Tensor gradient velocity

K set of referential internal state variables I velocity gradient tensor... 

Flows that produce an exponential increase in length with time are referred to as strong flows, and this behavior results if the symmetric part of the velocity gradient tensor (D) has at least one positive eigenvalue. For example, 2D flows with K > 0 and uniaxial extensional flow are strong flows simple shear flow (K = 0) and all 2D flows with K < 0 are weak flows. 

The rate of separation of the fragments depends on the functions A r), C(r), Fc> and the fragmentation number, while the rate of rotation depends only on the function B(r). Further, it is apparent that the separation between the fragments increases only when the hydrodynamic force exceeds the binding physicochemical force. The pair of fragments rotates as a material element in an apparent flow with an effective velocity gradient tensor... 

The extra terms in the bottom row are a result of nonvanishing unit-vector derivatives. The tensor products of unit vectors (e.g., ezer) are called unit dyads. In matrix form, where the unit vectors (unit dyads) are implied but usually not shown, the velocity-gradient tensor is written as... 

It should be noted that the velocity-gradient tensor is not symmetric. The matrix form of the velocity-gradient tensor for other coordinate systems is stated in Section A.8. 

In this expression, I is the identity tensor, W and (VV)T are, respectively, the velocity-gradient tensor and its transpose (Appendix B.2). 

The velocity-gradient tensor is VV and the operator indicates the dyadic product of two tensors, which produces a scalar. Work is a scalar quantity. 

The physical quantities commonly encountered in polymer processing are of three categories scalars, such as temperature, pressure and time vectors, such as velocity, momentum and force and tensors, such as the stress, momentum flux and velocity gradient tensors. We will distinguish these quantities by the following notation,... 

The dynamics of rigid, isolated spheroids was first analyzed for the case of shear flow by Jeffery[95]. When subject to a general linear flow with velocity gradient tensor G, the time rate of change of the unit vector defining the orientation of the symmetry axis of such a particle will have the following general form,... 

J. J. Wang, D. Yavich, and L. G. Leal, Time-resolved measurement of velocity gradient tensor in linear flow by photon correlation spectroscopy, Phys. Fluids A, submitted (1995). 

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. 

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

Indeed, we can obtain a relation between the stress tensor and the velocity gradient tensor if we exclude tensor from the set of equations (8.32)-(8.33). This can be done in two different ways. 

We shall consider the case of shear stress when one of the components of the velocity gradient tensor has been specified and is constant, namely v 2 0. 

Further on, we shall consider the case of shear stress when one of the components of the velocity gradient tensor has been specified and is constant, namely V12 0. This situation occurs in experimental studies of polymer solutions (Ferry 1980). In order to achieve such a flow, it is necessary that the stresses applied to the system should be not only the shear stress a 12, as in the case of a linear viscous liquid, but also normal stresses, so that the stress tensor is... 

In steady-state shear, when the only component of the velocity gradient tensor differs from zero is z/12, equation (9.19) is followed by... 

We shall assume that the dumbbell is situated in the stream of viscous fluid characterised by the mean velocity gradient tensor Vij. According to (2.8), the resistance force for every particle of the dumbbell can be written as... 

In the case of low velocity gradients, the time-independent distribution function may be found in the form of an expansion in terms of the invariant combinations of vector p and the symmetrical and anti-symmetrical velocity-gradient tensors 7. and w. In the steady-state case, one has, to within the second-order terms in the velocity gradients,... 

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... 

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