Viscosity shear rate dependence

For polymer melts or solutions, Graessley [40-42] has shown that for a random coil molecule with a Gaussian segment distribution and a uniform number of segments per unit volume, a shear rate dependent viscosity arises. This effect is attributed to shear-induced entanglement scission. 

Polymers in solution or as melts exhibit a shear rate dependent viscosity above a critical shear rate, ycrit. The region in which the viscosity is a decreasing function of shear rate is called the non-Newtonian or power-law region. As the concentration increases, for constant molar mass, the value of ycrit is shifted to lower shear rates. Below ycrit the solution viscosity is independent of shear rate and is called the zero-shear viscosity, q0. Flow curves (plots of log q vs. log y) for a very high molar mass polystyrene in toluene at various concentrations are presented in Fig. 9. The transition from the shear-rate independent to the shear-rate dependent viscosity occurs over a relatively small region due to the narrow molar mass distribution of the PS sample. 

For a precise analysis of the shear rate dependent viscosity it is necessary to know at which critical rate of deformation shear-induced disturbance can no longer be leveled out by the recoil of the polymers. 

It is possible to approximate the shear rate dependent viscosity at any rate of deformation (y> ycrit), to such an extent that degradation may be neglected. 

The coordinates (x, y, z) define the (velocity, gradient, vorticity) axes, respectively. For non-Newtonian viscoelastic liquids, such flow results not only in shear stress, but in anisotropic normal stresses, describable by the first and second normal stress differences (oxx-Oyy) and (o - ozz). The shear-rate dependent viscosity and normal stress coefficients are then [1]... 

The usual approach for non-Newtonian fluids is to start with known results for Newtonian fluids and modify them to account for the non-Newtonian properties. For example, the definition of the Reynolds number for a power law fluid can be obtained by replacing the viscosity in the Newtonian definition by an appropriate shear rate dependent viscosity function. If the characteristic shear rate for flow over a sphere is taken to be V/d, for example, then the power law viscosity function becomes... 

Now for a power iaw fluid the viscosity n is repiaced by the shear rate dependent viscosity q. 

In the in situ consolidation model of Liu [26], the Lee-Springer intimate contact model was modified to account for the effects of shear rate-dependent viscosity of the non-Newtonian matrix resin and included a contact model to estimate the size of the contact area between the roller and the composite. The authors also considered lateral expansion of the composite tow, which can lead to gaps and/or laps between adjacent tows. For constant temperature and loading conditions, their analysis can be integrated exactly to give the expression developed by Wang and Gutowski [27]. In fact, the expression for lateral expansion was used to fit tow compression data to determine the temperature dependent non-Newtonian viscosity and the power law exponent of the fiber-matrix mixture. 

Fig. 3.5 Logarithmic plot of the shear rate-dependent viscosity of a narrow molecular weight distribution PS (A) at 180°C, showing the Newtonian plateau and the Power Law regions and a broad distribution PS( ). [Reprinted with permission from W. W. Graessley et al., Trans. Soc. Rheol., 14, 519 (1970).]...

EitherEq. E3.1-9or Eq. E3.1-10, known as the Rabinowitsch or Weissenberg-Rabinowitsch equations, can be used to determine the shear rate at the wail yw by measuring Q and AP or r and Tw (21). Thus, in Eq. E3.1-4 both xw and yw can be experimentally measured for any fluid having a shear rate-dependent viscosity as long as it does not slip at the capillary wall. Therefore, the viscosity function can be obtained. 

Kalashnikov VN (1994) Shear-rate dependent viscosity of dilute polymer solutions. J Rheol... 

SHEARING FLOW. Figure 3-14 shows the shear-rate-dependent viscosities of polystyrenes of various molecular weights in a couple of low-viscosity solvents, decalin and toluene (Noda et al. 1968 see also Kotaka et al. 1966). Plotted is the intrinsic relative viscosity, [r] l[r] o, against a dimensionless shear rate,... 

Velocity maps were measured with How-compensated spin-echo sequences [Rof l], The expefimental data were analysed in terms of a power law for the shear stress a[2 which solves the constitutive equation (T 2 = f)(dy/d/)" for the shear-rate-dependent viscosity rj and the applied shear strain y. The azimuthal velocity at radius r and rotation speed wr of the inner cylinder can be expressed as... 

Internal Viscosity and Cerf-Peterlin Theory. The concept of the internal viscosity was first employed by Kuhn and Kuhn 120) in an attempt to describe the shear-rate dependent viscosity with a dumbbell model or a bead-spring model with N = 1. They assumed that a force proportional to the relative velocity of the beads is exerted on the bead from the connector (spring) in addition to the spring force which is proportional to the relative position of the beads. This force intrinsic to the polymer molecule is compared with the frictional force from the viscous medium and is associated with the term internal viscosity . 

An important distinction between polymeric liquids and suspensions arises from their different microstructures and is evidenced by the elastic recoil phenomena that polymers exhibit but suspensions do not. The polymeric or macromolecular system when deformed under stress will recover from very large strains because like an elastic material the restoring force increases with the deformation. With a suspension, however, the forces between the particles decrease with increasing separation so that there is limited mechanism for recovery. There are, however, a variety of rheological properties common to polymeric liquids that suspensions will exhibit including shear rate dependent viscosity and time-dependent behavior. We shall discuss these differences in more detail in the following section. 

To accommodate non-Newtonian effects, a realistic constitutive equation is used for the shear-rate dependent viscosity (21-23). 

Y) where ti(y) is the shear rate dependent viscosity of the is, it follows ... 

The rheological parameters of primary scientific and practical concern are the static and dynamic shear modulus, the yield stress, and the shear rate-dependent viscosity. The aim is to understand and predict how these depend on the system parameters. In order to accomplish this with any hope of success, there are two areas that need to be emphasized. First, the systems studied must be characterized as accurately as possible in terms of the volume fraction of the dispersed phase, the mean drop size and drop size distribution, the interfacial tension, and the two bulk-phase viscosities. Second, the rheological evaluation must be carried out as reliably as possible. 

Numerous microscale devices and techniques have been developed to characterize the flow behavior of polymer solutions. The principal motivation for this broad class of techniques is to enable characterization of tiny liquid volumes for which samples are costly or difficult to obtain in large quantities. These microfluidic rheometers fall into three categories, organized in order of increasing De devices to measure intrinsic viscosity, shear-rate-dependent viscosity, and non-Newtonian behavior for a range of flow types. 

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