Solid-phase shear viscosity

The gas phase viscosity is defined by the temperature and the gas composition through a semi-empirical function. For the solid phase shear viscosity, we (cf., 14) use semi-empirical relations based upon the viscometric measurements of Schugerl (21). The solid phase bulk viscosity is, at present, inaccessible to measurement consequently, we define it to be a multiple of the shear viscosity. 

Fig. 40. Steady shear viscosities of aqueous dispersions of polystyrene latices in nonadsorbing dextran solutions (Patel and Russel, 1989b) (a) a/r, = 6.9, 0 = 0.20. A, single phase, 4nR J/3pb = 0.15 B, two-phase, 4jtR /3pb = 0.30 C, two-phase, 4jtRj/3pb = 0.45 D, two-phase, 4jtRj/3pb = 0.65. (b) a/R, = 1.9, 0 = 0.10. F, single phase, 4jtR /pb = 0.65 G, fluid-fluid, 4jtR /3pb = 0.75 H, fluid-solid, 4nR /3p = 0.95 I, fluid-solid, 4jiR3/3p = 1.25.

The rheological properties of the complex network structure of tomato pastes may be assumed to be made up of two contributions (1) one network structure contributed by the solids phase, in proportion to 0s, and (2) another network structure contributed by the liquid (continuous phase), in proportion to 0i = 1 - 0s- However, the effective continuous phase is not the low-viscosity serum itself, but a highly viscous liquid that is an integral part of the tomato paste. Also implicit is that the solids fraction plays a major role in the structure of the TP samples, that is, it can be considered to be the structuring component. These assumptions are also in line with the weak gel behavior indicated by the dynamic shear data. 

The first term on the right-hand side represents momentum exchange between solid phases I and s and Kis is the solid-solid exchange coefficient. The last term represent additional shear stresses, which appear in granular flows (due to particle translation and collisions). Expressions for solids pressure, solids viscosity (shear and bulk) and solid-solid exchange coefficients are derived from the kinetic theory of granular flows. 

Liquid viscosities have been observed to increase, decrease, and remain constant in microfluidic devices as compared to viscosities in larger systems. ° Deviations from the no-slip boundary condition have been observed to occur at high shear rates. One important deviation from no-slip conditions occurs at moving contact lines, such as when capillary forces pull a liquid into a hydrophilic channel. The point at which the gas, liquid, and solid phases move along the channel wall is in violation of the no-slip boundary condition. Ho and Tai review discrepancies between classical Stokes flow theory and observations of flow in microchannels. No adequate theory is yet available to explain these deviations from classical behavior. ... 

A particle migration model was proposed by Gadala-Maria and Acrivos to describe experimental shear-induced migration observations. This model allows for a better understanding of the shear effects on particle diffusion for concentrated suspensions. Based on these studies, a conservation equation for the solid phase was established by Phillips, Amstrong, and Brown, which takes into account convective transport, diffusion due to particle-particle interactions, and the variation of viscosity within the suspension, namely ... 

Figure 9.9 Exceptional physical properties of liquid water (solid lines) temperature dependences (upper diagrams) of the density d (45) and isothermal compressibility Xt (adapted from Refs. (45 7)) pressure dependences (lower drawings) of the shear viscosity 7] at various temperatures (adapted from Ref. (48)) and of the isothermal diffusion coefficient Z) at 0 (adapted from Ref. (49)). Dashed lines sketch typical dependences displayed by almost all other liquids. Note that at —15 °C no value is given for 17 at/ > 300MPa, because of a phase transition towards ice V (Figure 8.5).

In continuous systems consisting of solid phases, the parameter G is the modulus of elasticity of rigid body. Its values may fall in the range between 109 and 10" N m 2. The elasticity modulus of common liquids under the conditions of uniform (hydrostatic) compression is also of the same order of magnitude. However, due to low viscosity, the shear elasticity of liquids may only be observed by rapid tests in which the time of stress action is close to the relaxation period. For this reason at typical times of mechanical action liquids with low r values behave as viscous media. 

The shear moduli are Pf for the fluid shear viscosity, p, for the solid shear modulus, and Pm for the macroscopic (matrix) shear modulus. Under the conditions of porosity and pressure variations, the effective densities of the liquid phase are ... 

For pol5rmer microcomposites (i.e., composites with micron sized filler), two simple relationships between melt viscosity (h), shear modulus (G) in solid-phase state and degree of filling volume (tpj were obtained. The relationship between h and G has the following form ... 

For polymer microcomposites, that is, composites with filler of micron sizes, two simple relations between melt viscosity t], shear modulus G in solid-phase state and filling volume degree (p were obtained [63]. The relationship between Tj and G has the following form [63] ... 

The turbulent stresses, Xk, in the momentum equations for the A -phase might be calculated by using the Boussinesq turbulent-viscosity model [8] for both phases or by applying a model of a Newtonian fluid for the gas phase and a granular shear stress for the solid phase [19]. 

Molecular dynamics has proved to be a powerful method for simulating and/or predicting several features of polymer systems. Properties on either side of the glass transition temperature (see Section 1.5) have been successfully simulated, as has the solid-to-liquid transition, and provided descriptions of the dynamics (segmental motions, chain diffusion, conformational transitions, etc.) that are in accord with relaxation measurements and such bulk properties as shear viscosities and elastic moduli. The method may also provide a good description of the variation in heat capacity and other thermodynamic fimctions across a phase transition. Several collections of these investigations have recently been published. ... 

In the solid phase momentum balance there are physical models needed to describe the solid pressure p, the shear viscosity /i, the bulk viipcosity A, and the drag function (3. These models are briefly described in the following chapters. A more comprehensive discussion and a comparison with experiments can be found in Boemer et al. (1995b). 

DOrc monolayers, due to the unsaturation, i.e. kinks of the alkyl chains, are in the liquid expanded phase, which is a fluid phase at all film pressures FI [3,13,15]. At 21 °C and T1 >25 mN m DPPC monolayers are in the solid analogous phase [3,13,16], which is highly incompressible and condensed [13,16]. Shah and Schulman [13] show that the effect of cholesterol on either saturated or unsaturated phospholipids is strikingly different. Cholesterol increases the surface elasticity, the dilational and the shear viscosity of unsaturated phospholipid monolayers [3,13,14,17]. In saturated monolayers cholesterol disturbs the order between phospholipid molecules fluidifying the solid monolayer [13,14,18] and lowering its shear viscosity [18]. Pure cholesterol monolayers are liquid [13] and have very low surface shear viscosities which are hardly detectable [18]. 

Knowing 0, the solid phase pressure and the solid phase bulk and shear viscosities can be calculated from formulae derived by Tun et al. [1984]. The granular temperature conductivity, k, has also been formulated by Tun et al. [1984]. y has been modeled in terms of 0 by Jenkins and Savage [1983]. For dense regimes, the interphase momentum transfer coefficient, / , can be calculated from the Ergun equation already encountered in Chapter 11 on fixed bed reactors [Gidaspow, 1994]. For dilute regimes, a correlation has been proposed by Wen and Yu [1966]. 

---