Substantial differential derivative

Equation (3.133) can be applied to a fluid element moving with the mass average velocity v. After replacing the differential operators with substantial time derivative operators inEq. (3.133), we have... 

The terms in brackets equal an operator analogous to the substantial time derivative known from the transport phenomena literature. The total differential of / (r, c, t) is given by ... 

The titaniated (25)-2,5-dihydro-2-isopropyl-3,6-dimethoxypyrazines derived from cyclo(L-Val, Gly) or cyclo(L-Val, Ala) (1, R1 = H, CH3) react with a,/I-unsaturatcd aldehydes exclusively by 1.2-addition (cf. nearly exclusive 1,4-addition of ,//-unsaturated ketones with cuprate complexes of 2,5-dialkoxy-3,6-dihydropyrazines, see Section D. 1.5.2.3.1.4.) in a highly diastereoselective mode to give virtually only the (l S,2R)-diastereoniers 2 ". In reactions with the corresponding lithiated pyrazines both regioselectivity and diastereofacial differentiation at C-2 are also remarkably high (dc 95 %), but the diastereomeric excess at C-l is substantially smaller (30 50%) ... 

The formation of the Wheland intermediate from the ion-radical pair as the critical reactive intermediate is common in both nitration and nitrosation processes. However, the contrasting reactivity trend in various nitrosation reactions with NO + (as well as the observation of substantial kinetic deuterium isotope effects) is ascribed to a rate-limiting deprotonation of the reversibly formed Wheland intermediate. In the case of aromatic nitration with NO, deprotonation is fast and occurs with no kinetic (deuterium) isotope effect. However, the nitrosoarenes (unlike their nitro counterparts) are excellent electron donors as judged by their low oxidation potentials as compared to parent arene.246 As a result, nitrosoarenes are also much better Bronsted bases249 than the corresponding nitro derivatives, and this marked distinction readily accounts for the large differentiation in the deprotonation rates of their respective conjugate acids (i.e., Wheland intermediates). 

Chiral butyrolactones of type 27 and 28 have substantial value in asymmetric synthesis because they contain readily differentiable difunctional group relationships e.g. 1,5-di-carboxylic acid, 1,4-hydroxy carboxylic acid, 1,6-hydroxy-carboxylic acid, 1,6-diol etc.) that would be difficult to assemble by existing asymmetric condensation and pericyclic processes. Applications of these chiral derivatives of glutaric acid to syntheses of indole, indoline and quinolinone alkaloids are illustrated in Schemes 16-18. 

The data here related on the kinetics of the propylene polymerization and of the transfer processes and the studies of the catalysts carried out with C-labelled alkylaluminums, derive from a series of researches mostly carried out some time ago, when the knowledge of the mechanism of the considered catalytic processes was still rather limited. Nevertheless, it helped remarkably to know these new processes of anionic coordinated polymerization their true catalytic nature (which regard to a-TiCU) differentiates them from the more usual polymerization processes (radicalic) which, actually, are not catalytic. They substantially contributed to demonstrate that the anionic coordinated polymerization is a step-wise addition process in which each monomeric unit inserts itself into a metal carbon bond of the catalytic complex. 

The net flow of solvent into the solution will continue unless a pressure equal to II is applied in the reverse direction (for example, as a result of the buildup of a hydrostatic head on the solution side of the membrane as the dilution process proceeds). Indeed, if a pressure greater than II is deliberately applied on the solution side, the net flow of solvent will be out of the solution and into the pure solvent (Fig. 14.2b). This is reverse osmosis, which obviously can be used to derive pure water from solutions. The semipermeable membrane must be able to withstand a substantial pressure differential, as well as to resist passage of ions and other solutes... 

The differential operator in the integrand represents the substantial derivative for a flowing system, although in this form it is somewhat disguised. 

Overall our objective is to cast the conservation equations in the form of partial differential equations in an Eulerian framework with the spatial coordinates and time as the independent variables. The approach combines the notions of conservation laws on systems with the behavior of control volumes fixed in space, through which fluid flows. For a system, meaning an identified mass of fluid, one can apply well-known conservation laws. Examples are conservation of mass, momentum (F = ma), and energy (first law of thermodynamics). As a practical matter, however, it is impossible to keep track of all the systems that represent the flow and interaction of countless packets of fluid. Fortunately, as discussed in Section 2.3, it is possible to use a construct called the substantial derivative that quantitatively relates conservation laws on systems to fixed control volumes. 

This equation represents the rate of change of the system s total stored energy in terms of the substantial derivative for a flowing system applied to an Eulerian control volume fixed in space. Differentiating the definition of total energy yields an expression for the substantial derivative of the total energy... 

Substantiation for this hypothesis is constructed on the availability in the initial set of differential equations with a small parameter s standing before some derivatives. We will write this set as... 

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