Upper-convected time derivative

Ata = -l,b = c = 0, equations [7.2.24] correspond to the Maxwell liquid with a discrete spectrum of relaxation times and the upper convective time derivative. For solution of polymer in a pure viscous liquid, it is convenient to represent this model in such a form that the solvent contribution into total stress tensor will be explicit ... 

The upper-convected time derivative is a time derivative in a special coordinate system whose base coordinate vectors stretch and rotate with material lines. With this definition of the upper-convected time derivative, stresses are produced only when material elements are deformed mere rotation produces no stress (see Section 1.4). Because of the way it is defined, the upper-convected time (teriva-tive of the Finger tensor is identically zero (see eqs. 2.2.3S and 1.4.13) ... 

Perhaps the simplest way to combine time-dependent phenomena and rheological nonlinearity is to incorporate nonlinearity into the simple Maxwell equation, eq. 3.2.18. This can be done by replacing the substantial time derivative in a tensor version of eq 3.2.18 with the upper-convected time derivative of r, using eq. 4.3.2 (Oldrpyd, 1950) ... 

Since K is homogeneous, vK = 0. Thus, the left-hand side is the upper-convected time derivative referred to in Chapter 4 (eq. 4.3.2), which is often designated by v over the tensor... 

Just as there are various possible finite strain tensors, there are various time derivatives that can be used in place of the ordinary derivative of stress in Eq. 10.21 to satisfy the continuum mechanics requirements for a model to be able to describe large, rapid deformations in arbitrary coordinate systems. The derivative that yields a differential model equivalent to Lodge s Eq. 10.6 is the upper convected time derivative (defined in Eq. 11.19), and the resulting model is called the upper-convected Maxwell model. Other possibilities include the lower-convected derivative and the corotational derivative. Furthermore, a weighted-sum of two of these derivatives can be used to formulate a differential constitutive equation for polymeric liquids. In particular, the Gordon-Schowalter convected derivative [7] is defined in this manner. 

Other combinations of upper- and lower-convected time derivatives of the stress tensor are also used to construct constitutive equations for viscoelastic fluids. For example, Johnson and Segalman (1977) have proposed the following equation... 

The time derivative is the linear combination of the upper emd lower-convected derivative of, namely the Giordon - Schowalter derivative [43] ... 

Note first that if the fluid is at a state of equilibrium with no flow, then the time derivative d is equal to zero, and the velocity gradient Vv is also zero. This implies from the above equations that = G8. Hence cr, i = <7 2 = ct t, = G at equilibrium, and aj = 0, for i j. Thus, although the diagonal stress components are not zero at equilibrium, they are all equal to each other, and the nondiagonal components are all equal to zero. Hence, the stress tensor is isotropic, but nonzero at equilibrium. (If one redefines the stress tensor as H = a — G8, then S " = 0 at equilibrium. The upper-convected Maxwell equation can then be rewritten in terms of Z .)... 

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)... 

Time derivatives play a central role in rheology. As seen above, the upper and lower convected derivatives fall out naturally from the deformation tensors. The familiar partial derivative, 8/9t, corresponds to an observer with a fixed position. The total derivative, d/dt, allows the observer to move freely in space, while if the observer follows a material point we have the material , or substantial derivative, denoted variously by the symbols d(m)/dr, D/Dr or ( ). We could expect that these different expressions could find their way into constitutive relations (see Section 5) as time rates of change of quantities that are functions of spatial position and time. However, only certain rate operations can be used by themselves in constitutive relations. This will depend on how two different observers who are in rigid motion with respect to each other measure the same quantity. The expectation is that a valid constitutive relation should be invariant to such changes in observer. This principle is called material frame indifference or material objectivity , and constitutes one of the main tests that a proposed constitutive relation has to pass before being considered admissible. 

This is the constitutive equation or rheological equation of state for the elastic dumbbell suspensions. It is identical to the upper-convected Maxwell model, eq. 4.3.7. The molecular dynamics have led to a proper (frame-indifferent) time derivative and to a definition... 

To explain the imbalance, O Nions and Oxburgh (1983) and Oxburgh and O Nions (1987) proposed that a barrier, which is suggested to exist between the upper and the lower mantle from seismic observation, has trapped helium in the lower mantle and retarded the heat transport from the lower mantle to the upper mantle. O Nions et al. (1983) suggested, from a semiquantitative discussion, that delayed heat transfer from the lower mantle to the upper mantle with a time constant of about 2Ga would enhance the present heat flow by a factor of two. McKenzie and Richter (1981) made numerical calculation on a two-layered mantle convection and showed that heat transfer from the lower mantle to the upper mantle is considerably retarded to give rise to an enhancement of the present surface heat flow up to a factor of two. If the thermal barrier not only retards the heat transfer and hence enhances the present surface heat flow but also essentially prevents the 4He flux from the lower to the upper mantle, this would qualitatively explain the imbalance. If this indeed were the case, we would expect a large amount of 4He accumulation in the lower mantle. However, it is difficult to conclude such a large accumulation of 4He in the lower mantle from the currently available scarce noble gas data derived from mantle-derived materials. 

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