Upper convected rate

As observed in (2.159), 8 v/8t, 8cv/8t denotes a part of v excluding the change of basis, which corresponds to the change of v when the observer is moving along the same coordinate system of the deformed body. 8 v/8t is known as the contravariant derivative or upper convected rate, and 8cv/8t is known as the covariant derivative or lower convected rate. ... 

A review by Bird and Wiest [6] gives a more complete list of existing viscoelastic models. The upper convective model and the White-Metzner model are very similar with the exception that the White-Metzner model incorporates the strain rate effects of the relaxation time and the viscosity. Both models provide a first order approximation to flows, in which shear rate dependence and memory effects are important. However, both models predict zero second normal stress coefficients. The Giesekus model is molecular-based, non-linear in nature and describes thepower law region for viscosity andboth normal stress coefficients. The Phan-Thien Tanner models are based on network theory and give non-linear stresses. Both the Giesekus and Phan-Thien Tanner models have been successfully used to model complex flows. 

The rheological constitutive equation of the Rouse model is that of an upper-convected Maxwell model, with the consequence that steady-state elongational flow only exists for strain rates lower than l/(2A,i). The steady-state elongational wscosity depends then on strain rate ... 

Figure 2.4 depicts the growth rates predicted from (2.26) for two jets of the upper-convected Maxwell liquids, which are shown by curves 1 and 3. They correspond to different values of the relative gas velocity (U for curve 1 is higher... 

Comparing eq. 4.3.10 with eq. 4.3.1, we see that to second order in the velocity gradient the upper-convected Maxwell equation for small strain rates reduces to a special case of the equation of the second-order fluid with V i.o = 2kr]o and V 2,o = 0. All properly formulated constitutive equations for which the stress is a smooth functional of the strain history reduce at second order in the velocity gradient to the equation of the second-order fluid. Example 4.3.3, however, illustrates that the equation of the second-order fluid cannot be trusted except for slow nearly steady flows. 

Calculate the predictions of the upper-convected Maxwell equation in (a) start Up of steady shear and (b) steady state uniaxial extension for arbitrary shear rate y and extension rate e, and compare these predictions with those for the Newtonian and second-order fluids. 

This model, often referred to as the upper convective Maxwell model, is weakly non-linear in that it predicts a first normal stress, but no shear thinning effects, i.e, the shear stress increases linearly with shear rate so that the viscosity is independent of shear rate. Combining Eqs. 2, 4, 5 and 6, we see that the tube model predicts the viscosity to be. 

At low shear rates i.e., yr < 1, the second term on the left hand side of Eq. 92 can be neglected, and the expression becomes an exact differential version of the upper convected Maxwell model, hi steady state shear flow Eq. 92 gives each component first normal stress as. 

It is seen that the material functions obtained from the covariant convected derivative of a are different from those obtained from the contravariant convected derivative of a. Experimental results reported to date indicate that the magnitude of N2 is much smaller than that of (say -A 2/ i 0.2-0.3). Therefore, the rheology community uses only the contravariant convected derivative of a when using Eq. (3.4), which is referred to as the upper convected Maxwell model. However, the limitations of the upper convected Maxwell model lie in that, as shown in Eq. (3.6), (1) it predicts shear-rate independent viscosity (i.e., Newtonian viscosity, t]q), (2) is proportional to over the entire range of shear rate, and (3) N2 = 0. There is experimental evidence (Baek et al. 1993 Christiansen and Miller 1971 Ginn and Metzner 1969 Olabisi and Williams 1972) that suggests Nj is negative. Also, as will be shown later in this chapter, and also in Chapter 5, in steady-state shear flow for many polymeric liquids, (1) l (k) follows Newtonian behavior at low y and then decreases as y increases above a certain critical value, and (2) increases with at low y and then increases with y (l < n < 2) as y increases further above a certain critical value. 

Let us consider the upper convected Maxwell model given by Eq. (3.4). For steady-state uniaxial elongation flow, for which the rate-of-strain tensor d is defined by Eq. (2.15), we have (see Appendix 3E)... 

Koscher and Fulchiron [140] assumed to be proportional to the first normal stress difference Ni, which they calculated with an upper-convected Maxwell model. Although a larger HMW tail leads to longer time scales in the relaxation spectrum, and therefore enhances the development of the first normal stress difference, this approach does not emphasize the importance of the longest molecules as much as when the nucleation rate... 

Xs is the Newtonian part of the extra stress, De is the Deborah number, T(x,y) is film temperature, and To is a reference temperature. Also in (3), a is the mobility parameter, p is the ratio of the solvent viscosity to zero-shear-rate viscosity, and f(T) accounts for temperature dependence of zero-shear-rate viscosity. The upper-convected derivative in equation (3) is defined for an arbitrary second-rank tensor ct as... 

In all of these systems, the rate of generation at the gas-solid interface is so rapid that only a small fraction is canied away from the particle surface by convective heat uansfer. The major source of heat loss from the particles is radiation loss to tire suiTounding atmosphere, and the loss per particle may be estimated using unity for both the view factor and the emissivity as an upper limit from tlris source. The practical observation is that the solids in all of these methods of roasting reach temperatures of about 1200-1800 K. 

Figure 3 illustrates some additional capability of the flow code. Here no pressure gradient is Imposed (this is then drag or "Couette flow only), but we also compute the temperatures resulting from Internal viscous dissipation. The shear rate in this case is just 7 — 3u/3y — U/H. The associated stress is.r — 177 = i/CU/H), and the thermal dissipation is then Q - r7 - i/CU/H). Figure 3 also shows the temperature profile which is obtained if the upper boundary exhibits a convective rather than fixed condition. The convective heat transfer coefficient h was set to unity this corresponds to a "Nusselt Number" Nu - (hH/k) - 1. 

In the example above, if the drop of dye is not carefully placed at the bottom of the beaker, the water is disturbed and convection currents are set up. Consequently, it can be observed that the dye is transported to the upper part at a much faster rate. It generally takes less time for the solution to achieve a uniform color because of convection caused by external forces. 

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