Upper convected derivative

The first kind of modification to the UCM model that may be conceivable is that of the convected derivative. This leads one to consider that the motion of the network junctions is no more that of the continuum and thus, the afiine assumption of the Lodge model is removed. Among the various possibilities, Phan Thien and Tanner suggested the use of the (Jordon-Schowalter derivative [47], which is a linear combination of the upper- and lower-convected derivatives, instead of the upper-convected derivative ... 

One may try to avoid the problem by the use of the upper-convected derivative, which ensures the coincidence of the principal axes of stress and strain. But doing that, it appears that any kinetics based on the stress amplitude is improper, since materials which exhibits thickening behaviour in elongation are, to the contrary, shear-thinning. Consequently no unique dependence can be expected for these two kinematics. The determination of a single set of parameters in various flows in then bound to be a compromise. 

In [62] Renardy proves the linear stability of Couette flow of an upper-convected Maxwell fluid under the 2issumption of creeping flow. This extends a result of Gorodtsov and Leonov [63], who showed that the eigenvalues have negative real parts (I. e., condition (S3) holds). That result, however, does not allow any claim of stability for non-zero Reynolds number, however small. Also it uses in a crucial way the specific form of the upper-convected derivative in the upper-convected Maxwell model, aind does not generalize so far to other Maxwell-type models. 

The triangular superscript denotes the upper-convected derivative and (... 

At steady state, the upper-convected derivative, defined by Eq. (3-33), reduces to... 

Parameter P governs the eontribution of the Maxwell element to effective viscosity, T, (Newtonian viscosity of the solution). Equation [7.2.26] is similar to the Oldroyd-type equation [7.2.15] with the only difference that in the former the upper convective derivative is used to account for nonlinear effects instead of partial derivative, d/dt. 

A and are phenomenological coefficients characteristic of the fluid, not the turbulence. The time derivative in Eq. (3) is the upper convected derivative of Oldroyd (7). In what follows,.A.and.. are assumed to be independent of the... 

The above equation is generalized to three dimensions by replacing the stress component T by the stress matrix and the strain component 7 by the strain matrix. This procedure works well as long as one confines oneself to small strains. For large strains, the time derivative of the stress requires special treatment to ensure that the principle of material objectivity(78) is not violated. This principle requires that the response of a material not depend on the position or motion of the observer. It turns out that one can construct several different time derivatives all of which satisfy this requirement and also reduce to the ordinary time derivative for infinitesimal strains. By experience over many years, it has been found (see Chapter 3 of Reference 79) that the Oldroyd contravariant derivative also called the codeformational derivative or the upper convected derivative, gives the most realistic results. This derivative can be written in Cartesian coordinates as(79)... 

Hereafter in constitutive equations such as eq 4.3.1 we will use the stress tensor r, which does not contain the isotropic pressure term pi. In eq 4.3.1 we have introduced the upper-convected derivative, denoted by V, which when acting on an arbitrary tensor A gives by definition... 

Note that it is possible to define other ctmvected derivatives (Bird etai, 1987 Larson, /988j. The upper-convected derivative arises most naturally from molecular theory. We will see this in Chapter II with the elastic dumbbell model. Note also V... 

Using the definition of eq. 4.3.2 for die upper-convected derivative, the upper-convected Maxwell equation (eq. 4.3.7) can be written in expanded form as follows ... 

Use of Eq. (3.33) into Eq. (3.25) yields the material functions that are exactly the same as Eq. (3.17) obtained from the ZFD model, Eq. (3.15). However, as will be shown, the Giesekus model with a = 1 is not the same as the ZFD model. Substituting the following relationship between the upper convected derivative and the Jaumann... 

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... 

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