Tensor rate-of-strain

The equation of momentum conservation, along with the linear transport law due to Newton, which relates the dissipative stress tensor to the rate of strain tensor = 1 (y. 4, and which introduces two... 

In generalized Newtonian fluids, before derivation of the final set of the working equations, the extra stress in the expanded equations should be replaced using the components of the rate of strain tensor (note that the viscosity should also be normalized as fj = rj/p). In contrast, in the modelling of viscoelastic fluids, stress components are found at a separate step through the solution of a constitutive equation. This allows the development of a robust Taylor Galerkin/ U-V-P scheme on the basis of the described procedure in which the stress components are all found at time level n. The final working equation of this scheme can be expressed as... 

Note that convected derivatives of the stress (and rate of strain) tensors appearing in the rheological relationships derived for non-Newtonian fluids will have different forms depending on whether covariant or contravariant components of these tensors are used. For example, the convected time derivatives of covariant and contravariant stress tensors are expressed as... 

The turbulent energy dissipation rate can be expressed in terms of the fluctuating rate-of-strain tensor ... 

Note that 7Zu = 0 due to the continuity equation. Thus, the pressure-rate-of-strain tensor s role in a turbulent flow is to redistribute turbulent kinetic energy among the various components of the Reynolds stress tensor. The pressure-diffusion term T is defined... 

Thus, only the normal Reynolds stresses (i = j) are directly dissipated in a high-Reynolds-number turbulent flow. The shear stresses (i / j), on the other hand, are dissipated indirectly, i.e., the pressure-rate-of-strain tensor first transfers their energy to the normal stresses, where it can be dissipated directly. Without this redistribution of energy, the shear stresses would grow unbounded in a simple shear flow due to the unbalanced production term Vu given by (2.108). This fact is just one illustration of the key role played by 7 ., -in the Reynolds stress balance equation. 

The second-order fluctuating rate-of-strain tensor is real and symmetric. Thus, its three eigenvalues are real and, due to continuity, sum to zero. The latter implies that one eigenvalue (a) is always positive, and one eigenvalue (y) is always negative. In the turbulence literature (Pope 2000), y is referred to as the most compressive strain rate. 

Experience with applying the Reynolds-stress model (RSM) to complex flows has shown that the most critical term in (4.52) to model precisely is the anisotropic rate-of-strain tensor 7 .--1 (Pope 2000). Relatively simple models are thus usually employed for the other unclosed terms. For example, the dissipation term is often assumed to be isotropic ... 

Returning to (4.52), it should be noted that many Reynolds-stress models have been proposed in the literature, which differ principally by the closure used for the anisotropic rate-of-strain tensor. Nevertheless, almost all closures can be written as (Pope 2000)... 

Now that we have discussed the geometric interpretation of the rate of strain tensor, we can proceed with a somewhat more formal mathematical presentation. We noted earlier that the (deviatoric) stress tensor t related to the flow and deformation of the fluid. The kinematic quantity that expresses fluid flow is the velocity gradient. Velocity is a vector and in a general flow field each of its three components can change in any of the three... 

In the previous section we discussed the nature and some properties of the stress tensor t and the rate of strain tensor y. They are related to each other via a constitutive equation, namely, a generally empirical relationship between the two entities, which depends on the nature and constitution of the fluid being deformed. Clearly, imposing a given stress field on a body of water, on the one hand, and a body of molasses, on the other hand, will yield different rates of strain. The simplest form that these equations assume, as pointed out earlier, is a linear relationship representing a very important class of fluids called Newtonian fluids. 

From Eq. E2.5-17 we can calculate the total viscous dissipation between the parallel plates. The second invariant of the rate of strain tensor multiplied by the viscosity gives the viscous dissipation per unit volume. From Table 2.3 we find that, for the case at hand, the second invariant reduces to y2z therefore, the total viscous energy dissipation (VED) between the plates will be given by... 

The Rate of Strain Tensor general flow held... 

The Invariants of the Rate of Strain Tensor in Simple Shear and Simple Elogational Flows Calculate the invariants of a simple shear flow and elonga-tional flow. 

Finally, a number of commonly used constitutive equations are derived from Eq. 3.3-13 by specifying G1, G2,... instead of specifying only G1 and settingG2,... equal to zero. Moreover, in these equations, M, are allowed to be functions of the invariants of the strain or rate-of-strain tensors, since there is experimental evidence supporting this dependence (35). Examples of such usable integral co-deformational constitutive equations are ... 

GNF-based constitutive equations differ in the specific form that the shear rate dependence of the viscosity, t](y), is expressed, but they all share the requirement that the non-Newtonian viscosity t](y) be a function of only the three scalar invariants of the rate of strain tensor. Since polymer melts are essentially incompressible, the first invariant, Iy = 0, and for steady shear flows since v = /(x2), and v2 V j 0 the third invariant,... 

According to our model, these are the only components of the rate of strain tensor. Thus, we can write an expression for the magnitude of the rate of strain tensor (cf. Eq. 2.7-11)... 

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