Shear flow between parallel plates

Using (5.137) to substitute for 71 in (5.122) and then multiplying throughout by d(t)/dy gives an equation analogous to (5.126) with 0oo replaced by j . Integrating this resultant equation from y to zero and appl3dng the conditions (5.136) leads to a result similar to equation (5.128), namely, 

By symmetry, the solution for the range y h obeys the relation (5,138) with the negative square root being chosen, and this ultimately leads, by the same reasoning, to the solution on this interval being obtained by simply replacing y with —y in the solution (5.139). Therefore 

As in the previous subsection, the constant c is essentially the shear stress iu because a = 0 in (5.108), so that equation (5.131) holds as before. When c is known, then the angle 0o in the middle of the sample at y = 0 is determined via (5.139) and (5.140) by the relation 

This means that the maximum angle 0o of orientation of the director within the sample depends upon the quantity h /c = 

After the solution (j) has been determined by equations (5.139) and (5.141), the solution for the velocity can be evaluated by integrating equation (5.121) to give 

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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. ... 

Figure 10,17 Birefringence interference patterns in torsional shearing flow between parallel plates of MBBA, illuminated by monochromatic light and observed between crossed polarizers. The images on the left were obtained at a rotation speed, of 2.09 x 10 sec , while those on the right were obtained at 4.18 x 10 sec >, From top to bottom, the gap is 44, 94, and 194 ixm. (From Wahl and Fischer, reprinted with permission from Mol. Cryst. Liq. Cryst. 22 359, Copyright 1973, Gordon and Breach Publishers.)...

A smaller r results in better distributive mixing in the laminar flow field. The shear rate and residence time in shear flow between parallel plates where one plate is moving can be written as ... 

In this Section we describe in detail the shear flow examples discussed by Leslie [163] for nematic liquid crystals. We first make some comments on Newtonian and non-Newtonian behaviour of fluids in Section 5.5.1 before going on to derive the general explicit governing equations for shear flow, equations (5.121) and (5.122), in Section 5.5.2. We then specialise in Sections 5.5.3 and 5.5.4 to the specific problems of shear flow near a boundary and shear flow between parallel plates. Section 5.5.5 discusses some scaling properties for nematics. 

FIGURE 7.1 Arrangement for producing laminar, unidirectional shear flow between parallel plates (see Table 7.1). 

FIGURE 8.11 Torsional shear flow between parallel plates. 

Note 4 Some experimental methods, such as capillary flow and flow between parallel plates, employ a range of shear rates. The value of tj evaluated at some nominal average value of Y is termed the apparent viscosity and given the symbol /app. It should be noted that this is an imprecisely defined quantity. 

The velocity field between the cone and the plate is visualized as that of liquid cones described by 0-constant planes, rotating rigidly about the cone axis with an angular velocity that increases from zero at the stationary plate to 0 at the rotating cone surface (23). The resulting flow is a unidirectional shear flow. Moreover, because of the very small i//0 (about 1°—4°), locally (at fixed r) the flow can be considered to be like a torsional flow between parallel plates (i.e., the liquid cones become disks). Thus... 

The preceding relationship establishes that the cone-and-plate flow is viscometric, where cf> is direction 1, that is, the direction of motion, 6 is direction 2, that is, the direction in which the velocity changes, and r is direction 3, that is, the neutral direction. Furthermore, the flow field is such that shear rate is constant in the entire flow field, as it is in the flow between parallel plates. 

Inherent Errors in Using the Power Law Model in Pressure Flows The shear rate during pressure flow between parallel plates varies from zero at the center to maximum shear rate at the wall, yw. Most polymer melts show Newtonian behavior at low shear rates, hence using the Power Law model for calculating flow rate introduces a certain error. How would you estimate the error introduced as a function of C, where C is the position below which the fluid is Newtonian [See Z. Tadmor, Polym. Eng. Sci., 6, 202 (1966).]... 

Pressure Flow Calculations Using the Equivalent Newtonian Viscosity6 Consider fully developed isothermal laminar pressure flow between parallel plates of a shear-thinning liquid with a flow curve fitted to the following polynomial relationship above the shear rate )>0 ... 

The following example examines the SDF in drag flow between parallel plates. In this particular flow geometry, although the shear rate is constant throughout the mixer, a rather broad SDF results because of the existence of a broad residence time distribution. Consequently, a minor component, even if distributed at the inlet over all the entering streamlines and placed in an optimal orientation, will not be uniformly mixed in the outlet stream. 

For values of 0.4 < Ri/R0 < 1.0, which represent relatively narrow annuli, the function F becomes independent of the degree of shear thinning of the melt. At the limit / —> 1.0, Table 12.2 can be used to relate the volumetric flow rate to the axial pressure drop, since the geometrical situation corresponds to the flow between parallel plates. 

FIG. 15.2 Types of simple shear flow. (A) Couette flow between two coaxial cylinders (B) torsional flow between parallel plates (C) torsional flow between a cone and a plate and (D) Poisseuille flow in a cylindrical tube. After Te Nijenhuis (2007). 

W. H. Suckow, P. Hrycak, and R. G. Griskey, Heat Transfer to Non-Newtonian Dilatant (Shear-Thickening) Fluids Flowing Between Parallel Plates, AIChE Symp. Ser. (199/76) 257,1980. 

Very narrow gap Ri,IRc > 0.99). This is similar to flow between parallel plates. Taking the shear rate at radius (/ , + Rc)f2,... 

Most rheological measurements measure quantities associated with simple shear shear viscosity, primary and secondary normal stress differences. There are several test geometries and deformation modes, e.g. parallel-plate simple shear, torsion between parallel plates, torsion between a cone and a plate, rotation between two coaxial cylinders (Couette flow), and axial flow through a capillary (Poiseuille flow). The viscosity can be obtained by simultaneous measurement of the angular velocity of the plate (cylinder, cone) and the torque. The measurements can be carried out at different shear rates under steady-state conditions. A transient experiment is another option from which both y q and ]° can be obtained from creep data (constant stress) or stress relaxation experiment which is often measured after cessation of the steady-state flow (Fig. 6.10). 

At medium shear rates director orientation and velocity profde can only be obtained by numerical calculations. In the following the flow between parallel plates with a surface alignment parallel to the pressure gradient is calculated [37]. The orientation of the capillary is the same as in Fig. 10. The equation for linear momentum is ... 

Similarly, the velocity gradient is constant across the gap as in flow between parallel plates (eq. 5.2.2), and thus the shear rate is the average in the gap... 

The theoretical basis for spatially resolved rheological measurements rests with the traditional theory of viscometric flows [2, 5, 6]. Such flows are kinematically equivalent to unidirectional steady simple shearing flow between two parallel plates. For a general complex liquid, three functions are necessary to describe the properties of the material fully two normal stress functions, Nj and N2 and one shear stress function, a. All three of these depend upon the shear rate. In general, the functional form of this dependency is not known a priori. However, there are many accepted models that can be used to approximate the behavior, one of which is the power-law model described above. 

For simple shear flow (as between parallel plates in relative motion) Eq. 2.7-9 reduces to ... 

Example 7.3 Effect of Viscosity Ratio on Shear Strain in Parallel-Plate Geometry Consider a two-parallel plate flow in which a minor component of viscosity /t2is sandwiched between two layers of major component of viscosities /q and m (Fig. E7.3). We assume that the liquids are incompressible, Newtonian, and immiscible. The equation of motion for steady state, using the common simplifying assumption of negligible interfacial tension, indicates a constant shear stress throughout the system. Thus, we have... 

Fig. E7.ll SDFs for fully developed Newtonian, isothermal, steady flows in parallel-plate (solid curves) and tubular (dashed curve) geometries. The dimensionless constant qp/qd denotes the pressure gradient. When qp/qd = —1/3, pressure increases in the direction of flow and shear rate is zero at the stationary plate qpjqd = 0 is drag flow when qp/qd = 1/3, pressure drops in the direction of flow and the shear rate is zero at the moving plate. The SDF for the latter case is identical to pressure flow between stationary plates. (Note that in this case the location of the moving plate at / — 1 is at the midplane of a pure pressure flow with a gap separation of H = 2H.)...

We have already discussed confinement effects in the channel flow of colloidal glasses. Such effects are also seen in hard-sphere colloidal crystals sheared between parallel plates. Cohen et al. [103] found that when the plate separation was smaller than 11 particle diameters, commensurability effects became dominant, with the emergence of new crystalline orderings. In particular, the colloids organise into z-buckled" layers which show up in xy slices as one, two or three particle strips separated by fluid bands see Fig. 15. By comparing osmotic pressure and viscous stresses in the different particle configurations, tlie cross-over from buckled to non-buckled states could be accurately predicted. 

Figure 2.2.1 Shear flow between two parallel plates.

All details of the pure continents relative molecular mass and rheology, and experimental setup of the ear cell are in reference (5). PIB and PDMS samples behave as Newtonian liquids under the experimental conditions hoe with shear viscosities of 10 Pa-s each (25 °C). The gap width between parallel plates in the Linkam CSS-450 shear cell was consistently set to 36 pm (standard uncertainty 3 pm verified optically by microscope stage translation), with all observations in the vorticity-flow plane. 

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