Shear strain rate, viscosity

Glasses, like metals, are formed by deformation. Liquid metals have a low viscosity (about the same as that of water), and transform discontinuously to a solid when they are cast and cooled. The viscosity of glasses falls slowly and continuously as they are heated. Viscosity is defined in the way shown in Fig. 19.7. If a shear stress is applied to the hot glass, it shears at a shear strain rate 7. Then the viscosity, ij, is defined by... 

Dilute polyelectrolyte solutions, such as solutions of tobacco mosaic virus (TMV) in water and other solvents, are known to exhibit interesting dynamic properties, such as a plateau in viscosity against concentration curve at very low concentration [196]. It also shows a shear thinning at a shear strain rate which is inverse of the relaxation time obtained from the Cole-Cole plot of frequency dependence of the shear modulus, G(co). 

The viscosity of a fluid, rj, is defined in terms of a test in which it is sheared. The viscosity is the ratio of the shear stress to the shearing strain rate y,r] = x/y. The strain rate, y, is the rate of shearing between two planes divided by the distance between them. Determine the SI units for viscosity. 

According to the change of strain rate versus stress the response of the material can be categorized as linear, non-linear, or plastic. When linear response take place the material is categorized as a Newtonian. When the material is considered as Newtonian, the stress is linearly proportional to the strain rate. Then the material exhibits a non-linear response to the strain rate, it is categorized as Non Newtonian material. There is also an interesting case where the viscosity decreases as the shear/strain rate remains constant. This kind of materials are known as thixotropic deformation is observed when the stress is independent of the strain rate [2,3], In some cases viscoelastic materials behave as rubbers. In fact, in the case of many polymers specially those with crosslinking, rubber elasticity is observed. In these systems hysteresis, stress relaxation and creep take place. 

For elastic bodies, the shear stress is related to the shear strain by the shear modulus. For viscous fluids, the shear stress is related to the shear strain rate by the viscosity. We note that for laminar viscous flow in a Margules viscometer (Figure 10.7), radial fluid displacement is zero (gr = 0). Thus, differentiating with respect to time ... 

Systems approach borrowed from the optimization and control communities can be used to achieve various other tasks of interest in multiscale simulation. For example, Hurst and Wen (2005) have recently considered shear viscosity as a scalar input/output map from shear stress to shear strain rate, and estimated the viscosity from the frequency response of the system by performing short, non-equilibrium MD. Multiscale model reduction, along with optimal control and design strategies, offers substantial promise for engineering systems. Intensive work on this topic is therefore expected in the near future. 

The samples were sheared using a rotational viscometer with a coaxial cylinder system, based on the Searle-type, where the inner cylinder (connected to a sensor system) rotates while the outer cylinder remains stationary. The outer cylinder surrounding the inner one was jacketed, allowing good temperature control, and the annular gap was of constant width. The sensor system used was the NV type, with a rotor with a recommended viscosity range of 2x10 mPa, a maximum recommended shear stress of 178 Pa, and a maximum recommended shear strain rate of 2700 s this rotor could work with volumes from 10-50 ml. Flow was laminar. 

Rheocalic V2.4. The Bingham mathematic model was used to determine viscosity. The Bingham equation is t = Tq + ly. Where t is the shear stress applied to the material, y is the shear strain rate (also called the strain gradient). To is the yield stress and p is the plastic viscosity. 

The flow of liquids or semisolids is described by viscosity, or, more precisely, by shear viscosity (unit Pa sec). The viscosity defines the resistance of the material against flow. Viscosity is not a coefficient, because it is a function of the shear strain rate y [ti = /(y)]. In the classical fluid mechanics, the dynamic viscosity is obtained using a viscometer. (A viscometer is a rheometer, i.e., an instrument for the measurement of rheological properties, limited to... 

The relationships between stress, strain, and viscosity are usually depicted in the so-called rheograms. In the pharmaceutical sciences, typical flow curves are presented, i.e., x = /(y). In the engineering sciences, the viscosity is usually drawn as a function of the shear stress [vj = f(x)]. This is sensible as most viscometers control the shear stress applied rather than the shear strain rate. However, the entity of interest is the viscosity as a function of the shear strain rate [i] = /(y)]. 

Newtonian behavior can be observed only for ideally viscous bodies. The flow curve shows a direct proportionality between shear stress and shear strain rate (Fig. 4) with the straight line going through zero. The viscosity remains constant over the complete range of shear stresses applied and is independent of the shear strain rate (Fig. 5). The stress in the material goes back... 

Fig. 5 Viscosity as a function of shear strain rate for a Newtonian (a) and a Bingham body (b).

Shear thickening materials show an increase in viscosity with increasing shear strain rate. An idealized flow curve is presented in Fig. 6, and the viscosity as a function of shear strain rate is depicted in Fig. 7. The shear thinning region usually extends only over about one decade of shear rate (power law index n > 1) in contrast to shear thinning, which usually covers at least two or three decades. Also, in many cases, shear thickening is preceded by a short phase of shear thinning at low shear strain rates. ° ... 

The Bingham body model describes materials with an apparent yield strength above which Newtonian flow is observed. This is illustrated in Figs. 4 and 5, which show a typical flow curve and viscosity as a function of shear strain rate, respectively. 

Flocculated suspensions often have high viscosities at low shear strain rates, which give the impression of the existence of a yield stress. The nearly Newtonian flow observed is due to a dramatic reduction in the effective number of particles as a result of the particle association process. 

Materials can show linear and nonlinear viscoelastic behavior. If the response of the sample (e.g., shear strain rate) is proportional to the strength of the defined signal (e.g., shear stress), i.e., if the superposition principle applies, then the measurements were undertaken in the linear viscoelastic range. For example, the increase in shear stress by a factor of two will double the shear strain rate. All differential equations (for example, Eq. (13)) are linear. The constants in these equations, such as viscosity or modulus of rigidity, will not change when the experimental parameters are varied. As a consequence, the range in which the experimental variables can be modified is usually quite small. It is important that the experimenter checks that the test variables indeed lie in the linear viscoelastic region. If this is achieved, the quality control of materials on the basis of viscoelastic properties is much more reproducible than the use of simple viscosity measurements. Non-linear viscoelasticity experiments are more difficult to model and hence rarely used compared to linear viscoelasticity models. 

However, the use of the parallel plate geometry is not recommended for viscosity measurements, because the shear strain rate variation along the gap between the plates is larger than that experienced in concentric cylinder systems.However, there might be advantages when using the geometry in oscillatory studies. ... 

Shear thinning of concentrated suspensions is typical for submicron particles dispersed in a low viscosity Newtonian fluid.At low shear strain rates. Brownian motion leads to a random distribution of the particles in the suspension, and particle collision will result in viscous behavior. At high shear strain rates, however, particles will arrange in layers, which can slide over each other in the direction of flow. This results in a reduced viscosity of the system in agreement with the principles of shear thinning. A pro-noimced apparent yield stress can be found for shear thinning suspensions, if the Brownian motion is suppressed by electrostatic repulsion forces, which result in three-dimensional crystal-like structures of the particles with low mobility. 

After shearing for a longer time, a steady state interfacial stress was obtained and from this value and the average shear strain rate over the interface an apparent surface shear viscosity was calculated by ... 

Appendix B explains how polymer melt flow curves can be derived, and defines apparent (shear) viscosity. It is difficult to correlate the apparent viscosity with a single molecular weight average, because it depends on the width of the molecular weight distribution. However, in the limit of very low shear strain rates 7, when the entanglements between polymer chains produce negligible molecular extension, the apparent viscosity approaches a limiting value... 

The apparent shear viscosity decreases as the shear strain rate increases. 

To represent the above-named Bingham viscosity in the simulation model the possibility to characterize Non-Newtonian fluids in the CFD-code is necessary. Most commercial CFD-Software allocate a so-called Power-Law-Model whereby (26) may be possibly specified. Otherwise the Bingham model has to be implemented in the actual internal expression language, e.g. an executive Fortran-Routine in ANSYS/FLOTRAN or a definition in the CFX-Expression Language (CEL). In either case it is inevitable for the CFD-code to have dynamic access to the (system-) variable for the shear strain rate y. 

These relationships are known as Newton s Law of viscous flow a is termed the fluidity and -q the dynamical shear viscosity. Newton s Law is analogous to Hooke s Law, except shear strain has been replaced by shear strain rate and the shear modulus by shear viscosity. As shown later, this analogy is often very important in solving viscoelastic problems. In uniaxial tension, the viscous equivalent to Hooke s Law would be a=7] ds/dt), where q is the uniaxial viscosity. As v=0.5 for many fluids, this equation can be re-written as <7-=3Tj(de/dO using t7=t /[2(1+v)], the latter equation being the equivalent of the interrelationship between three engineering elastic constants, (fi=E/[2il + v)]). 

From the data of Figure 7.13, the apparent viscosity of potyCmetltyl methacrylate) at 190°C and at a shear stress of 1(X) kPa is 3.9x 10 Pas. The shear stress is uniform through the layer of melt and hence, even for this non-Newtonian fluid, the shear strain-rate y will be uniform with a value... 

Shear rate viscosity (shear strain or stress) behavior is fundamental to controlling melt flow rheology. At low strain rate viscosity, the zero strain rate viscosity is directly related to extensional (stretching) strain... 

At low shear rates, polymeric liquid properties are characterized by two constitutive parameters zero shear rate viscosity t]o and recoverable shear compliance Jq, which indicates fluid elasticity. At higher shear strain rates, rheological behavior is measured with a viscometer. Extensional strain viscosity, associated with extensional flow, occurs with film extrusion. 

Most polymer processes are dominated by the shear strain rate. Consequently, the viscosity used to characterize the fluid is based on shear deformation measurement devices. The rheological models that are used for these types of flows are usually termed Generalized Newtonian Fluids (GNF). In a GNF model, the stress in a fluid is dependent on the second invariant of the stain rate tensor, which is approximated by the shear rate in most shear dominated flows. The temperature dependence of GNF fluids is generally included in the coefficients of the viscosity model. Various models are currently being used to represent the temperature and strain rate dependence of the viscosity. 

Here, tmr H) is the yield stress caused by the apphed magnetic field H, j is the shear rate and r/p is the field-independent plastic viscosity defined as the slope of the measured shear stress against the shear strain rate. 

The most important flow process in polymer liquids is shear flow. Polymer liquids differ from simple liquids, first in that the shear viscosity is invariably extremely large, and second in that Newton s empirical equation giving a linear relationship between shear stress r and shear strain rate y with constant shear viscosity ft... 

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