Shear gradients, flow/viscosity

When the shear stress changes in Newtonian, dilatant, or pseudoplastic liquids, as well as in Bingham bodies or fluids above the flow limit, the corresponding shear gradient or the corresponding viscosity is reached almost instantaneously. In some liquids, however, a noticeable induction time is necessary, i.e., the viscosity also depends on time. If, at a constant shear stress or constant shear gradient, the viscosity falls as the time increases, then the liquid is termed thixotropic. Liquids are termed rheopectic or antithixotropic, on the other hand, when the apparent viscosity increases with time. Thixotropy is interpreted as a time-dependent collapse of ordered structures. A clear molecular picture for rheopexy is not available. 

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]... 

Utilization of a microfabricated rf coil and gradient set for viscosity measurements has recently been demonstrated [49]. Shown in Figure 4.7.9 is the apparent viscosity of aqueous CMC (carboxymethyl cellulose, sodium salt) solutions with different concentrations and polymer molecular weights as a function of shear rate. These viscosity measurements were made using a microfabricated rf coil and a tube with id = 1.02 mm. The shear stress gradient, established with the flow rate of 1.99 0.03 pL s-1 was sufficient to observe shear thinning behavior of the fluids. 

The Taylor vortices described above are an example of stable secondary flows. At high shear rates the secondary flows become chaotic and turbulent flow occurs. This happens when the inertial forces exceed the viscous forces in the liquid. The Reynolds number gives the value of this ratio and in general is written in terms of the linear liquid velocity, u, the dimension of the shear gradient direction (the gap in a Couette or the radius of a pipe), the liquid density and the viscosity. For a Couette we have ... 

A specially designed thin-film machine can be used to process very viscous, non-Newtonian materials. The apparatus can also be used to remove solvents from polymers and polycondensation processes having viscosities exceeding 10,000 poises. The Luwa thin-film machine has a small clearance between the heated wall and rotor blade. This clearance results in high shear gradients and considerably reduces apparent viscosity. The increased turbulence and improved surface renewal that ensue improve reaction velocities and aid the required forced product flow on the walls of the apparatus. 

In this thin-film machine, the small clearance between heated wall and rotor blade, together with the high peripheral blade velocity, results in high shear gradients, whereby the apparent viscosity in the film is considerably reduced. The resulting increased turbulence and better surface renewal improve heat transfer, increase reaction velocities, and aid the required forced product flow on the wall. On the basis of test... 

The remaining six quantities are called shear stresses. They have two subscripts associated with the coordinates, and are referred to as the components of the molecular momentum flow tensor, or the components of the molecular stress tensor, as they are associated with molecular motion. Usually, the viscous stress tensor, t, and the molecular stress tensor, it, are simply referred to as stress tensors. For a Newtonian fluid, we may express the stresses in terms of velocity gradients and viscosities in Cartesian coordinates as follows ... 

The flow of gases and simple liquids can be described by a single property the shear viscosity or viscosity for short. You can measure the viscosity by shearing the liquid between two parallel plates (as in Figure C4-1). This causes a velocity gradient normal to the direction of the motion (also known as the shear rate). The viscosity is the ratio of the shear stress to the... 

In fact, the fiber contribution to the shear viscosity of a fiber suspension at steady state is modest, at most. The reason is that, without Brownian motion, the fibers quickly rotate in a shear flow until they come to the flow direction in this orientation they contribute little to the viscosity. Of course, the finite aspect ratio of a fiber causes it to occasionally flip through an angle of n in its Jeffery orbit, during which it dissipates energy and contributes more substantially to the viscosity. The contribution of these rotations to the shear viscosity is proportional to the ensemble- or time-averaged quantity (u u ), where is the component of fiber orientation in the flow direction and Uy is the component in the shear gradient direction. Figure 6-21 shows as a function of vL for rods of aspect... 

The issue of slip at the solid-liquid interface has been a topic of much debate [103]. The influence of slip on the frequency of the QCM is discussed in detail in the chapter by M. Urbakh et al. 2006, in this volume. Shp can be very easily integrated into the framework of the multilayer formahsm and we briefly show this connection. We represent slip by a layer close to the solid surface (a film ) with a reduced viscosity. Inside this layer, the shear gradient is increased, leading to the flow profile indicated in Fig. 10. The slope of the profile dv(z)/dz is proportional to o (z)- The slip length, hs, is the difference between the location of the surface and the extrapolated plane of zero shear. One can show that the slip length, b, is given by ... 

Fig. 10 Flow profile above a solid surface with slip. Dashed line the viscosity r] z) increases continuously from a small value at the surface to a somewhat higher value in the bulk. At the surface, the shear gradient is correspondingly increased. Solid line the viscosity is reduced inside a hypothetical discrete layer of thickness df...

Shear stress results as the flowing layers of macromolecules slip past one another. The shear rate is the difference in the rates of flow or the shear gradient-that is, the change in the rates of the flowing layers across the radius of the cross section. The viscosity decreases with increasing shear rate. 

Owing to experimental difficulties, steady-state shear measurements of Ni and 0 2 are relatively rare. Their rate of shear gradients, Ni/y,r] = 012/y usually show a similar dependence ]322]. The value of the complex viscosity ist] >t]. In the steady shear flow of a two-phase system, the stress is continuous across the interphase, but the rate of deformation is not. Thus, for polymer blends, plots of the rheological functions versus stress are more appropriate than those versus rate, that is, a Ni = Ni oi2) plot is similar to G = G (G"). 

B) Newtonian liquid. This type of liquid flows under the influence of stress, and the flow gradient is proportional to the applied stress. The viscosity q of such a liquid, which is defined as the ratio of the stress and the shear gradient, is thus constant and independent of the stress. Many solvents with low molecular weight, among these is water, are Newtonian liquids. 

C) Dilatant liquid (shear thickening). This type of liquid also flows under the influence of stress, but the viscosity increases with increasing stress or shear gradient. For example, several polymer solutions show dilatant properties. 

It is instructive to consider the three simple geometries for plane parallel shear flow, the Miesovicz geometries [73], corresponding to the orientations of the director relative to the flow axis and the shear gradient. Choosing v = u(z)x one has the effective shear viscosities ... 

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