Convected coordinate

In this connection, a procedure is of interest, which has been followed in the treatment of dilute polymer solutions. In this theory a convective coordinate system is used which moves with the chain molecule under consideration. In this way, the indicated difficulty is overcome when a certain simplified model is used for the chain molecule. The set of coordinates of this model, as related to the convective coordinate system, is then transformed into a set of normal coordinates which behave independently. For a more detailed treatment and discussion of this matter the reader is referred to Chapters 3 and 5 of this review. 

Equation (6.57) is an inhomogeneous first-order linear differential equation of the tensor X[oj in the range —oo < t < t. With the condition that X[o] is finite at t = —oo, we may obtain the solution for X[oj. X[oj(t,t) = x(i), as the convected coordinates coincide with the fixed coordinates when... 

We consider the convected coordinate system embedded in a flowing fluid and deforming with it. The convected coordinates denoted by... 

An important concept in continuum mechanics is the objectivity, or admissibility, of the constitutive equation. There are the covariant and contravariant ways of achieving objectivity. The molecular theories the elastic dumbbell model of this chapter, the Rouse model to be studied in the next chapter, and the Zimm model which includes the preaveraged hydrodynamic interaction, all give the result equivalent to the contravariant way. In this appendix, we limit our discussion of continuum mechanics to what is needed for the molecular theories studied in Chapters 6 and 7. More detailed discussions of the subject, particularly about the convected coordinates, can be found in Refs. 5 and 6. 

Concept of convected coordinates, in both mathematics and continuum mechanics, has wider meaning (see [13,20]). 

In describing the kinematics of a deformable body, instead of using a coordinate system fixed in space, it is convenient to use a coordinate system embedded in the moving object. This is frequently referred to as a convected coordinate system, and was first introduced by Oldroyd (1950). Any measure of deformation (strain) defined relative to such a coordinate system always refers to the same element of materials, and therefore should be independent of the local rate of translation or rotation. As will be shown in this section, if they are going to be useful, all kinematic variables defined in terms of the convected coordinates must be transformed to a fixed coordinate system as all physical measurements are made relative to the fixed coordinate system. 

The definition of the convected strain tensor involves the difference between two quantities associated with a given material point at different times, and it refers to the same material point in convected coordinates. Now, we must transform the quantities i ,y( , t) and V y( , t ) (also v (, t ), and v 9(, t)) in such a manner that they both refer to the same point in a coordinate system fixed in space, because physical quantities (kinematic and dynamic variables) can only be measured relative to a frame of reference fixed in space. This can be done by making use of the transformation relations between two coordinate systems. 

Remembering that the coordinate systems x and are arbifiary (except that c are fixed in space and in the material), let us choose an arbitrary spatial coordinate system x and then choose a convected coordinate system that coincides with the spatial coordinate system at present time t. Note that, in this choice, the present configuration is a reference configuration, so that all other configurations at time t < t) are compared with the present one. Erom Eq. (2.84) we then have... 

We have shown how the strain tensors in the spatial coordinates may be obtained from those in the convected coordinates,... 

Having defined strain tensors in convected coordinates, we now describe the rate-of-strain (or rate-of-deformation) tensor. This may be obtained by taking the derivative of a strain tensor with time, with the convected coordinates held constant. Such a derivative is commonly referred to as the material derivative, which may be considered as the time rate of change as seen by an observer in a convected coordinate system. Using the notation D/Dr for the substantial (material) time derivative, we have from Eq. (2.86)... 

Since every material point always has the same convected coordinate position at all times, regardless of the extent of deformation of the medium, the relative coordinate displacements between any two points must be constant, so that any change in the actual distance between the points must be reflected by a change in the metric v-j. That is, if the distance between two points d changes with time, the convected metric v-j must change accordingly with time since, by definition, the convected coordinates f of a material point are independent of time. Therefore Eq. (2.99) may be rewritten with the aid of Eq. (2.86) as... 

It should be remembered that the metric tensor v j and in a convected coordinate system are related to the metric tensors gij and g J in a spatial coordinate system by Eqs. (2.84) and (2.85). 

Here, b/br is called the convected derivative due to Oldroyd (1950), and it is the fixed coordinate equivalent of the material derivative of a second-order tensor referred to in convected coordinates. The physical interpretation of the right-hand side of Eq. (2.104) may be given as follows. The first two terms represent the derivative of tensor a j with time, with the fixed coordinate held constant (i.e., Da /Dr), which may be considered as the time rate of change as seen by an observer in a fixed coordinate system. The third and fourth terms represent the stretching and rotational motions of a material element referred to in a fixed coordinate system. This is because the velocity gradient dv fdx (or the velocity gradient tensor L defined by Eq. (2.59)) may be considered as a sum of the rate of pure stretching and the material derivative of the finite rotation. For this reason, the convected derivative is sometimes referred to as the codeformational derivative (Bird et al. 1987). 

Now we seek the rule of transformation of the material derivative of tensor which is a type of time derivative following the motion of the material element. It is important to realize, however, that all tensor quantities described in reference to the convected coordinate system f must be transformed into tensor quantities fixed in space (i.e fixed coordinate system x ) because all physical measurements are made relative to the fixed coordinafe sysfem. 

In this section definitions are given of the main kinematic tensors (see ref. 5, chapter 9) that are needed for the continuum mechanics description of viscoelastic materials as well as for the presentation of kinetic theory results. Some of these tensors are defined naturally in terms of the velocity field of the material, whereas others are defined easily in terms of the displacement functions that describe the motion of fluid particles. Inasmuch as the velocity field and the displacement functions are themselves interrelated, it is possible to interrelate the two groups of kinematic tensors. Here the emphasis is on working definitions of the kinematic tensors and not on their derivation from the motion of a convected coordinate system, which is a standard starting point for the discussion of continuum mechanics an important basic reference for the kinematics and dynamics of continuous media is a paper by Oldroyd. ... 

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