Shear flow definition

FIGURE 8.5 Sample shear flow definition of shear stress, strain, and shear rate. 

In the semi-dilute regime, the rate of shear degradation was found to decrease with the polymer concentration [132, 170]. By extrapolation to the dilute regime, it is frequently argued that chain scission should be nonexistent in the absence of entanglements under laminar conditions. No definite proof for this statement has been reported yet and the problem of isolated polymer chain degradation in simple shear flow remains open to further investigation. 

This definition gives y = G when applied to a simple shear flow (i.e., K = 0). 

For the case that no convection stream is applied, the time, which elapses between the start of shear flow and the establishment of the temperature maximum in the center of the gap has been reported to be useful for the measurement. This measurement could be carried out within 10 sec (202). The possibility of such a procedure, however, follows from the theoretical treatment only for relatively wide gaps of d > 2 mms. For metal cylinders (199) as well as for glass cylinders (203) times of establishment can be calculated which are definitely too short for the application of this method, when smaller gap widths are used. 

Another explanation of the lithium gap in the Hyades could be found in terms of turbulent diffusion and nuclear destruction. Turbulence is definitely needed to explain the lithium abundance decrease in G stars. If this turbulence is due to the shear flow instability induced by meridional circulation (Baglin, Morel, Schatzman 1985, Zahn 1983), turbulence should also occur in F stars, which rotate more rapidly than G stars. Fig. 2 shows a comparison between the turbulent diffusion coefficient needed for lithium nuclear destruction and the one induced by turbulence. Li should indeed be destroyed in F stars This effect gives an alternative scenario to account for the Li gap in the Hyades. The fact that Li is normal in the hottest observed F stars could be due to their slow rotation. 

The governing DE and boundary conditions for the temperature field are again (9 1) and (9-2), and the dimensionless equation and boundary conditions are (9-7) and (9-8), with the same definition for the dimensionless temperature (9-3) and the sphere radius as a characteristic length scale. Only the form of the velocity field, u, and the choice of characteristic velocity are different for this case of a linear shear flow far from the sphere. The appropriate choice for the characteristic velocity is... 

Simple shear flow causes rotation, and also the liquid inside a drop subjected to shear flow does rotate. The drop is also deformed, and the relative deformation (i.e., the strain) is defined as D = (L — B)/ L + B) see Figure 11.7a for definition of L and B. For small Weber numbers, we simply... 

Here the absolute value of the velocity gradient is called the shear rate. For a newtonian fluid it is known that in this simple shear flow only the shear stress zxy is nonzero. However, it is possible that all six independent components of the stress tensor may be nonzero for a non-newtonian fluid according to its definition. For simple shearing flow of an isotropic fluid it can be proven [6] that the total stress tensor can have the general form... 

We can extend the mixing-length concept to the turbulent heat flux. We consider the same shear flow as above, in which buoyancy effects are for the moment neglected. By analogy to the definition of the eddy viscosity, we can define an eddy viscosity for heat transfer by... 

Rheological measurements were carried out using a capillary viscometer and the studies were conducted with and without application of nltrasonic oscillation. The results obtained are shown in Figure 5.3 and indicate the dependence of the effective flow state (p f on the stress shear. The definition of the toughness depends on the arguments about the influence of the ultrasonic oscillations using the expression ... 

Fig. 7.2 Illustration of the definition to the Newtonian fluid in the shear flow field. / is the shear force, and A is the shear area...

There are three types of melt behavior in a simple shear flow dilatant (D) (shear thickening) Newtonian (N), and pseudoplastic (P) (shear thinning). Similarly, in an extensional flow, the liquids may be stress hardening (SH), Troutonian (T), or stress softening (SS). By definition, the response considered here is taken at sufficiently... 

The Weissenberg number compares the elastic forces to the viscous effects. It is usually used in steady flows. One can have a flow with a small Wi number and a large De number, and vice versa. Sometimes the characteristic time of the flow in the deflnition of the Deborah number has been taken to be the reciprocal of a characteristic shear rate of the flow in these cases, the Deborah number and the Weissenberg number have the same definition. Pipkin s diagram (see Fig. 3.9 in Tanner 2000) classifies shearing flow behavior in terms of De and Wi, and provides a useful guide for the choice of constitutive equations. 

The dielectric constant of poly(lithium methacrylate) dispersed in Cereclor oil at a volume fraction of 0,3 vs. the frequency of electric field under different shear rates is shown in Figure 7. The dielectric constant reaches a maximum value when Eq. (5) is satisfied. This phenomenon is called the FMP resonance between a shear field and an electric field. The flow field definitely has an influence on the particle polarization and hence on the dielectric constant. The dielectric constant thus becomes a function of both the shear rate and the frequency of the applied electric field. Especially when the flow field is rotational and strong enough to such an extent that the particle is able to spin, it may compete with the applied electric field for particle polarization. In other words, the particles or particle clusters can be orientated not only under an electric field, but also under a shear flow field. FMP was also observed in rigid or flexible polymer solutions [25-271. 

In most treatises,"- 3 the strain tensor is defined with all components smaller by a factor of 2 than inequation 3, so that 711 = dui/dxi and 721 = du2/bx + bui/bx ). However, such a definition makes discussion of shear or shear flow somewhat clumsy either a practical shear strain and practical shear rate must be introduced which are twice 721 and 721 respectively, or else a factor of 2 must be carried in the constitutive equations. Since most of the discussion in this book is concerned with shear deformations, we use the definition of equation 3 which follows Bird and his school" and Lodge. - This does cause a slight inconvenience in the discussion of compressive and tensile strain, where a practical measure of strain is subsequently introduced (Section F below). In older treatises on elasticity, strains are defined without the factor of 2 appearing in the diagonal components of equation 3, but with the other components the same. 

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