Shear Viscosity Coefficients

The quantities a, c, f, F, r, and p are the thermal diffusivity, sound speed, heat capacity ratio, bulk viscosity coefficient, shear viscosity coefficient, and density of the sample, respectively and Eo, a, P and Cp are the energy fluence of the laser beam, the optical absorption coefficient, the volume expansion coefficient, and the isobaric heat capacity, respectively, of the fluid. Tlie first and second terms in Eq. 2 describe the time dependences of the thermal and acoustic modes of wave motion, respectively. Since the decays of the acoustic and thermal mode densities back to their ambient values take place on such different time scales (microsecond time scale for acoustic mode and millisecond time scale for thermal mode), they were recorded on the oscilloscope using different time bases. 

Viscosity is equal to the slope of the flow curve, Tf = dr/dj. The quantity r/y is the viscosity Tj for a Newtonian Hquid and the apparent viscosity Tj for a non-Newtonian Hquid. The kinematic viscosity is the viscosity coefficient divided by the density, ly = tj/p. The fluidity is the reciprocal of the viscosity, (j) = 1/rj. The common units for viscosity, dyne seconds per square centimeter ((dyn-s)/cm ) or grams per centimeter second ((g/(cm-s)), called poise, which is usually expressed as centipoise (cP), have been replaced by the SI units of pascal seconds, ie, Pa-s and mPa-s, where 1 mPa-s = 1 cP. In the same manner the shear stress units of dynes per square centimeter, dyn/cmhave been replaced by Pascals, where 10 dyn/cm = 1 Pa, and newtons per square meter, where 1 N/m = 1 Pa. Shear rate is AH/AX, or length /time/length, so that values are given as per second (s ) in both systems. The SI units for kinematic viscosity are square centimeters per second, cm /s, ie, Stokes (St), and square millimeters per second, mm /s, ie, centistokes (cSt). Information is available for the official Society of Rheology nomenclature and units for a wide range of rheological parameters (11). 

On the other hand, when a liquid is sheared between two planes, and there is bonding with the planes, the bonds transfer momentum from the faster plane to the slower one. This is the liquid-like mode. In this case, the viscosity coefficient decreases with increasing temperature. 

The coefficient, p, of the viscosity resisting dislocation motion is the shear stress at the glide plane, x divided by the frequency of momentum transfer, v. The maximum value that x can have is about Coct/47i, and as mentioned above v = 1013/sec for the Al atoms,so p = Coct./47rv = 4x 10 3 Poise.This is comparable to the dislocation viscosity coefficients in other metallic systems. Another view of the viscosity is Andrade s theory in which ... 

POISE (P). A unit of dynamic viscosity. The unit is expressed in dyne second per square centimeter The centipoise (cP) is more commonly used The formal definition of viscosity arises from the concept put forward by Newton that under conditions of parallel flow, the shearing stress is proportional to the velocity giadieut. If lire force acting on each of two planes of aiea A parallel to each oilier, moving parallel lo each other with a relative velocity V, and separated by a perpendicular distance X, be denoted by F. the shearing stress is F/A and the velocity gradient, which will be linear for a true liquid, is V/X. Thus, Ft A = q V/X, where the constant if is the viscosity coefficient or dynamic viscosity of the liquid. The poise is the CGS unit of dynamic viscosity. 

The molecular theory of Doi [63,166] has been successfully applied to the description of many nonlinear rheological phenomena in PLCs. This theory assumes an un-textured monodomain and describes the molecular scale orientation of rigid rod molecules subject to the combined influence of hydrodynamic and Brownian torques, along with a potential of interaction (a Maier-Saupe potential is used) to account for the tendency for nematic alignment of the molecules. This theory is able to predict shear thinning viscosity, as well as predictions of the Leslie viscosity coefficients used in the LE theory. The original calculations by Doi for this model employed a preaveraging approximation that was later... 

The quantity rj in set (9.58) represents the shear viscosity coefficient and depends on the invariant of the anisotropy tensor in the same way as the relaxation time i... 

Equations (9.63)-(9.65) were used, in fact, to evaluate the shear viscosity coefficient and the relaxation times which reveal the nature of dependence on the velocity gradient v 12 or shear stress oyi (Isayev 1973). It is convenient to consider the shear viscosity coefficient and the relaxation time as a generalised function of the first invariant of the tensor of additional stresses... 

Figure 19. The ratio of elongational to shear viscosities The theoretical dependence of the ratio of elongational to shear viscosity coefficients on the invariant of the additional stress tensor is calculated according to equation (9.71) and depicted by the dashed curve. The solid curves represent experimental data for systems listed in Table 3. Adapted from the paper of Pokrovskii and Kruchinin (1980).

The relationship between shearing stress and rate of shear can be used to define the flow properties of materials. In the simplest case, the shearing stress is directly proportional to the mean rate of shear x = fly (Figure 8-5). The proportionality constant T is called the viscosity coefficient, or dynamic viscosity, or simply the viscosity of the liquid. The metric unit of viscosity is the dyne.s cm-2, or Poise (P). The commonly used unit is 100 times smaller and called centiPoise (cP). In the SI system, t is expressed in N.s/m2. or... 

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. 

Temperature displacements of curves flow along the viscosity and shear stress axles Concentration displacements of curves flow along the viscosity and shear stress axles empiristic coefficient (Ex 3)... 

By Newton s definition the viscosity or, more appropriately, the viscosity coefficient, jj, of a fluid in a laminar steady-state flow is expressed as the tangential force, F, per unit area. A, required to maintain a unit rate of shear (or velocity gradient), G, in the liquid. If the liquid fills the space between two parallel planes of area. A, one of which moves at a constant distance, r, from the other with a relative velocity, u, then we have... 

Sarman and Evans [24, 32] performed a comprehensive study of the flow properties of a variant of the Gay-Beme fluid. In order to make the calculations faster the Lennard-Jones core of the Gay-Beme potential was replaced by a 1/r core. This makes the potential more short ranged thereby decreasing the number of interactions and making the simulation faster. The viscosity coefficients were evaluated by EMD Green-Kubo methods both in the conventional canonical ensemble and in the fixed director ensemble. The results were cross checked by shear flow simulations. The studies covered nematic phases of both prolate ellipsoids with a length to width ratio of 3 1 and oblate ellipsoids with a length to width ratio of 1 3. The complete set of potential parameters for these model systems are given in Appendix II. 

Newtonian Fluid or Emulsion A fluid or emulsion whose rheological behavior is described by Newton s law of viscosity. Here shear stress is set proportional to the shear rate. The proportionality constant is the coefficient... 

To close this Section we comment on two papers that do not fit under any neat heading. The first of these is by Xiao et al,261 who study the final stages of the collapse of an unstable bubble or cavity using MD simulations of an equilibrated Lennard-Jones fluid from which a sphere of molecules has been removed. They find that the temperature inside this bubble can reach up to an equivalent of 6000 K for water. It is at these temperatures that sonolumines-cence is observed experimentally. The mechanism of bubble collapse is found to be oscillatory in time, in agreement with classical hydrodynamics predictions and experimental observation. The second paper, by Lue,262 studies the collision statistics of hard hypersphere fluids by MD in 3, 4 and 5 dimensions. Equations of state, self-diffusion coefficients, shear viscosities and thermal conductivities are determined as functions of density. Exact expressions for the mean-free path in terms of the average collision time and the compressibility factor in terms of collision rate are also derived. Work such as this, abstract as it may appear, may be valuable in the development of microscopic theories of fluid transport as well as provide insight into transport processes in general. 

Equilibrium MD simulations of self-diffusion coefficients, shear viscosity, and electrical conductivity for C mim][Cl] at different temperatures were carried out [82] The Green-Kubo relations were employed to evaluate the transport coefficients. Compared to experiment, the model underestimated the conductivity and self-diffusion, whereas the viscosity was over-predicted. These discrepancies were explained on the basis of the rigidity and lack of polarizability of the model [82], Despite this, the experimental trends with temperature were remarkably well reproduced. The simulations reproduced remarkably well the slope of the Walden plots obtained from experimental data and confirmed that temperature does not alter appreciably the extent of ion pairing [82],... 

Assuming that Ci, C2, and C3 are not affected by temperature, pressure, and dissolution of CO2, they can be determined from a viscosity-shear rate curve of the neat polymer. Namely, the coefficient, Ci, which is equivalent to n - 1, can be determined by the slope of the viscosity and shear rate curve. The values of C2 and C3 can be determined from data of viscosity vs. free volume fraction of the neat polymer. The data of free volume fraction required for determining C2 and C3 can be obtained from PVT data of the neat polymer at temperatures and pressures where the viscosity measurements of the neat polymer are performed. 

In this test, the viscosity remained higher than 40,000 mPa s after being aged at 90°C for 100 hours. That means the gel solutions were thermally stable by 90°C. Being sheared at 3000 r/min for 15 minutes, the gel solutions lost 87 to 89% of their viscosity. After shearing was stopped, the gel viscosities were restored to 70 to 85% of the unsheared viscosity. Using the reservoir cores of 973 md, the flood tests showed that the plugging rate of 88 to 96% and the residual resistance coefficient of 16.2 to 28.6 were obtained after 10 PV of gel injection. In a three-layered artificial core of 1000 md permeability and 0.72 permeability variation coefficient, the incremental recovery factor of gel treatment was 0.4 to 0.93% OOIP. 

Upon introducing the Kirkwood Eq. 40 into Eq. 39, and comparing with like terms in the phenomenological pressure tensor, the shear and volume viscosity coefficients are immediately obtained ... 

It is now possible to calculate a shear viscosity coefficient for actual liquids composed of spherical molecules. By assuming an... 

It will be noted that a function analogous to ipz which arises in the shear viscosity, does not appear in the thermal conductivity. Its absence, of course, is due to the lack of second order surface harmonic perturbation of the radial distribution function g0m in the case of heat conduction. It may be anticipated that this difference in the form of the number density perturbation might lead to the thermal conductivity coefficient leaving a functional dependence on the temperature which is quite different from that of the shear viscosity coefficient. However, the exact temperature dependence of the two coefficients (Eqs. 42 and 47) has not yet been explored. 

Zwanzig, Kirkwood, Oppenheim, and Alder38 have numerically evaluated Eq. 47 for liquid argon at its normal boiling point, using the same Bom-Green radial distribution function g0<2) and the same numerical values for the frictional coefficient and for the constants in the intermolecular potential function value obtained was x — 4.1 x 10-4 cal/g—sec—°K which is in reasonable agreement with the experimental value of 2.9 x 10-4. 

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