Tensor vorticity

Voigt-Kelvin element Voigt-Kelvin model Voigt element Voigt model volume compression vorticity tensor width of the resonance curve Young s modulus zero-shear viscosity... 

Here X is an important parameter which determine whether the time evolution of n is dominated by the strain tensor A or by the vorticity tensor fi (cf. Eq. (42)). If X is larger than unity, the former dominates and n tends to assume a steady-state angle 0 relative to the flow direction which is determined by the equation... 

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... 

Table 3.3 Components of the Vorticity Tensor to in Three Coordinate Systems ...

For future reference, we also define the antisymmetric part of the velocity gradient, also called the vorticity tensor, as... 

The vorticity tensor is related to the angular velocity of the fluid element. For flows with no rotation, such as the extensional flow depicted in Figure 1-12, Vv is symmetric and the vorticity tensor is zero. 

The antisymmetric contribution to Vu, which we have denoted in (2 49) as f2, is known as the vorticity tensor. Again, more is said about the vorticity tensor later in this chapter. 

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 ... 

To prove our assertions about the physical significance of the rate-of-strain and vorticity tensors, we consider the relative motion of two nearby material points in the fluid P, initially at position x and Q, which is at x + Sx. We denote the velocity of the material point P as... 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

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. 

There is one new factor that needs to be considered for this problem. The vector harmonic functions are irreducible tensors. Hence, for any sum of such terms to satisfy the boundary conditions, (8-28), we should also express the tensors, T and K, in terms of irreducible tensors. Now, we have seen previously that T can be expressed in terms of the rate-of-strain and vorticity tensors ... 

Although the rate-of-strain tensor is irreducible, the antisymmetric vorticity tensor is not. A well-known result from tensor calculus3 is that an arbitrary second-order tensor M can... 

Here, and 2 the first and second stress difference coefficient functions, and the derivative of the strain rate is the Jaumann derivative, which is related to a frame of reference that translates and rotates with the local velocity of the fluid (this relationship can be numerically evaluated from the deformation and vorticity tensors). 

Jaumann s derivative ordinary total derivative trace of the tensor, trx = Xu, vorticity tensor transpose of the tensor Vv... 

Thus Aij is the stretching tensor or rate of strain and cuy is the rotation or vorticity tensor and are illustrated below. 

To help determine the flight flank geometry and clearance, a two-dimensional BEM analysis was initiaiiy performed. To evaluate the strength of the elongational flow versus the shear flow, the flow number [76] was analyzed. The flow number is the ratio of the magnitude of the rate of deformation tensor y to the sum of y + co, where CO is the magnitude of the vorticity tensor. 

The symmetric part is called the rate of deformation tensor, and the antisymmetric part Wij is called the vorticity tensor. The rate of deformation tensor describes the rate at which neighboring particles move relative to each other, irrelevant to superposed rigid rotation. The vorticity is a measure of the local rate of rigid rotation. 

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