Shear rate definition

Obviously shear rate in different parts of a mixing tank are different, and therefore there are several definitions of shear rate (/) for average shear rate in the impeller region, oc V, the proportionaUty constant varies between 8 and 14 for all impeller types (2) maximum shear rate, oc tip speed (%NU), occurs near the blade tip (3) average shear rate in the entire tank is an order of magnitude less than case / and (4) minimum shear rate is about 25% of case 3. 

This leads to a definition of apparent viscosity as the ratio of shear stress to apparent shear rate... 

Working in terms of the apparent viscosity /rw, at the wall shear rate, by definition ... 

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... 

The shear rate y can be extracted from equation 3.17 by differentiating with respect to r. Moreover, if a definite integral is differentiated wrt the upper limit (here O, the result is the integrand evaluated at the upper limit. It is convenient first to multiply equation 3.17 by throughout, then differentiating wrt tw gives... 

Another definition is based, not on the true shear rate at the wall, but on the flow characteristic. This quantity, which may be called the apparent viscosity for pipe flow, is given by... 

The science that deals with the deformation and flow of matter is called rheology. An important rheological concept is the shear force, sometimes called the shear stress, or the force that causes a layer of a fluid material to flow over a layer of stationary material. The rate at which a layer of a fluid material flows over a layer of stationary material is called the shear rate. A fluid flowing through a tube, for example, would be the fluid material, while the tube wall would be the stationary material. An important rheological measurement that is closely related to the resistance to flow is called viscosity. The technical definition of viscosity is the ratio of shear stress to shear rate ... 

Liu and Neeld (78) used VisiMix software to calculate shear rates in laboratory, pilot plant, and production scale vessels. Their results (Table 3) showed marked differences, by as much as two orders of magnitude, in the shear rates calculated in the conventional manner [from tip speed and the distance from impeller tip to baffle, i.e., y = ND/ T — Z))] and the shear rates computed by VisiMix. The latter s markedly higher shear rates resulted from VisiMix s definition of the shear rate in terms of Kolmogorov s model of turbulence and the distribution of flow velocities. Note that VisiMix s estimates of the respective shear rates in the vicinity of the impeller blade are comparable at all scales while the shear rates in the bulk volume or near... 

By definition the term non-Newtonian encompasses all materials which do not obey the direct proportionality between shear stress and shear rate depicted by Eq. (1). 

Figure 8.14 shows the result of combining the correlations of /30 (Fig. 8.13) and JeR (Fig. 5.18). The product (30Jcr is remarkably independent of concentration and molecular weight from infinite dilution all the way to the undiluted melt state and shows no substantial variation from one polymer to another. For the particular definition of critical reduced shear rate y0 used here, the experimental result for narrow distribution linear chains can be expressed as...

When this expression is derived, it is notized that sin 2 becomes equal to unity in the limit of zero shear rate (cf. Fig. 1.2). It should be clear that the indicated limits of zero shear rate and concentration are bom out of the conventional definitions of [n and [rj] and are superfluous for the present purpose. 

Data from viscometers are often presented as a linear plot of shear stress versus shear rate, sometimes called a rheogram (Figure HI.1.2). This type of plot allows the viewer to see directly if there is Newtonian behavior because the plot will take the form of a straight line through the origin. A non-Newtonian response is, by definition, nonlinear and may or may not pass through the origin. If the sample has an apparent yield stress, then the line or curve will... 

Continuous Mixers In continuous mixers, exiting fluid particles experience both different shear rate histories and residence times therefore they have acquired different strains. Following the considerations outlined previously and parallel to the definition of residence-time distribution function, the SDF for a continuous mixer/(y) dy is defined as the fraction of exiting flow rate that experienced a strain between y and y I dy, or it is the probability of an entering fluid particle to acquire strain y. The cumulative SDF, F(y), defined by... 

FIGURE 10 Illustration of average velocity from the radial discharge of a radial flow impeller, showing the definition of fluid shear rate (AV/AY). 

The equation leads to the definition of a time and strain-dependent memory fimction which can be further factorized into a time-dependent part (the linear memory function) and a strain-dependent damping function. Though on one hand, there is some experimental evidence for this in limited time ranges, on the other hand, a few experiments might question this strong hypothesis since, for example, the damping function obtained fi um step shear rate data is found to be different from that in step shear strain. 

As per the definition of a Newtonian fluid, the shear stress, ct, and the shear rate, y, are proportional to each other, and a single parameter, t], the viscosity. 

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