dilute suspensions apparent viscosity

Since n is less than unity, the apparent viscosity decreases with the deformation rate. Examples of such materials are some polymeric solutions or melts such as rubbers, cellulose acetate and napalm suspensions such as paints, mayonnaise, paper pulp, or detergent slurries and dilute suspensions of inert solids. Pseudoplastic properties of wallpaper paste account for good spreading and adhesion, and those of printing inks prevent their running at low speeds yet allow them to spread easily in high speed machines. 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

In this section we examine the flow of a suspension of particles, particularly the apparent viscosity coefficient of the suspension. Our interest is in calculating the convective mass flux of a suspension as distinct from the diffusive flux of Brownian motion. As previously, we shall assume a very dilute suspension in which each particle behaves as if it were in a liquid of infinite extent. To simplify the calculation, we neglect Brownian motion, although, as we discuss later, in the very dilute limit considered and for spherical particles it has no effect on the suspension viscosity. 

With the above solution for the additional flow produced by a single particle, the total perturbation can now be calculated for a dilute suspension in shear flow. The calculation was first made by Einstein, and we follow an approach given by J.M. Burgers as outlined by Sadron (1953). In this approach the apparent, or effective, viscosity is found from the shearing stress on the... 

A systematic review on the primary, secondary and tertiary electroviscous effect has been presented by Conway and Dobry-Duclaux [IJ in 1960. A brief review on some of those cITccts has been given by Dukhin [50] and Saville [51], and a more unified review has been presented by Hunter [4], In a dilute suspension, the apparent viscosity will increase with the particle volume fraction and the surface charge of the particle. A viscosity equation first published by Smoluchowski without proof [52] for describing such a system is... 

The zeta potential values ( ) were measured using diluted suspensions, whereas the ionic strength (/), interparticle separation distance (IPS), yield stress (t ), and apparent viscosity data (rj pp) were either estimated or measured in 58 vol% solid-loading slurries (from Ref. 12). 

For more dilute multiphase systems, the correlations developed by Bamea and Mizrahi (1973) are to be preferred because of the underlying physical arguments. Not only did they present a detailed review of the many multiparticle models available at the time, but they also distinguished between three independent effects, viz., (a) a hydrostatic effect, taken into account by using an apparent suspension density rather than the fluid density (b) a momentum transfer hindrance effect due to increased momentum transfer within the fluid due to the presence and motion of adjacent particles taken into account by an apparent viscosity and (c) a wall effect as the adjacent particles have a similar restraining effect on the flow field as the pipe wall for a single particle setding in a pipe. 

Eq. (1) is valid for a particle settling without the interference of other particles, i.e., for diluted systems. The particle settling velocity decreases as particle concentration increases. This phenomenon is known as hindered settling. Several equations can be found in the literature to account for this phenomenon, but a simple method to calculate the hindered settling velocity is to use Eq. (1) replacing the liquid density and viscosity by the apparent density and viscosity of the suspension [24]. 

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