The Material or Substantial Derivative

It is possible to describe a flowing system from a fixed or moving observer point of view. A fixed observer, such as described in Fig. 5.2, feels the transient effects a change in time before the system reaches steady state. 

Once the system reaches steady state, the fixed observer feels a constant velocity, temperature and other field variables. 

On the other hand, a moving observer, such as the one shown in Fig. 5.3, not only feels the transient effects but also the changes that it undergoes as it travels through a gradient of velocity, temperature, concentration, etc. 

The moving observer, described by a fluid particle, feels 

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Batchelor [6] and Ghil and Childress [28] examine the transformation of the governing Eulerian equations for compressible flows from an inertial to a rotational frame. As it turns out, only the momentum equation is actually effected by this transformation because the material or substantial derivative operator of scalar functions is invariant to rotation. 

Yi = pjp is the mass fraction of species i and j, is its diffusirm mass flux, where the subscript /, i = 1,..N, denotes species i. The specific enthalpy is h = e + pip, where e denotes the specific internal energy and p is the pressure q is the mass specific heat flux, a is the extra stress tensor and d/dt denotes the material or substantial derivative. 

The transformation between Lagrangian and Eulerian variables is determined by the motion vector x = r(X, t). Considering the rate of change of a physical variable G at a fixed material point, X, leads to the material or substantial derivative... 

Time derivatives play a central role in rheology. As seen above, the upper and lower convected derivatives fall out naturally from the deformation tensors. The familiar partial derivative, 8/9t, corresponds to an observer with a fixed position. The total derivative, d/dt, allows the observer to move freely in space, while if the observer follows a material point we have the material , or substantial derivative, denoted variously by the symbols d(m)/dr, D/Dr or ( ). We could expect that these different expressions could find their way into constitutive relations (see Section 5) as time rates of change of quantities that are functions of spatial position and time. However, only certain rate operations can be used by themselves in constitutive relations. This will depend on how two different observers who are in rigid motion with respect to each other measure the same quantity. The expectation is that a valid constitutive relation should be invariant to such changes in observer. This principle is called material frame indifference or material objectivity , and constitutes one of the main tests that a proposed constitutive relation has to pass before being considered admissible. 

When applying the conservation laws to fluid matter treated as a continuum, the question arises as to the amount of matter to be considered. Typically this decision is based on convenience and/or level of detail required. There is no single choice. Many possibilities exist that can lead to the same useful predictions. The conservation laws of continuum dynamics can be applied to the fluid contained in a volume of arbitrary size, shape, and state of motion. The volume selected is termed a control volume. The simplest is one where every point on its surface moves with the local fluid velocity. It is called a material control volume since, in the absence of diffusion across its interface, it retains the material originally present within its control surface. Although conceptually simple, they are not readily used since they move through space, change their volume, and deform. An analysis of the motion of material control volumes is usually termed Legrangian and time derivatives are termed material or substantial derivatives. 

Polychlorinated Pesticides. A once substantial but now diminished use for DCPD is in the preparation of chlorinated derivatives for further use or synthesis into pesticide compounds (see Insectcontrol technology). Soil permanence and solubiUty of the products in human fatty tissues have considerably restricted the use of these compounds. The more prominent chlorinated pesticides were aldrin, dieldrin, chlordane, and heptachlor, all of which use hexachorocyclopentadiene as a starting material. Aldrin and dieldrin are no longer used in the U.S. Chlordane and heptachlor are stiU produced, but only for export use. 

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