Tensor translation

The use of these resistance tensors is developed in detail by Happel and Brenner (H3). While enabling compact formulation of fundamental problems, these tensors have limited application since their components are rarely available even for simple shapes. Here we discuss specific classes of particle shape without recourse to tensor notation, but some conclusions from the general treatment are of interest. Because the translation tensor is symmetric, it follows that every particle possesses at least three mutually perpendicular axes such that, if the particle is translating without rotation parallel to one of these axes, the total... 

The symmetric tensor K = [Kij] is called translational. It characterizes the drag of a body under translational motion and depends only on the size and shape of the body. In the principal axes, the translation tensor is reduced to the diagonal form... 

A symmetric tensor ft is called a rotational tensor. It depends both on the shape and size of the particle and on the choice of the origin. The rotational tensor characterizes the drag under rotation of the body and has the diagonal form with entries fii, Q2, 3 in the principal axes (the positions of the principal axes of the rotational and translational tensors in space are different). For axisymmetric bodies, one of the major axes (for instance, the first) is parallel to the symmetry axis, and in this case = O3. For a spherical particle, we have fii = fl2 = flj. 

In problems of gravity settling of particles, the translational tensor is most important. 

Experimental data and numerical results for principal values of the translational tensor for some axisymmetric and orthotropic bodies (cylinders, doubled cones, parallelepipeds) were discussed in [94], It was established that the results are well approximated by the following dependence for the relative coefficient of the axial drag ... 

If the velocity of a spherical particle in Stokes settling is always codirected with the gravity force, even for homogeneous axisymmetric particles the velocity is directed vertically if and only if the vertical coincides with one of the principal axes of the translational tensor K. If the angle between the symmetry axis and the vertical is [Pg.85]

The tensor R,-, termed the translation tensor, for a rigid body depends solely on the size and shape of the body. The translation tensor has the dimensions of length and may be interpreted as an equivalent radius. In the polymer literature the force is usually expressed in terms of a translational friction tensor T. The components are called translational friction coefficients... 

The tensor Rij is called the translational tensor, or the resistance tensor. Its components depend on particle s size and shape and have the dimensionality of length. They can be interpreted as equivalent radii of the body. The tensor is called the friction tensor, and the values Vy are known as mobilities. They are similar to mobilities introduced in Section 4.5. Therefore the tensor with components Vij is called the mobility tensor. 

Many partides have a non-spherical shape. Sometimes their shape can be approximated by an ehipsoid. Let a, U2, be the semi-axes of this ellipsoid, and i i, i 2, 1 3 - the components of the translational tensor along these semi-axes. In the special case when a = a, U2 = = b we have... 

The elements in the mean-square translational tensor and the mean-square rotational tensor are related to the mean-square displacements by following equations,... 

Certain special features to be imposed on a model may be expressed by more complicated constraint equations. We note as an example the assumption of a rigid molecule with prescribed dimensions whose position and orientation are to be refined. The position may be described by the coordinates of the centre of mass and the orientation by three Euler angles with respect to a unitary coordinate system. The atomic coordinates and thus the structure factor. Equation [1], are expressed as functions of these six parameters. The latter may then be adjusted to optimize the deviance. A similar procedure can be used to constrain the atomic displacement parameters of a molecule to rigid-body movements described by a translation tensor, a libration tensor and a transla-tion/libration-correlation tensor (TLS model). This model neglects intramolecular vibrations. 

---