Shearing flow defined

Consider the linear response of the affine network (9.26) to a small oscillatory shear flow defined by the deformation tensor... 

Polymer melts in shear flow exhibit normal stresses as well as shear stresses. These are associated with the melt elasticity (nonlinear viscoelasticity) and are the cause of the Weissenberg rod climbing effect [46]. The stress tensor in the shear flow defined by Eq. 1.26 is of form ... 

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. 

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. 

The Giesekus criterion for local flow character, defined as

extensional flow, 0 in simple shear flow and — 1 in solid body rotation [126]. The mapping of J> across the flow domain provides probably the best description of flow field homogeneity current calculations in that direction are being performed in the authors laboratory. 

The shear flow factor represents additional flow transport introduced by the roughness in relative sliding. It is defined as the mean flow in an element bearing where the two surfaces are in relative sliding, but the mean pressure gradient would be zero. 

Viscoelasticity illustrates materials that exhibit both viscous and elastic characteristics. Viscous materials tike honey resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain instantaneously when stretched and just as quickly return to their original state once the stress is removed. Viscoelastic materials have elements of both of these properties and, as such, exhibit time-dependent strain. Viscoelasticity is the result of the diffusion of atoms or molecules inside an amorphous material. Rubber is highly elastic, but yet a viscous material. This property can be defined by the term viscoelasticity. Viscoelasticity is a combination of two separate mechanisms occurring at the same time in mbber. A spring represents the elastic portion, and a dashpot represents the viscous component (Figure 28.7). 

Fig. 4.3.3 (a) Shear flow of a Newtonian fluid defined as the ratio of the shear stress and trapped between the two plates (each with a shear rate, (b) A polymeric material is being large area of A). The shear stress (a) is defined stretched at both ends at a speed of v. The as F/A, while the shear rate (y) is the velocity material has an initial length of L0 and an gradient, dvx/dy. The shear viscosity (r s) is (instantaneous) cross-sectional area of A. 

Viscosity, defined as the resistance of a liquid to flow under an applied stress, is not only a property of bulk liquids but of interfacial systems as well. The viscosity of an insoluble monolayer in a fluid-like state may be measured quantitatively by the viscous traction method (Manheimer and Schechter, 1970), wave-damping (Langmuir and Schaefer, 1937), dynamic light scattering (Sauer et al, 1988) or surface canal viscometry (Harkins and Kirkwood, 1938 Washburn and Wakeham, 1938). Of these, the last is the most sensitive and experimentally feasible, and allows for the determination of Newtonian versus non-Newtonian shear flow. 

The phase angle changes with frequency and this is shown in Figure 4.7. As the frequency increases the sample becomes more elastic. Thus the phase difference between the stress and the strain reduces. There is an important feature that we can obtain from the dynamic response of a viscoelastic model and that is the dynamic viscosity. In oscillatory flow there is an analogue to the viscosity measured in continuous shear flow. We can illustrate this by considering the relationship between the stress and the strain. This defines the complex modulus ... 

With increasing flow rate, the orientational state in the nematic solution should change. Larson [154] solved numerically Eqs. (39) and (40b) with Vscf(a) given by Eq. (41) for a homogeneous system (T[f ] = 0) in the simple shear flow to obtain the time-dependent orientational distribution function f(a t) as a function of k. The non-steady orientational state in the nematic solution can be described in terms of the time-dependent (dynamic) scalar order parameter S[Pg.149]

In a smectic A liquid crystal one can easily define two directions the normal to the layers p and an average over the molecular axes, the director, h. In the standard formulation of smectic A hydrodynamics these two directions are parallel by construction. Only in the vicinity of phase transitions (either the nematic-smectic A or smectic A-smectic C ) has it been shown that director fluctuations are of physical interest [33, 44, 45], Nevertheless h and p differ significantly in their interaction with an applied shear flow. 

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... 

Normal Stresses in Shear Flow. The tendency of polymer molecules to curl-up while they are being stretched in shear flow results in normal stresses in the fluid. For example, shear flows exhibit a deviatoric stress defined by... 

For a small amplitude oscillatory shearing flow, the strain is defined as,... 

In example 2.2 we obtained that for steady shearing flows the viscometric functions for this constitutive equation are defined by... 

The cross channel flow is derived in a similar fashion as the down channel flow. This flow is driven by the x-component of the velocity, which creates a shear flow in that direction. However, since the shear flow pumps the material against the trailing flight of the screw channel, it results in a pressure increase that creates a counteracting pressure flow which leads to a net flow of zero1. The flow rate per unit depth at any arbitrary position along the 2-axis can be defined by... 

Fiber motion — Jeffery orbits. The motion of ellipsoids in uniform, viscous shear flow of a Newtonian fluid was analyzed by Jeffery [32, 33] in 1922. For a prolate spheroid of aspect ratio a (defined as the ratio between the major axis and the minor axis) in simple shear flow, u°° = (zj), the angular motion of the spheroid is described... 

Viscoelastic materials are those which exhibit both viscous and elastic characterists. Viscoelasticity is also known as anelasticity, which is present in systems when undergoing deformation. Viscous materials, like honey, polymer melt etc, resist shear flow (shear flow is in a solid body, the gradient of a shear stress force through the body) and strain, i.e. the deformation of materials caused by stress, is linearly with time when a stress is applied [1-4]. Shear stress is a stress state where the stress is parallel or tangencial to a face of the material, as opposed to normal stress when the stress is perpendicular to the face. The variable used to denote shear stress is r which is defined as ... 

Oblique incidence of light through a shear flow. The axes defining the components n j and n2 lie in the (JC, y) plane. 

The dynamics of rigid, isolated spheroids was first analyzed for the case of shear flow by Jeffery[95]. When subject to a general linear flow with velocity gradient tensor G, the time rate of change of the unit vector defining the orientation of the symmetry axis of such a particle will have the following general form,... 

It is instructive to present the solution to equation (7.106) for the case of simple shear flow. For a spheroid oriented with its symmetry axis defined by the polar angle 9 relative to the z axis, and azimuthal axis <(> measured in the (x, y) plane relative to the x axis, equation (7.106) produces the following two equations for a simple shear flow of the form, v = G (0, x, 0) ... 

Non-Newtonian Viscosity In the cone-and-plate and parallel-disk torsional flow rheometer shown in Fig. 3.1, parts la and 2a, the experimentally obtained torque, and thus the % 2 component of the shear stress, are related to the shear rate y = y12 as follows for Newtonian fluids T12 oc y, implying a constant viscosity, and in fact we know from Newton s law that T12 = —/ . For polymer melts, however, T12 oc yn, where n < 1, which implies a decreasing shear viscosity with increasing shear rate. Such materials are called pseudoplastic, or more descriptively, shear thinning Defining a non-Newtonian viscosity,2 t],... 

Next we define the two normal stress difference functions that arise in simple shear flows... 

For the simple shear flow, the only one component of the velocity gradient tensor differs from zero, namely, v 2 0. The shear stress and the differences of the normal stresses are defined by equation (9.61) as... 

Many of the comments in the previous chapter about the selection of grade, additives and mixing before moulding apply equally in preparation for extrusion. It is important of course that the material should be appropriate for the purpose, uniform, dry, and free from contamination. It should be tested for flow and while many tests have been devised for this it is convenient to classify them as either for low or high rates of shear. The main terms used in such testing ( viscosity , shear rate , shear strain , etc.) are defined in words and expressed as formulae in ISO 472, and it is not necessary to repeat them here. Viscosity may be regarded as the resistance to flow or the internal friction in a polymer melt and often will be measured by means of a capillary rheometer, in which shear flow occurs with flow of this type—one of the most important with polymer melts—when shearing force is applied one layer of melt flows over another in a sense that could be described as the relationship between two variables—shear rate and shear stress.1 In the capillary rheometer the relationship between the measurements is true only if certain assumptions are made, the most important of which are ... 

Extensional flow (also called elongational flow) is defined as a flow where the velocity changes in the direction of the flow dvi/ dxy in contradistinction with shear flow where the velocity changes normal to the direction of flow (dv1/dx2). In uniaxial flow in the x1 direction the extensional rate of strain is defined as ... 

Eq. (15.98) strongly resembles Eq. (15.61), that we obtained for shear stress build up after starting a steady shear flow with shear rate q at time t = 0. Accordingly, the elongational or extensional viscosity, which is defined to be... 

The functions g(9) and h 6) represent two separate effects of disentanglement g(6) is the fractional reduction of entanglement density due to steady shear flow and h(ff) is the fractional reduction in energy dissipation rate per molecule due to dis-entanglement in steady shear flow (for details, see Graessley, 1974, Chap. 8). Eqs. (16.52)-(16.55) define an implicit expression for the master curve rj/rj0 vs. qzn, the reduced viscosity also being present in the arguments of the functions g(6) and h(6). 

Plastic processing is primarily the flow and shaping of viscous liquids. The scientific study of this flow is called rheology. Assuming laminar shear flow, viscosity is defined as the ratio of shear stress to shear rate. 

The orientation of anisotropic particles during dip coating can be analyzed by considering the rotational diffusion of these particles in shear. Rotational diffusion in shear flow has been reviewed by Van de Ven [58]. The ratio of the shear rate, y, to the rotational diffusion coefficient, , defines the rotational Peclet number (Pe = y/ ). When the rotational Peclet number is small (i.e., near zero), the anisotropic particles are randomly oriented by diffusion. When the rotational Peclet number is large, the particles rotate but have a preferential orientation aligned with the shear. The period of rotation is given by... 

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