Tensor, deformation velocity

The arithmetic mean of both angular velocities is called the strain tensor, often also called the deformation velocity tensor... 

The partial derivatives of x are the velocity vector y and the deformation gradient tensor f, respectively. 

Wall thickness Channel width Acoustic velocity Friction coefficient Conductance Capillary number Discharge coefficient Drag coefficient Diameter Diameter Dean number Deformation rate tensor components Elastic modulus Energy dissipation rate Eotvos number Fanning friction factor Vortex shedding frequency Force... 

The individual fluid elements of a flowing fluid are not only displaced in terms of their position but are also deformed under the influence of the normal stresses tu and the shear stresses T (i j)- The deformation velocity depends on the relative movement of the individual points of mass to each other. It is only in the case when the points of mass in a fluid element do not move relatively to each other that the fluid element behaves like a rigid solid and will not be deformed. Therefore a relationship between the velocity field and the deformation, and with that also between the velocity field and the stress tensor must exist. This relationship is required if we wish to express the stress tensor in terms of the velocities in Cauchy s equation of motion. 

L represents the tensor of the deformation velocities, q is the heat flow, r is the energy supply due to the external energy or heat sources, and e and are the specific internal energy and the specific internal energy production of the constituent y , respectively. 

In Chapter I (recall Figure 1.4.1) we were also concerned with describing how points separate. There we used the displacement of (mints to find F here we need rate of displacement Clearly F, the deformation gradient tensor, and L, the velocity gradient tensor, are related. Recall eq. 1.4.3... 

It is seen in Eq. (2.53) that the velocity gradient tensor L(t) can be determined from the rate of deformation gradient tensor t(t) and the inverse of the deformation gradient tensor that is,... 

The velocity gradient tensor L(r) can be determined from the relative deformation gradient tensor F (r ) also, since we have... 

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. 

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

Several generalizations of the inelastic theory to large deformations are developed in Section 5.4. In one the stretching (velocity strain) tensor is substituted for the strain rate. In order to make the resulting constitutive equations objective, i.e., invariant to relative rotation between the material and the coordinate frame, the stress rate must be replaced by one of a class of indifferent (objective) stress rates, and the moduli and elastic limit functions must be isotropic. In the elastic case, the constitutive equations reduce to the equation of hypoelastidty. The corresponding inelastic equations are therefore termed hypoinelastic. 

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

In this section, we use the Cartesian force of Section VI to derive several equivalent expressions for the stress tensor of a constrained system of pointlike particles in a flow field with a macroscopic velocity gradient Vv. The excess stress of any system of interacting beads (i.e., point centers of hydrodynamic resistance) in a Newtonian solvent, beyond the Newtonian contribution that would be present at the applied deformation rate in the absence of the beads, is given by the Kramers-Kirkwood expression [1,4,18]... 

Stress and Strain Rate The stress and strain-rate state of a fluid at a point are represented by tensors T and E. These tensors are composed of nine (six independent) quantities that depend on the velocity field. The strain rate describes how a fluid element deforms (i.e., dilates and shears) as a function of the local velocity field. The stress and strain-rate tensors are usually represented in some coordinate system, although the stress and strain-rate states are invariant to the coordinate-system representation. 

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,... 

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... 

The spatial velocity gradient a = grad va can be decomposed into symmetric and skew-symmetric parts as la = sym a + skw 1 = dQ + wQ., where da and wa are the deformation rate and the spin tensors, respectively. 

The title of the book, Optical Rheometry of Complex Fluids, refers to the strong connection of the experimental methods that are presented to the field of rheology. Rheology refers to the study of deformation and orientation as a result of fluid flow, and one principal aim of this discipline is the development of constitutive equations that relate the macroscopic stress and velocity gradient tensors. A successful constitutive equation, however, will recognize the particular microstructure of a complex fluid, and it is here that optical methods have proven to be very important. The emphasis in this book is on the use of in situ measurements where the dynamics and structure are measured in the presence of an external field. In this manner, the connection between the microstructural response and macroscopic observables, such as stress and fluid motion can be effectively established. Although many of the examples used in the book involve the application of flow, the use of these techniques is appropriate whenever an external field is applied. For that reason, examples are also included for the case of electric and magnetic fields. 

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... 

A fluid in motion may simultaneously deform and rotate. Decomposing the velocity gradient tensor into two parts can separate these motions ... 

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. 

For uniform uniaxial elongational deformation along axis 1, the tensor of the velocity gradients, taking into account the condition of incompressibility, can be written in the form... 

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