Flow shear

Consider now the steady laminar flow of a nematic fluid between two parallel plates. If the flow is along x and the velocity gradient along y the components of the velocity and the director are 

Gahwiller assumed that the velocity profile may be approximated by the usual parabolic dependence 

The phase difference between the two perpendicularly polarized components when light is incident normal to the plates is then 

At high magnetic fields the experiment reduces in effect to Miesowicz s method except that Gahwiller extended it to arbitrary orientations of the magnetic field. If 6 is the angle between the director and the flow direction and (p that between the projection of the director on the yz plane and the velocity gradient, then, neglecting secondary flow (see 3.6.5), one may write approximately 

By choosing 0 and p appropriately, one can determine i/j, 173 and Using these two sets of data, Gahwiller was able to determine all five independent viscosity coefficients as well as x - Some of his results are presented in figs. 3.6.6 and 3.6.7. 

The most important flow process in polymer liquids is shear flow. Polymer liquids differ from simple liquids, first in that the shear viscosity is invariably extremely large, and second in that Newton s empirical equation giving a linear relationship between shear stress r and shear strain rate y with constant shear viscosity ft 

The determination of /i, for polymer liquids is most commonly obtained by capillary flow (see Fig. 7.8). We examine capillary flow for this reason and 

A layer of molten poly(methyl methacrylate) at 190 C is of uniform thickness 3 mm, and is sandwiched between two flat, parallel plates. A shear stress of 100 kPa is applied to the melt. Find the relative sliding velocity of the plates, using data for apparent viscosity from Fig. 7.13. 

From the data of Fig. 7.13, the apparent viscosity of poly (methyl methacrylate) at 190 C and at a shear stress of 100 kPa is 3.9 x 10 N s/m. The shear stress is uniform through the layer of melt and hence, even for this non-Newtonian fluid, the shear strain-rate V will be uniform with a value 

The shear strain-rate in the melt is identically equal to the transverse velocity gradient. Hence, the relative sliding velocity v of the plates is given by 

All experiments were carried out on sample SI. The radius of gyration of an isotropic sample, determined from the Zimm plot in the low scattering vector range, is close to 82 A. 

Specimens were sheared by using the above described sliding plate rheometer at two different shear stresses Oxyi 0.05 and 0.2 MPa. For axy=O.05 MPa, the behaviour of the melt was found to be nearly Newtonian. One sample was sheared at the highest shear stress up to a final shear strain of y=2.4. For the lowest shear stress, two samples with different shear strains were prepared y=2.8 and y=4. These samples are referenced in Table 7. 

Parameters of the shear flow, characteristic angles of orientation and experimental root mean 

The angle Xo can be compared on the one hand to the extinction angle of birefringence, and on the other hand to the orientation of the principal directions of the Cauchy deformation tensor, which would correspond to a molecular deformation purely affrne with the macroscopic deformation shear strain. For a simple shear deformation y, is given by  

Another interesting result is obtained from the comparison of the scattering of the sheared samples in the z direction to that of an isotropic sample. Fig. 30 shows the Zimm plot of sample C in the directions x and z for a specimen cut in the x-z plane. The corresponding curve for an isotropic sample, which has also been plotted in the same figure, is found to be identical to within experimental error to the curve in the z direction. This result indicates that the position correlations within the chain in the neutral direction of the shear flow are not affected by the flow, at least up to the values of stress and strain used in the present study. In particular. Table 7 shows that the mean square chain dimension in that direction, Rg 2 which has been determined either in the x-z or in the y-z planes for the various samples, is round to be equal to the radius of gyration of an isotropic sample to within experimental error (Rg 2=82 lA). The same result has been found by Lindner in dilute solutions [31]. 

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Figure A3.1.5. Steady state shear flow, illustrating the flow of momentum aeross a plane at a height z.

Theoretically the apparent viscosity of generalized Newtonian fluids can be found using a simple shear flow (i.e. steady state, one-dimensional, constant shear stress). The rate of deformation tensor in a simple shear flow is given as... 

In addition to the apparent viscosity two other material parameters can be obtained using simple shear flow viscometry. These are primary and secondary nomial stress coefficients expressed, respectively, as... 

Kaye, A., Lodge, A. S. and Vale, D. G., 1968. Determination of normal stress difference in steady shear flow. Rheol. Acta 7, 368-379. 

Non-Newtonian Fluids Die Swell and Melt Fracture. Eor many fluids the Newtonian constitutive relation involving only a single, constant viscosity is inappHcable. Either stress depends in a more complex way on strain, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known coUectively as non-Newtonian and are usually subdivided further on the basis of behavior in simple shear flow. 

Fig. 10. Fluid behavior in simple shear flow where A is Bingham B, pseudoplastic C, Newtonian and D, dilatant.

Pseudoplastic fluids are the most commonly encountered non-Newtonian fluids. Examples are polymeric solutions, some polymer melts, and suspensions of paper pulps. In simple shear flow, the constitutive relation for such fluids is... 

Consistent with this model, foams exhibit plug flow when forced through a channel or pipe. In the center of the channel the foam flows as a soHd plug, with a constant velocity. AH the shear flow occurs near the waHs, where the yield stress has been exceeded and the foam behaves like a viscous Hquid. At the waH, foams can exhibit waH sHp such that bubbles adjacent to the waH have nonzero velocity. The amount of waH sHp present has a significant influence on the overaH flow rate obtained for a given pressure gradient. 

Hardness. The Knoop indentation hardness of vitreous sihca is in the range of 473—593 kg/mm and the diamond pyramidal (Vickers) hardness is in the range of 600—750 kg/mm (1 4). The Vickers hardness for fused quartz decreases with increasing temperature but suddenly decreases at approximately 70°C. In addition, a small positive discontinuity occurs at 570°C, which may result from a memory of quartz stmcture (165). A maximum at 570°C is attributed to the presence of small amounts of quartz microcrystals (166). Scanning electron microscopic (sem) examination of the indentation area indicates that deformation is mainly from material compaction. There is htfle evidence of shear flow (167). 

Plastic Forming. A plastic ceramic body deforms iaelastically without mpture under a compressive load that produces a shear stress ia excess of the shear strength of the body. Plastic forming processes (38,40—42,54—57) iavolve elastic—plastic behavior, whereby measurable elastic respoase occurs before and after plastic yielding. At pressures above the shear strength, the body deforms plastically by shear flow. 

The force is direcdly proportional to the area of the plate the shear stress is T = F/A. Within the fluid, a linear velocity profile u = Uy/H is estabhshed due to the no-slip condition, the fluid bounding the lower plate has zero velocity and the fluid bounding the upper plate moves at the plate velocity U. The velocity gradient y = du/dy is called the shear rate for this flow. Shear rates are usually reported in units of reciprocal seconds. The flow in Fig. 6-1 is a simple shear flow. 

The Prandtl mixing length concept is useful for shear flows parallel to walls, but is inadequate for more general three-dimensional flows. A more complicated semiempirical model commonly used in numerical computations, and found in most commercial software for computational fluid dynamics (CFD see the following subsection), is the A — model described by Launder and Spaulding (Lectures in Mathematical Models of Turbulence, Academic, London, 1972). In this model the eddy viscosity is assumed proportional to the ratio /cVe. 

The drag force is exerted in a direction parallel to the fluid velocity. Equation (6-227) defines the drag coefficient. For some sohd bodies, such as aerofoils, a hft force component perpendicular to the liquid velocity is also exerted. For free-falling particles, hft forces are generally not important. However, even spherical particles experience lift forces in shear flows near solid surfaces. 

Secondly, the pressure drop, P, in the above expression is the pressure drop due to shear flow along the die. If a pressure transducer is used to record the... 

David.son, L, Large eddy simulation A dynamic one-equation subgrid model for three-dimensional recirculating flow. In llth Int. Symp. on Turbulent Shear Flow, vol. 3, pp. 26.1-26.6, Grenoble, 1997. 

Adler, P.M., 1981. Heterocoagulation in shear flow. Journal of Colloid and Interface Science, 83, 106-115. 

Pope, S.B., 1979. Probability distribution in turbulent shear flows. In Turbulent shear flows 2. Berlin Springer, pp. 7-16. 

P. A. Thompson, M. O. Robbins. Shear flow near solids epitaxial order and flow boundary conditions. Phys Rev A 47 6830-6837, 1990. 

Molecular dynamics, in contrast to MC simulations, is a typical model in which hydrodynamic effects are incorporated in the behavior of polymer solutions and may be properly accounted for. In the so-called nonequilibrium molecular dynamics method [54], Newton s equations of a (classical) many-particle problem are iteratively solved whereby quantities of both macroscopic and microscopic interest are expressed in terms of the configurational quantities such as the space coordinates or velocities of all particles. In addition, shear flow may be imposed by the homogeneous shear flow algorithm of Evans [56]. 

M. Kroger, R. Makhloufi. Wormlike micelles under shear flow A microscopic model studied by nonequihbrium molecular dynamics computer simulations. Phys Rev E 55 2531-2536, 1996. 

A. Milchev, J. Wittmer, D. P. Landau. A Monte Carlo study of equilibrium polymers in a shear flow. Europ Phys J B 1999 (in press). 

Liquids are able to flow. Complicated stream patterns arise, dependent on geometric shape of the surrounding of the liquid and of the initial conditions. Physicists tend to simplify things by considering well-defined situations. What could be the simplest configurations where flow occurs Suppose we had two parallel plates and a liquid drop squeezed in between. Let us keep the lower plate at rest and move the upper plate at constant velocity in a parallel direction, so that the plate separation distance keeps constant. Near each of the plates, the velocities of the liquid and the plate are equal due to the friction between plate and liquid. Hence a velocity field that describes the stream builds up, (Fig. 15). In the simplest case the velocity is linear in the spatial coordinate perpendicular to the plates. It is a shear flow, as different planes of liquid slide over each other. This is true for a simple as well as for a complex fluid. But what will happen to the mesoscopic structure of a complex fluid How is it affected Is it destroyed or can it even be built up For a review of theories and experiments, see Ref. 122. Let us look into some recent works. 

FIG. 15 A colloidal suspension subject to shear flow. The arrows indicate the mean particle velocities. 

Figure 10 shows that upon cessation of shear flow of the melt, shear stress relaxation of LLDPE is much faster than HP LDPE because of the faster reentangle-... 

Viscosities of the blends and composites were measured in shear flow with a Gottfert Rheograph 2002 capillary viscosimeter. The shear rate was investigated from 100-10000 s" . The L D ratio of the capillary die was 30 mm 1 mm. Rabinowitch correction was made to the measurements, but Bagley correction was not applied. 

Concerning a liquid droplet deformation and drop breakup in a two-phase model flow, in particular the Newtonian drop development in Newtonian median, results of most investigations [16,21,22] may be generalized in a plot of the Weber number W,. against the vi.scos-ity ratio 8 (Fig. 9). For a simple shear flow (rotational shear flow), a U-shaped curve with a minimum corresponding to 6 = 1 is found, and for an uniaxial exten-tional flow (irrotational shear flow), a slightly decreased curve below the U-shaped curve appears. In the following text, the U-shaped curve will be called the Taylor-limit [16]. 

To constitute the We number, characteristic values such as the drop diameter, d, and particularly the interfacial tension, w, must be experimentally determined. However, the We number can also be obtained by deduction from mathematical analysis of droplet deforma-tional properties assuming a realistic model of the system. For a shear flow that is still dominant in the case of injection molding, Cox [25] derived an expression that for Newtonian fluids at not too high deformation has been proven to be valid ... 

Similar folds—fo ds that have the same geometric form, but where shear flow in the plastic beds has occurred (sec Figure 2-4.5). 

From the results obtained in [344] it follows that the composites with PMF are more likely to develop a secondary network and a considerable deformation is needed to break it. As the authors of [344] note, at low frequencies the Gr(to) relationship for Specimens Nos. 4 and 5 (Table 16) has the form typical of a viscoelastic body. This kind of behavior has been attributed to the formation of the spatial skeleton of filler owing to the overlap of the thin boundary layers of polymer. The authors also note that only plastic deformations occurred in shear flow. 

During dynamic measurements frequency dependences of the components of a complex modulus G or dynamic viscosity T (r = G"/es) are determined. Due to the existence of a well-known analogy between the functions r(y) or G"(co) as well as between G and normal stresses at shear flow a, seemingly, we may expect that dynamic measurements in principle will give the same information as measurements of the flow curve [1],... 

At least, in absolute majority of cases, where the concentration dependence of viscosity is discussed, the case at hand is a shear flow. At the same time, it is by no means obvious (to be more exact the reverse is valid) that the values of the viscosity of dispersions determined during shear, will correlate with the values of the viscosity measured at other types of stressed state, for example at extension. Then a concept on the viscosity of suspensions (except ultimately diluted) loses its unambiguousness, and correspondingly the coefficients cn cease to be characteristics of the system, because they become dependent on the type of flow. 

If we consider a shear flow of a diluted suspension of noninteracting particles, then substitution of spheres by particles of ellipsoidal form leads only to a variation of... 

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