The Rate of Strain Tensor

In an important class of materials, called Newtonian, this relationship is linear and one parameter—the viscosity—specifies the constitution of the material. Water, low-viscosity fluids, and gases are Newtonian fluids. However, most polymeric melts are non-Newtonian and require more complex constitutive equations to describe the relationship between the stress and the rate of strain. These are discussed in Chapter 3. 

We first consider a small rectangular element at time t in shear flow, as shown in Fig. 2. 4. This element is a vanishingly small differential element, and therefore without loss of generality we can assume that the local velocity field is linear, as shown in the figure. 

At time t + At the rectangular fluid element is translated in the x direction and deformed into a parallelogram. We define the rate of shear as -dS/dt, where 5 is the angle shown in the figure. 

Out of simple geometrical considerations, we express the rate of shear in terms of velocity gradients as follows  

we find that the rate of shear (or shear rate, as it is commonly referred to), or the rate of change of the angle 8, simply equals the velocity gradient. 

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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... 

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. 

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)... 

At least for weak strain and weak curvature, the influences of these strain and curvature phenomena on the flame structure can be characterized in terms of a single quantity, an effective curvature of the flame with respect to the flow or the total stretch of the flame surface produced by the flow with respect to the moving, curved flame. By evaluating b from the rate-of-strain tensor in the products just behind the flame, the last of these quantities may be expressed nondimensionally as... 

Here, v(r) is the velocity field at position r, p(r) the pressure field, and o(r) the rate-of-strain tensor defined as the symmetric part of the velocity gradient tensor. In the calculation below, n(r) is assumed to be spherically symmetric around a solute. v(r) around a rotating sphere can be expressed in the form... 

The flow field in Eq. (Al-7) is really just a solid-body rotation which rotates, but does not deform, the fluid element. As a result, the rate-of-strain tensor D is the zero tensor, and the Finger strain tensor is the unit tensor. 

The stress tensor t in Eq. (6) is related to the rate-of-strain tensor by a rheological equation of state such as ... 

As is well known, the components of the rate-of-strain tensor are defined by... 

The shear rate is often calculated as the second invariant (the first invariant is the trace) of the rate-of-strain tensor ... 

There are two proper explanations, one based on physical intuition and the other based on the principle of material objectivity. The latter is discussed in many books on continuum mechanics.19 Here, we content ourselves with the intuitive physical explanation. The basis of this is that contributions to the deviatoric stress cannot arise from rigid-body motions -whether solid-body translation or rotation. Only if adjacent fluid elements are in relative (nonrigid-body) motion can random molecular motions lead to a net transport of momentum. We shall see in the next paragraph that the rate-of-strain tensor relates to the rate of change of the length of a line element connecting two material points of the fluid (that is, to relative displacements of the material points), whereas the antisymmetric part of Vu, known as the vorticity tensor 12, is related to its rate of (rigid-body) rotation. Thus it follows that t must depend explicitly on E, but not on 12 ... 

Thus the rate of change of the distance between two neighboring material points depends on only the rate-of-strain tensor E, i.e., on the symmetric part of Vu. It can be shown in a similar manner that the contribution to the relative velocity vector Su that is due to the... 

Problem 2-7. Vorticity Tensor. Consider two nearby material points, P and Q. In Section H, we demonstrated that the distance between these two points increases (or decreases) at a rate that depends on the rate-of-strain tensor E. Show that the rate of angular rotation of the vector 6x between these points depends on the vorticity tensor f2. 

Problem 2-23. Constitutive Equations. As a model of a nonpolar microstructured fluid, consider the material to be described by a single unit vector p. Construct the most general relation between the stress tensor T and the rate-of-strain tensor e that is linear in e and depends on p. Note when e = 0, the stress is not necessarily isotropic. 

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