Rate-of-deformation tensor

Rate of Deformation Tensor For general three-dimensional flows, where all three velocity components may be important and may vary in all three coordinate directions, the concept of deformation previously introduced must be generalized. The rate of deformation tensor l)y has nine components. In Cartesian coordinates, 

Vorticity The relative motion between two points in a fluid can be decomposed into three components rotation, dilatation, and deformation. The rate of deformation tensor has been defined. Dilatation refers to the volumetric expansion or compression of the fluid, and vanishes for incompressible flow. Rotation is described by a tensor Ciy, = driJdXj - dvj/dx,. The vector of vorticity given by one-half the 

Here CO is the magnitude of the vorticity vector, which is directed along the z axis. An irrotational flow is one with zero vorticity. Irro-tational flows have been widely studied because of their useful mathematical properties and applicability to flow regions where viscous effects may be neglected. Such flows without viscous effects are called inviscid flows. 

We can remove rotation from the velocity gradient by recalling the definition of F in terms of the stretch and rotation tensors. 

The time derivative of the stretching tensor V is usually called the rate of deformation tensor 2D. It is symmetric. 

R is antisymmetric (or skew-symmetric) and is called W, the vor-ticity tensor. 

A number of other symbols are frequently used in rheological literature to represent the rate of deformation and vorticity tensors, particularly A or ] for 2D and Q for W. Bird et al. (1987) call y the rate-of-strain tensor. Example 2.2.2 should help to illustrate 2D and 2W.  

For the flows given in Example 2.2.1, determine the components of 2D and 2W. Also calculate their invariants. 

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Theoretically the apparent viscosity of generalized Newtonian fluids can be found using a simple shear flow (i.e. steady state, one-dimensional, constant shear stress). The rate of deformation tensor in a simple shear flow is given as... 

The rate of deformation tensor in a pure elongatioiial flow has the following form... 

The elongation viscosity defined by Equation (1.19) represents a uni-axial extension. Elongational flows based on biaxial extensions can also be considered. In an equi-biaxial extension the rate of deformation tensor is defined as... 

All of the described differential viscoelastic constitutive equations are implicit relations between the extra stress and the rate of deformation tensors. Therefore, unlike the generalized Newtonian flows, these equations cannot be used to eliminate the extra stress in the equation of motion and should be solved simultaneously with the governing flow equations. 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977)... 

Ec[uation (4.91) is written using the components of the rate of deformation tensor D as ... 

Non-dimensionalization of the stress is achieved via the components of the rate of deformation tensor which depend on the defined non-dimensional velocity and length variables. The selected scaling for the pressure is such that the pressure gradient balances the viscous shear stre.ss. After substitution of the non-dimensional variables into the equation of continuity it can be divided through by ieLr U). Note that in the following for simplicity of writing the broken over bar on tire non-dimensional variables is dropped. 

C - INITIALIZE TEMPERATURE SECOND IWARIANT OF RATE OF DEFORMATION TENSOR... 

V. Velocity relating to the first stage, cm/sec A Symmetrical rate of deformation tensor [Eq. (96)]... 

Non-Newtonian Viscosity. A Newtonian fluid is one where the deviatoric stresses that occur during deformation, t, are directly proportional to the rate of deformation tensor, 7,... 

These forms of the equation of motion are commonly called the Cauchy momentum equations. For generalized Newtonian fluids we can define the terms of the deviatoric stress tensor as a function of a generalized Newtonian viscosity, p, and the components of the rate of deformation tensor, as described in Table 5.3. 

Thus, we have uz = uz(r), ur = ug = 0 and p = p(z). With this type of velocity field, the only non-vanishing component of the rate-of-deformation tensor is the zr-component. It follows that for the generalized Newtonian flow, rzr is the only nonzero component of the viscous stress, and that Tzr = rZT r). The -momentum equation is then reduced to,... 

A common form of analyzing film blowing is by setting-up a coordinate system, , that moves with the moving melt on the inner surface of the bubble, and that is oriented with the film as shown in Fig. 6.21. Using the moving coordinates, we can define the three non-zero terms of the local rate of deformation tensor as... 

The velocity gradients were used to compute the rate of deformation tensor, the magnitudes of the rate of deformation and vorticity tensors. The magnitudes of the rate of... 

Furthermore, we can also use radial functions to interpolate the magnitude of the rate of deformation tensor using... 

As a reference to something more familiar, consider the case of a fluid where incompressibility is enforced via a Lagrange multiplier. For a Stokesian fluid, it is assumed that the constitutive variables (stress, energy, heat flux) are a function of density, p, temperature, T, rate of deformation tensor, d, and possibly other variables (such as the gradients of density and temperature). Exploiting the entropy inequality in this framework produces the following constitutive restriction for the Cauchy stress tensor [10]... 

Our starting point is the rate-of-deformation tensor given in Eq. E3.4-5... 

Torsional Flow of a CEF Fluid Two parallel disks rotate relative to each other, as shown in the following figure, (a) Show that the only nonvanishing velocity component is vg = flr(z/H), where ft is the angular velocity, (b) Derive the stress and rate of deformation tensor components and the primary and secondary normal difference functions, (c) Write the full CEF equation and the primary normal stress difference functions. 

In this case, as pointed out earlier, the extension is planar, but unequal in directions t and 3. In order to derive the rate of deformation tensor components, we need to define the flow field in terms of the dependent variables 8 and R. We note that in direction 2 at 2 = 8, we can write... 

Finally, we obtain the third component of the rate of deformation tensor yn from the equation of continuity Ea,- = 0 or Eytt = 0 to give... 

There are two general types of constitutive equations for fluids Newtonian and non-Newtonian. For Newtonian fluids, the relation between the stress tensor, t, and the rate of deformation tensor or the shear stress is linear. For non-Newtonian fluids the relation between the stress tensor and the rate of deformation tensor is nonlinear. The various Newtonian and non-Newtonian rheologies of fluids are shown in Figure 12.2. There are four types of behavior (1) Newtonian, (2) pseudo-plastic, (3) Bingham plastic, and (4) dilatent. The reasons for these different rheological behaviors will also be discussed in subsequent sections of this chapter. But first it is necessary to relate the stress tensor to the rate of deformation tensor. 

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