Convected time derivative operator

There are other time derivative operators that transform a tensor from convected to fixed coordinates, giving rise to equivalent expression of covariant and contravariant forms of a given tensor. One such time derivative operator may be formed by eliminating d-j after Eq. (2B.17) is substituted into Eqs. (2B.15) and (2B.16) and adding... 

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. 

A key assumption in deriving the SR model (as well as earlier spectral models see Batchelor (1959), Saffman (1963), Kraichnan (1968), and Kraichnan (1974)) is that the transfer spectrum is a linear operator with respect to the scalar spectrum (e.g., a linear convection-diffusion model) which has a characteristic time constant that depends only on the velocity spectrum. The linearity assumption (which is consistent with the linear form of (A.l)) ensures not only that the scalar transfer spectra are conservative, but also that if Scap = Scr in (A.4), then Eap ic, t) = Eyy k, t) for all t when it is true for t = 0. In the SR model, the linearity assumption implies that the forward and backscatter rate constants (defined below) have the same form for both the variance and covariance spectra, and that for the covariance spectrum the rate constants depend on the molecular diffusivities only through Scap (i.e., not independently on Sc or Sep). 

This differentiation is often called differentiation following the motion of the liquid and, to bring out both the time and space dependence, it is often written D//Df D/Dr is sometimes called the mobile operator or the convective derivative . If, for example, / denotes the density p of the liquid, t)p/Dr gives the rate of change of density of a particular liquid particle, but dp/df gives the rate of change of density at a particular point in space, i.e. part of the total change in p results from the lapse of time and part from the movement of the particle to a different place. 

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