Effective viscosity suspension

The effective viscosity of a suspension of particles in a fluid medium is greater than that of the pure fluid, owing to the energy dissipation within the electrical double layers. 

With some concentrated suspensions of solid particles, particularly those in which the liquid has a relatively low viscosity, the suspension appears to slip at the pipe wall or at the solid surfaces of a viscometer. Slip occurs because the suspension is depleted of particles in the vicinity of the solid surface. In the case of concentrated suspensions, the main reason is probably that of physical exclusion if the suspension at the solid surface were to have the same spatial distribution of particles as that in the bulk, some particles would have to overlap the wall. As a result of the lower concentration of particles in the immediate vicinity of the wall, the effective viscosity of the suspension near the wall may be significantly lower than that of the bulk and consequently this wall layer may have an extremely high shear rate. If this happens, the bulk material appears to slip on this lubricating layer of low viscosity material. 

In each of these cases, it is correctly assumed that the upthrust acting on the particles is determined by the density of the suspension rather than that of the fluid. The use of an effective viscosity, however, is valid only for a large particle settling in a fine suspension. For the sedimentation of uniform particles the increased drag is attributable to a steepening of the velocity gradients rather than to a change in viscosity. 

The effective viscosity depends on the solid hold-up, on particle size and distribution, on the surface properties, on the particle shape and density, on the properties of the liquid (p, p, d), on temperature, and the shear stress in the column. Depending on the solid concentration encountered in BSCR, we can classify the suspensions into "dilute" and "concentrated" groups. 

Tragacanth is widely used as a natural emulsifier in conjunction with acacia and is an effective viscosity modifier for suspension formulations. It contains a variety of methoxylated acids that upon contact with water become a gel. At around pH 5, it renders the maximum stable viscosity due to aging even though the maximum viscosity occurs at pH 8 with the freshly prepared solution. 

This brief section provides a historical and practical overview of useful empiricisms employed in suspension theories, including a few useful formulas. Early investigators were mainly concerned with the measurement and correlation of two fundamentally important, but apparently unrelated, quantities (i) The effective viscosity fi of sheared suspensions of neutrally buoyant particles and (ii) the sedimentation speed us of suspensions of non-neutrally buoyant particles. Upon appropriate normalization, both were regarded as being functions only of the volumetric solids concentration ... 

Cell models constitute a second major class of empirical developments. Among these, only two will be mentioned here as constituting the most successful and widely used. The first, due to Happel (1957,1958), is useful for estimating the effective viscosity and settling velocity of suspensions. Here, the suspension is envisioned as being composed of fictitious identical cells, each containing a single spherical particle of radius a surrounded by a concentric spherical envelope of fluid. The radius b of the cell is chosen to reproduce the suspension s volume fraction

[Pg.21]

Another fractal structure of interest is considered by Adler (1986). A three-dimensional fractal suspension may be constructed from a modified Menger sponge, as shown in Fig. 7(b). A scaling argument permitted calculating the effective viscosity of such a suspension however, this viscosity should be compared with numerical results for the solution of Stokes equations in such a geometry before this rheological result is accepted unequivocally. 

For Newtonian fluids, the effective viscosity is a constant rjly) = rj and Ni(y) = N y) = 0, su esting that the norma] stress differences arise from elasticity or memoiy of the material. For monosized ceramic suspensions, experimentally it has been found that > 0 andN2 < 0. 

The effective viscosity rj of cl dilute suspension of uncharged coUoidal particles in a liquid is greater than the viscosity t] of the original liquid. Einstein [1] derived the following expression for... 

Srmha [2] derived the following equation for the effective viscosity r] of a concentrated suspension of uncharged particles of volume fraction [Pg.515]

The effective viscosity of a suspension of particles of types other than rigid particles has also been theoretically investigated. Taylor [22] proposed a theory of the electroviscous effect in a suspension of uncharged liquid drops. This theory has been extended to the case of charged liquid drops by Ohshima [17]. Natraj and Chen [23] developed a theory for charged porous spheres, and Allison et al. [24] and Allison and Xin [25] discussed the case of polyelectrolyte-coated particles. 

In this chapter, we first present a theory of the primary electroviscous effect in a dilute suspension of soft particles, that is, particles covered with an ion-penetrable surface layer of charged or uncharged polymers. We derive expressions for the effective viscosity and the primary electroviscous coefficient of a dilute suspension of soft particles [26]. We then derive an expression for the effective viscosity of uncharged porous spheres (i.e., spherical soft particles with no particle core) [27]. 

Approximate results calculated via Eq. (27.57) are also shown as dotted lines in Fig. 27.2. It is seen that Ka > 100, the agreement with the exact result is excellent. The presence of a minimum of L Ka, la, alb) as a function of Ka can be explained qualitatively with the help of Eq. (27.57) as follows. That is, L Ka, la, alb) is proportional to 1/k at small Ka and to k at large Ka, leading to the presence of a minimum of L Ka, la, alb). As is seen in Fig. 27.3, for the case of a suspension of hard particles, the function L ko) decreases as Ka increases, exhibiting no minimum. This is the most remarkable difference between the effective viscosity of a suspension of soft particles and that for hard particles. It is to be noted that although L Ka, la, alb) increases with Ka at large Ka, the primary electroviscous coefficient p itself decreases with increasing electrolyte concentration. The reason is that the... 

EFFECTIVE VISCOSITY OF A CONCENTRATED SUSPENSION 527 surface potential J/o, which becomes for large Ka (Eq. (4.29))... 

EFFECTIVE VISCOSITY OF A CONCENTRATED SUSPENSION OF UNCHARGED POROUS SPHERES... 

We have shown that the effective viscosity of concentrated suspension of uncharged porous spheres of radius a and volume fraction in a liquid is given by Eq. (27.68) (as combined with Eq. (27.74)). The coefficient Q(2a, (f)) expresses how the presence of porous spheres affects the viscosity tjs of the original liquid. 

The suspension could be assigned an effective viscosity, rj, given by... 

The velocity field given by (7-185) will be used later to estimate heat transfer rates for spherical particles in a straining flow. Here, we focus on a different application of (7 185), namely, its use in predicting the effective viscosity of a dilute suspension of solid spheres. To carry out the calculation, it is first necessary to briefly discuss the properties of a suspension in a more general framework. 

It can, in fact, be proven that a dilute suspension of rigid spheres will always be Newtonian at the first 0(C) correction to the bulk stress, with an effective viscosity given by (7 196) of the form... 

Using this solution, determine the effective viscosity for a dilute suspension of spherical bubbles. Discuss the fact that the effective viscosity is smaller than for a dilute suspension of solid spherical particles. 

Problem 7-22. The Viscosity of a Multicomponent Membrane. An interesting generalization of the Einstein calculation of the effective viscosity of a dilute suspension of spheres is to consider the same problem in two dimensions. This is relevant to the effective viscosities of some types of multicomponent membranes. Obtain the Einstein viscosity correction at small Reynolds number for a dilute suspension of cylinders of radii a whose axes are all aligned. Although there is no solution to Stokes equations for a translating cylinder, there is a solution for a force- and torque-free cylinder in a 2D straining flow. The result is... 

It was shown in [179] that the effective viscosity is related to the ratio of the velocity of free sedimentation of a single particle according to the Stokes law to the velocity of particles in the suspension, that is, the effective viscosity is related to the correction factor A in the drag force. The expression... 

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