Divergence of a Tensor

The divergence of a second-order tensor produces a vector. 

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The second term in Eq. 2.33 requires taking the divergence of a tensor. This operation, V pVV which produces a vector, is expanded in several coordinate systems in Section A.ll. In noncartesian coordinate systems, since the unit-vector derivatives do not all vanish, the divergence of a tensor produces some unexpected terms. 

We note that the divergence of a tensor is a vector, whereas the divergence of a vector is a scalar... 

With the expressions relating charge and dipole density to field strength, e-quations 2.14 and 2.15, we can transform this expression into one involving the divergence of a tensor as... 

In general, as seen in Section 2.8.4, the vector of surface forces (per unit volume) on a differential element can be represented as the divergence of the tensor stress field... 

In noncartesian coordinates the divergence of a second-order tensor cannot be evaluated simply as a row-by-row operation as it can in a cartesian system. Hence some extra, perhaps unexpected, terms (e.g., rrg/r) appear in the direction-resolved force equations. General expressions for V-T in different coordinate systems are found in Section A.ll. 

When the gradient operator is dotted with a vector or a tensor, the divergence of the vector or tensor is obtained. The divergence of a vector produces a scalar... 

This equation shows that the entropy production is a quadratic form in all the forces. In continuous systems, the base of reference for diffusion flow affects the values of transport coefficients and the entropy due to diffusion. Prigogine proved the invariance of entropy for an arbitrary base of reference if the system is in mechanical equilibrium and the divergence of viscous tensors vanishes. 

By utilizing the expression for the divergence of a diagonal tensor in orthogonal, curvilinear coordinate systems, one can show that equation (2) reduces to... 

Two types of terms appear in the stress tensor a in eqn (5.28), terms of the form and The divergence of this tensor determines the force... 

The formulation of a divergence produces a tensor one order lower than the original tensor. A sensible operator is the Kronecker delta 6, defined by... 

C.2 Tensor Transformation Laws 1167 Then, if A is a second order tensor, or a dyad, the divergence of A is ... 

Generally, the divergence of a dyad or 2. order tensor, say a, is transformed... 

Both of the short-range terms may be transformed into the divergence of a local pressure tensor (cf. Appendix II), so that... 

Acceleration of the fluid elements is due to two types of forces, body forces (per unit of volume), F, acting within the whole fluid element volume (e.g. gravitational or electromagnetic forces) and forces acting on the surface of the small fluid element representing interaction with the rest of the fluid. The surface forces per unit of volume can be written as the divergence of a stress tensor [Pg.3]

This is a statement of the product rule for the divergence of the vector dot product of a tensor with a vector, which is valid when the tensor is symmetric. In other words, r = r, where is the transpose of the viscous stress tensor. Synunetry of the viscous stress tensor is a controversial topic in fluid dynamics, bnt one that is invariably assumed. is short-hand notation for the scalar double-dot product of two tensors. If the viscous stress tensor is not symmetric, then r must be replaced by in the second term on the right side of the (25-29). The left side of (25-29), with a negative sign, corresponds to the rate of work done on the fluid by viscous forces. The microscopic equation of change for total energy is written in the following form ... 

The second approach is continuous surface stress (CSS) model where the surface force is expressed as the divergence of a surface tension tensor ... 

Each of the three components / Fi dV of the resultant force (concerning all the internal stresses) can be transformed to a surface integral (Landau and Lifshitz, 1987). It follows from the general field theory that, in this case, the components of Fj must be the divergences of a certain second-rank tensor, i.e. 

It is assumed that the reader is familiar with some basic properties of Cartesian tensors, such as those that may be found in the book by Spencer [256]. In this book we shall define the divergence of a second order tensor Tij to be the tensor... 

If the stress is at the primary time step loeation and the veloeities are at the middle of the time step, then the resulting finite-difference equation is second-order accurate in space and time for uniform time steps and elements. If all quantities are at the primary time step, then a more complicated predictor-corrector procedure must be used to achieve second-order accuracy. A typical predictor-corrector scheme predicts the stresses at the middle of the time step and uses them to calculate the divergence of the stress tensor. 

The divergence of (4.21) yields a Poisson equation for p. However, the residual stress tensor r6 is unknown because it involves unresolved SGS terms (i.e., UfiJfi). Closure of the residual stress tensor is thus a major challenge in LES modeling of turbulent flows. 

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