Transpose of a tensor

To obtain the transpose of a tensor T the indices of its components (originally given a.s Tpg) are transposed such that... 

The inteifacial momentum balance equation (Equation 5.35) will now be written for the case of negligible siuface excess mass and momentum and two Newtonian fluids. The stress tensor in a Newtonian fluid is written following Bird et al. (2002) as pi - pt [Vv + (Vv) ], where I is the identity tensor and the superscript T represents the transpose of a tensor. When the inertial forces are considered as well, the total force on an interface exerted by the ith phase is n (Pi(v - d )v + pj - / [VVj + (Vv )T]). The dot product with n denotes that the forces act on a surface characterized by m. It is in making a force balance on the intraface that the effects of interfacial tension make themselves felt. The balance is known to be (see Chapter 5)... 

Here, a is the total stress, p the isotropic pressure, I the identity (imit) tensor, and t the extra stress (ie, the stress in excess of the isotropic pressure). V is the gradient differential operator, and v is the velocity vector denotes the transpose of a tensor. For a one-dimensional flow with a single velocity component V, in which v varies in a single spatial direction y that is transverse to the flow direction, equation 2 simplifies to the famihar form... 

Furthermore, traditional notation for scalars, vectors and variables will be adopted. A scalar of fixed value, e.g., the number of factors in a model, is represented by an italicized capitol, A. An italicized lowercase letter, e.g., the nth factor, represents a scalar of arbitrary value. All vectors are column vectors designated by lowercase bold, e.g., x. Matrices are given by uppercase bold, e.g., X, and cubes (third-order tensors by uppercase open-face letters, e.g., R. Transposes of matrices and vectors, defined by switching the row and column indices, is designated with a superscript T, e.g., xT. The transpose of a cube need not be defined for this chapter. Subscripts designate a specific element of a higher-order tensor, where the initial order is inferred by the number of subscripts associated with the scalar. 

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 ... 

Note that we have added the superscript T to indicate a column vector, written as transpose. Equation (1.75) is a special case of matrix multiplication. A tensor is symmetric, when Txy = Tyx, Txz = T x, and Tyz — T y. The product of multiplication of a tensor with a vector reduces to a multiplication of the vector with a scalar, if the diagonal elements are equal, i.e., Txx = Tyy = and all other elements of the tensor are zero. Moreover, if Txx = Tyy = = I and all other elements are... 

Momentum space averages Fourth-order tensor with components -F E Transpose of the tensor A... 

One can imagine limiting circumstances for which the latter equations are decoupled. Explicitly, for an almost ion-free (almost nonconductive) solution, one can see that 1 = 0, while Eq. (lb) reproduces Darcy s law. In the other extreme case of a solid (infinitely viscous) electrolyte, one has U = 0, and Eq. (la) reduces to the familar Ohm s law. In general, one can show that a = p, by virtue of the Onsager thermodynamic theorem [3] p denotes the transpose of the tensor p. 

The stretch tensor is not indifferent but invariant under a rotation of frame. Taking the material derivative and the transpose of the first of these, and using the results in (A.23)... 

Example 15.4-3 Both the piezoelectric effect and the Pockels effect involve coupling between a vector and a symmetric 7(2). The structure of K is therefore similar in the two cases, the only difference being that the 6 x 3 matrix [rqi is the transpose of the 3x6 matrix [diq where i 1, 2, 3 denote the vector components and q= l,. .., 6 denote the components of the symmetric 7(2) in the usual (Voigt) notation. Determine the structure of the piezoelectric tensor for a crystal of C3v symmetry. 

Equation 10.1 is a second-rank tensor with transpose symmetry. The normal components of stress are the diagonal elements and the shear components of stress are the nondiagonal elements. Although Eq. 10.1 has the appearance of a [3 x 3] matrix, it is a physical quantity that, for one set of axes, is specified by nine components, whereas a transformation matrix is an array of coefficients relating two sets of axes. The tensor coefficients determine how the three components of the force vector, /, transmitted across a small surface element, vary as different values are given to the components of a unit vector / perpendicular to the face (representing the face orientation) ... 

Adjoints of tensor operators require some discussion. The adjoint of a linear operator is the transposed complex conjugate. 

For such a flow field the rate of deformation tensor" y= P + (P ) (here means transpose ) is just y = + k, a quantity which may be... 

In this system of equations the piezoelectric charge constant d indicates the intensity of the piezo effect is the dielectric constant for constant T and is the elastic compliance for constant E eft is the transpose of matrix d. The mentioned parameters are tensors of the first to fourth order. A simplification is possible by using the symmetry properties of tensors. Usually, the Cartesian coordinate system in Fig. 6.12a is used, with axis 3 pointing in the direction of polarization of the piezo substance (see below) [5,6]. 

Please note the identity between the velocity v and the time derivative of u. Furthermore the quantities f, a, q and r stand for the sum of externally applied body forces, the transposed Cauchy stress tensor, the heat flux, and an arbitrary energy production term (e.g. due to latent heat during phase transitions). Eqs. (1-3) are equations of motion for the five unknown fields p, u and e. They are universal, namely material-independent. To solve these equations, the constitutive quantities, viz. heat flux and stress tensor, must be replaced by constitutive equations (cf. subsequent paragraph) q = (T, VT,...) and = (T,u,...). Moreover, up to now no temperature T occur in the balances (1-3). For this reason a caloric state equation, e = e T), must be introduced, which allows for replacing the internal energy e by temperature T. 

Sinee we will be working with a fixed basis, there is little point in making a formal distinetion between the tensor r and the 3x3 matrix of its components, so that imless otherwise stated, r will represent the tensor and the matrix of its components. With this understanding, it is merely neeessaiy to point out that all the usual matrix operations such as addition, transposition, multiplication, inversion, and so on, apply to tensors, and the standard notation is used for these operations. Thus, for example, / and r are, respectively, the transpose and inverse of the tensor (or matrix) r. Every second-order tensor r may be additively decomposed into a deviatoric part r and a spherical part these are defined by ... 

It is interesting to note that the diffusion equation contains the transpose of the velocity gradient tensor, but the solution is given in terms of one of the relative finite strain tensors. The tensor a plays an important role in the changes of the thermodynamic functions that occur when a polymer solution goes from a state of equilibrium to a state of flow. The changes in internal energy and entropy are ... 

For orientation measurements, this tensor also needs to be expressed in the coordinate system OXYZ, axrz, using the matrix transformation u.xyz = Oaxyz / where O is a matrix whose elements are the direction cosines of the coordinate axes and is its transposed matrix [44]. 

---