Planar shear flow

In momentum transfer, the constitutive relationship between shear stress, t, and shear rate for simple planar shear flow is given by Newton s law of viscosity ... 

We note that, in spite of the superficial similarity to a unidirectional flow, the solution for uo is actually quite different from the linear profile of a simple, planar shear flow. This is not at all surprising for an arbitrary ratio of the cylinder radii. However, we should expect that the approximation to a simple shear flow should improve as the gap width becomes... 

We have already seen in 3.6.4 that in ordinary nematics of rod-like molecules, the Leslie coefficients and are both negative and pj > l/ig]. Under planar shear flow, the director assumes an equilibrium orientation 00 given by... 

In order to extract both parameters, one can investigate the flow profiles of two qualitatively different tyjres of flows -planar shear flow (the Couette flow) and pressrue-driven flow (the Poiseuille flow) - in a thin film of thickness D. Both flows are characterized by a Unear stress profile such that nonlocal viscosity, which may arise from the extended molecirlar strac-ture,i2 i2 is not expected to affect the resirlts. At the center of the channel, the flow is described by the Navier-Stokes eqrra-tion yielding a linear velocity profile with shear ratey ... 

N. Xu, C. S. O Hern, and L. Kondic. Stabilization of nonlinear velocity profiles in athermal systems undergoing planar shear flow. Phys. Rev. E, 72 041504, 2005. 

Flow is generally classified as shear flow and extensional flow [2]. Simple shear flow is further divided into two categories Steady and unsteady shear flow. Extensional flow also could be steady and unsteady however, it is very difficult to measure steady extensional flow. Unsteady flow conditions are quite often measured. Extensional flow differs from both steady and unsteady simple shear flows in that it is a shear free flow. In extensional flow, the volume of a fluid element must remain constant. Extensional flow can be visualized as occurring when a material is longitudinally stretched as, for example, in fibre spinning. When extension occurs in a single direction, the related flow is termed uniaxial extensional flow. Extension of polymers or fibers can occur in two directions simultaneously, and hence the flow is referred as biaxial extensional or planar extensional flow. 

Simple shear (also known as planar Couette flow) is achieved when fluid is contained between two plane parallel plates in relative in-plane motion. If the velocity direction is taken to be x, one has x = y, all other xa 3 zero and... 

This flow field is somewhat idealized, and cannot be exactly reproduced in practice. For example, near the planar surfaces, shear flow is inevitable, and, of course, the range of % and y is consequently finite, leading to boundary effects in which the extensional flow field is perturbed. Such uniaxial flow is inevitably transient because the surfaces either meet or separate to laboratory scale distances. 

The degree of deformation and whether or not a drop breaks is completely determined by Ca, p, the flow type, and the initial drop shape and orientation. If Ca is less than a critical value, Cacri the initially spherical drop is deformed into a stable ellipsoid. If Ca is greater than Cacrit, a stable drop shape does not exist, so the drop will be continually stretched until it breaks. For linear, steady flows, the critical capillary number, Cacrit, is a function of the flow type and p. Figure 14 shows the dependence of CaCTi, on p for flows between elongational flow and simple shear flow. Bentley and Leal (1986) have shown that for flows with vorticity between simple shear flow and planar elongational flow, Caen, lies between the two curves in Fig. 14. The important points to be noted from Fig. 14 are these ... 

We start with the ground state (°), fi(° defined by the simple shear flow y(°), Fig. 17. The principal effect is, as expected, the appearance of a small tilt of the director from the layer normal (flow alignment), predominantly in z direction (Fig. 18). Note that the configuration of layers is also modified by the shear (Figs. 19 and 20), i.e., the cylindrical symmetry is lost. This is analogous to the shear-flow-induced undulation instability of planar layers (wave vector of undulations in the... 

Fig. 7.23 Critical capillary number for droplet breakup as a function of viscosity ratio p in simple shear and planar elongational flow. [Reprinted by permission from H. P. Grace, Chem. Eng. Commun., 14, 2225 (1971).]...

Winter et al. [119, 120] studied phase changes in the system PS/PVME under planar extensional as well as shear flow. They developed a lubrieated stagnation flow by the impingement of two rectangular jets in a specially built die having hyperbolic walls. Change of the turbidity of the blend was monitored at constant temperature. It has been found that flow-induced miscibility occurred after a duration of the order of seconds or minutes [119]. Miscibility was observed not only in planar extensional flow, but also near the die walls where the blend was subjected to shear flow. Moreover, the period of time required to induce miscibility was found to decrease with increasing flow rate. The LCST of PS/PVME was elevated in extensional flow as much as 12 K [120]. The shift depends on the extension rate, the strain and the blend composition. Flow-induced miscibility has been also found under shear flow between parallel plates when the samples were sheared near the equilibrium coexistence temperature. However, the effect of shear on polymer miscibility turned out to be less dramatic than the effect of extensional flow. The cloud point increased by 6 K at a shear rate of 2.9 s. ... 

Other forms of extensional flow are biaxial extensional flow and planar extensional flow or pure shear flow. 

Planar extensional flow or pure shear flow is extensional flow with the same but opposite rates of strain in two directions in the third direction, there is no flow ... 

Simple shear flow is the flow of the liquid of viscosity r]c over itself along a plane see Figure 15.6a. One encounters simple shear flow during flow through a tube, or flow over a planar surface. A droplet of size Rj subjected to simple shear flow, will be distorted due to the stress exerted on the droplet. The internal, coherent stress can be estimated with the help of the Laplace pressure in the droplets la/Rj, which is the pressure that the interface exerts and keeps the droplet together. The disruptive stress can be estimated through ... 

A few rheometers are available for measurement of equi-biaxial and planar extensional properties polymer melts [62,65,66]. The additional experimental challenges associated with these more complicated flows often preclude their use. In practice, these melt rheological properties are often first estimated from decomposing a shear flow curve into a relaxation spectrum and predicting the properties with a constitutive model appropriate for the extensional flow [54-57]. Predictions may be improved at higher strains with damping factors estimated from either a simple shear or uniaxial extensional flow. The limiting tensile strain or stress at the melt break point are not well predicted by this simple approach. 

While these functions have been adjusted to describe shear and uniaxial extensional flows, they seem to work poorly for planar extension of LDPE (Samurkas et al. 1989). Planar extensional flow represents a particularly difficult test for K-BKZ-type constitutive equations, since fits to shear data fix all the model parameters required for planar extension, and there is therefore no wiggle room left to obtain a fit to the latter. (This is because I = I2 in both shear and planar extension.) A recent non-K-BKZ molecular constitutive equation derived from reptation-related ideas shows improved qualitative agreement with planar extensional data (McLeish and Larson 1998). 

This formula applies to planar extensional flow as well as to shear, if the shear rate y in Eq. (9-11) is replaced by 2e, where is the extension rate. Taylor predicted that droplet breakup should occur when the viscous stresses that deform the droplet overwhelm the surface tension forces that resist deformation this occurs when D reaches a value Db given approximately by... 

It is also possible to calculate the shear viscosities and the twist viscosities by applying the SLLOD equations of motion for planar Couette flow, Eq. (3.9). If we have a velocity field in the x-direction that varies linearly in the z-direction the velocity gradient becomes Vu=ye ej, see Fig. 3. Introducing a director based coordinate system (Cj, C2, 63) where the director points in the e3-direction and the angle between the director and the stream lines is equal to 0, gives the following expression for the strain rate in the director based coordinate system. 

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