Normal stress suspensions

Filler orientation is not the only consequence of geometric effects in strained systems with an anisodiametric filler. The appearance of normal stresses has been named as another such phenomenon [169,171,185] which, like the abnormal viscosity behavior of suspensions, may not be due to the elasticity of system components. 

The refining process involves the circulation of the fibre suspension in such a way as to force the fibres between a stationary metal plate (the stator) and a moving metal plate (the rotor). As the fibres are wet at this stage, both mechanical and hydraulic forces are involved in altering fibre characteristics. Both shear and normal stresses (either tensional or compressive) are imposed on the fibres in this process, and the mechanical action is shown diagramatically in Figure 5.1. 

Concentrated emulsions can exhibit viscoelasticity, as can gelled foams and some suspensions. Compared with the previous equations presented, additional coefficients (including primary and secondary normal stress coefficients) are needed to characterize the rheology of viscoelastic fluids [376,382]. 

Normal stress differences do not exist in the absence of interparticle forces. Moreover, the relative viscosity of the suspension is a function of only particle densities approaching the maximum possible that still allow the suspension to flow, cluster size (and, as a result, the viscosity of the two-dimensional monolayer) appears to scale as... 

Repulsive interparticle forces cause the suspension to manifest non-Newtonian behavior. Detailed calculations reveal that the primary normal stress coefficient [cf. Eq. (8.7)] decreases like y 1. In contrast, the suspension viscosity displays shear-thickening behavior. This feature is again attributed to the enhanced formation of clusters at higher shear rates. 

Other dimensionless groups similar to the Deborah number are sometimes used for special cases. For example, in a steady shearing flow of a polymeric fluid at a shear rate y, the Weissenberg number is defined as Wi = yr. This group takes its name from the discoverer of some unusual effects produced by normal stress differences that exist in polymeric fluids when Wi 1, as discussed in Section 1.4.3. Use of the term Weissenberg number is usually restricted to steady flows, especially shear flows. For suspensions, the Peclet number is defined as the shear rate times a characteristic diffusion time to [see Eq. (6-12) and Section 6.2.2]. 

Experimental measurements of the normal stress differences A i and N2 for suspensions are rare, especially for hard spheres. According to a recent theory by Brady and Vicic (1995), in the dilute regime the low shear-rate values of the normal stress differences are M,o/hoK = O.8996 7r0 Pe and Af2,o/r/oy = —O.78867T0 Pe. These values are quite small at concentrations and shear rates where the theory might apply (0 0.15 Pe 0.1). The... 

Since at steady state the angular distribution of fiber orientations is predicted to be symmetric about the flow direction in a shearing flow, Eq. (6-50) implies that the normal stresses (e.g., a oc [u uy) will be identically zero. However, nonzero positive values of N have frequently been reported for fiber suspensions (Zimsak et al. 1994). Figure 6-24 shows normalized as discussed below, as a function of shear rate for various suspensions of high fiber aspect ratio. These normal stress differences are linear in the shear rate and can be quite large, as high as 0.4 times the shear stress, which is dominated by the contribution of the solvent medium, cr Fig- 6-24, the N] data are normalized... 

For semidilute suspensions, one expects n/cp to replace p in the logarithmic term in Eq, (6-55), but otherwise the expression for N] should be similar to that for dilute suspensions.] Hence, a plot of N / (prjsP ln(p)) versus y will be universal only if C scales as p and is independent of (p this is consistent with neither slender-body theory nor the simulations of Yamane et al. Hence, the effective diffusivity of rigid rods does not seem able to account for the behavior of the measured values of N in fiber suspensions. However, other possible sources may contribute to the first normal stress difference in these suspensions. For example, according to recent simulations, fiber flexibility produces a positive first normal stress difference (Yamamoto and Matsuoka 1995). Other possible sources of nonzero N include interactions of long fibers with rheometer walls, or streamline curvature. 

For very low particle concentrations, the shearing of colloidal crystals produces only a weak stress above that of the solvent. However, in more concentrated suspensions, the shear viscosity and normal stress differences have been found to have quite unusual behavior, which can, in part, be explained by (a) the formation of sliding layers and (b)... 

Laun (1994) has measured the first and second normal stress differences and N2 of suspensions at steady state in the shear-thickened state. He found, remarkably, that Ni is negative and N2 is positive, which are opposite the usual signs for these quantities He also found the following relationship between A i, N2, and the shear stress [Pg.308]

The viscous and elastic properties of orientable particles, especially of long, rod-like particles, are sensitive to particle orientation. Rods that are small enough to be Brownian are usually stiff molecules true particles or fibers are typically many microns long, and hence non-Brownian. The steady-state viscosity of a suspension of Brownian rods is very shear-rate- and concentration-dependent, much more so than non-Brownian fiber suspensions. The existence of significant normal stress differences in non-Brownian fiber suspensions is not yet well understood. 

Problem 6.7(a) (Worked Example) Estimate the first normal stress difference Ni for a suspension of long, thin particles (approximated as spheroids) with p = 100 and L = 0.1/ m, if the solvent viscosity is 1 P, the shear rate y is 100 sec, and the particle concentration is 0 = 0.001, which is in the dilute regime. 

The first of these relations was noted by Lodge [(46), Eq. (6.43)] and by Williams and Bird (77) as a result of the study of two different empirical constitutive equations. Later Spriggs (70) obtained Eqs. (7.22) and (7.23) from the Coleman and Noll (21) theory of second-order viscoelasticity. Eqs. (7.22) and (7.23) indicate that no additional information about fluids can be obtained from normal stress oscillatory measurements than has not already been obtained by shear stress oscillatory data. Eq. (7.24) seems to be new and is probably specific to rigid dumbbell suspensions. 

Fig. 6. Shear and normal stress relaxation for rigid dumbbell suspensions...

In this section we investigate some of the properties of mixtures undergoing steady shearing flow. Specifically we consider the viscosity and normal stress functions for suspensions of rigid dumbbells of various lengths which have the same zero shear rate viscosity as a solution containing dumbbells of length L only. 

Transient Effects In system where the structure changes with time upon imposition of stress, the transient effects are important. For example, semi-concentrated fiber suspensions in shear and extension show large transient peaks in the first and the second normal stress dilference [Dinh and Armstrong, 1984 Bibbo et al., 1985]. It is interesting that the peaks appear at dilferent times, first for N, then for Np and finally for... 

These two expressions correspond to a suspension of a in and a suspension of p in a, respectively. In each case, is the interfacial tension of a droplet of fluid i in the medium j y° is the interfacial tension in the absence of flow a is the droplet radius and 2 the second normal stress function of the fluid (Figure 9.10). 

Normal stresses originally recognized by Weissenberg [W3] through observation of rod climbing effects in soap-hydrocarbon liquid suspensions and polymer solution systems began to be measured on thermoplastics in the 1960s [C9, K9]. White and Tokita [W24] noted their occurrence in gum... 

Beginning with the paper by Jackson [20], disturbance stabilization in a fluidized bed is usually associated with the action of specific normal stresses inherent to the dispersed phase. These stresses impede volume deformations of the dispersed phase. Despite this fact having been understood for a long time, comprehensive development of a stability theory is hindered by the almost total absence of reliable information concerning the dependence of dispersed phase stresses (or of the corresponding bulk moduli of dispersed phase elasticity) on the suspension concentration and on the physical parameters. This lack of information partly invalidates all theoretical inferences bearing upon hydrodynamic stability in suspension flow. 

Thus, in order to render the stability theory completely determinate, we need to specify in an unequivocal form both the conservation equations governing macroscopic suspension flow and all the rheological equations of state. This is easily seen to be possible for coarse dispersions of small particles. For such dispersions, normal stresses in the dispersed phase may be approximately described in terms of the particulate pressure as explained in Section 4, and this pressure can be evaluated for uniform dispersion states with the help of Sections 7 and 8. As a result, particulate pressure appears to be a single-valued function of mean variables characterizing the uniform dispersion state under study and of the physical properties of its phases. This single-valued function involves neither unknown quantities nor arbitrary parameters. On the other hand, if the particle Reynolds number is small, all interphase interaction force constituents also can be expressed in an explicit consummate form with help from the theory in reference [24]. This expression for the fluid-particle interaction force recently has been employed as well in stability studies for flows of collisionless finely dispersed suspensions [15,60]. 

Housiadas and Tanner (2009), following the approach of Greco et al. (2005), have used a perturbation analysis to obtain the analytical solution for the pressure and the velocity field up to 0 (pDe) of a dilute suspension of rigid spheres in a weakly viscoelastic fluid, where

volume fraction of the spheres and De is the Deborah number of the viscoelastic fluid. The analytical solution was used to calculate the bulk first and second normal stress in simple shear flows and the elongational viscosity. The main results are... 

Mackley MR, Waimaborwom S, Gao P, Zhao F (1999) The optical microscopy of sheared liquids using a newly developed optical stage. J Microsc Anal 69 25—27 Mall-Gleissle SE, Gleissle W, McKinley GH, Buggisch H (2002) The normal stress behavior of suspensions with viscoelastic matrix fluids. Rheol Aeta 41 61—76 Marand H, Xu J, Srinivas S (1998) Determination of the equilibrium melting temperatme of polymer erystals linear and non-linear Hoffman-Weeks extrapolation. Macromol 31 8219-8229... 

Zarraga IE, Hill DA, Leighton D (2001) Normal stress and free surface deformation in concentrated suspensions of noncolloidal spheres in a viscoelastic fluid. J Rheol 45 1065-1084... 

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