Acceleration substantial derivative

The term on the left-hand side, arising from the product of mass and acceleration, can be expanded using the expression for the substantial derivative operator... 

The Eulerian acceleration derivative has a special name, called the substantial derivative. It is defined using a capital D as... 

The extra terms appear because in noncartesian coordinate systems the unit-vector derivatives do not all vanish. Only in cartesian coordinates are the components of the substantial derivative of a vector equal to the substantial derivative of the scalar components of the vector. The acceleration in the r direction is seen to involve w2, the circumferential velocity. This term represents the centrifugal acceleration associated with a fluid packet as it moves in an arc defined by the 9 coordinate. There is also a G acceleration caused by a radial velocity. In qualitative terms, one can visualize this term as being related to the circumferential acceleration (spinning rate) that a dancer or skater experiences as she brings her arms closer to her body. 

This chapter established three important concepts that are essential for the derivation of the conservation equations governing fluid flow. First, the Reynolds transport theorem was developed to relate a system to an Eulerian control volume. The substantial derivative that emerges from the Reynolds transport theorem can be thought of as a generalized time derivative that accommodates local fluid motion. For example, the fluid acceleration vector... 

Here, for a steady-state problem and parallel flow, there is no acceleration. In general, the substantial derivative for the cylindrical system is... 

Here v is the local fluid velocity, D/Dt is the substantial derivative or derivative following the motion , n is the pressure tensor, and g is the acceleration of gravity. It is customary to decompose the pressure tensor (or stress tensor) n as follows ... 

Here dv/dt is the substantial derivative of the velocity or the sum of derivatives with respect to time and space. The symbols g and p denote the acceleration due to gravity and the pressure, respectively. Dealing with ideal fluids the viscosity is zero and the Navier-Stokes equation can be simplified to the Euler equation ... 

In the equations of motion for the beads in Eq. (11.4), it seems reasonable to assume that the bead-acceleration terms on the left side (which have the general appearance of a substantial derivative ) are negligibly small compared to the individual terms on the right side. This assumption has been discussed in several publications [15-19]. When the acceleration terms are omitted, the bead equation of motion reduces to a simple force balance used in the publications of Kramers [1], Kirkwood [2], Rouse [3], Zimm [4], and others ... 

As recently as 1965, Thoma and Stewart predicted that alterations in reaction rates [in the presence of the cycloamyloses] should be anticipated whose magnitude and sign will fluctuate with the reaction type, and added that at the present juncture, it is impossible to sort out confidently. . . which factors may contribute importantly to raising or lowering the activation energy of the reaction. In the short interval between 1965 and the present, a wide variety of cycloamylose-induced rate accelerations and decelerations have, indeed, been revealed. More importantly, rate alterations imposed by the cycloamyloses can now be explained with substantially more confidence. The reactions of derivatives of carboxylic acids and organo-phosphorus compounds with the cycloamyloses, for example, proceed to form covalent intermediates. Other types of reactions appear to be influenced by the dielectric properties of the microscopic cycloamylose cavity. Still other reactions may be affected by the geometrical requirements of the inclusion process. 

The important effect of increasing pressure on the kinetics of chemical reactions has been noted since the hrst chemical experiments at high pressure. The simplest expectation derives from the observation that in liquids the viscosity rapidly increases with pressure. As a result, in strongly compressed liquids, and hnally in glasses, diffusion-controlled processes can be retarded. In contrast, however, other reaction pathways can be substantially accelerated. In general, the evolution of a reaction at high pressure can be heavily controlled by kinetic aspects, and these deeply involve intermolecular effects. 

A suitable stable transition state analog is hexachloronorbomene derivative 25, which mimics most of the geometrical features of the transition state, including the boat conformation of the cyclohexene ring. It was used to poduce antibodies that catalyse the reaction between 20 and 21 efficiently, with substantial rate acceleration and multiple turnovers. 

The water-soluble palladium complex prepared from [Pd(MeCN)4](Bp4)2 and tetrasulfonated DPPP (34, n=3, m=0) catalyzed the copolymerization of CO and ethene in neutral aqueous solutions with much lower activity [21 g copolymer (g Pd) h ] [53] than the organosoluble analogue in methanol. Addition of strong Brpnsted acids with weakly coordinating anions substantially accelerated the reaction, and with a catalyst obtained from the same ligand and from [Pd(OTs)2(MeCN)2] but in the presence of p-toluenesulfonic acid (TsOH) 4 kg copolymer was produced per g Pd in one hour [54-56] (Scheme 7.16). Other tetrasulfonated diphosphines (34, n=2, 4 or 5, m=0) were also tried in place of the DPPP derivative, but only the sulfonated DPPB (n=4) gave a catalyst with considerably higher activity [56], Albeit with lower productivity, these Pd-complexes also catalyze the CO/ethene/propene terpolymerization. 

The SiCaC reaction of 5-hexyn-l-al 23 gives the corresponding 2-(exo-silymethylene)-l-cyclopentanol 24 in high yield (Scheme 7.11) [20]. The reaction is accelerated by gem-disubstitution for example, the reaction of 3,3-gem-disubstituted 5-hexyn-l-al 23 (X = C(C02Et)2, C(CH20Me)2) is substantially faster and cleaner than the unsubstituted derivative (X=CH2), which is accompanied by a small amount of silylformylation product [21]. It should also be noted that the formation of l-siloxy-2-methylenecyclopentane is not observed, in sharp contrast with the nickel-catalyzed version of this reaction [20]. 

It should also be feasible to extend further the types of reaction that can be accelerated. For example, large solvent effects have been observed in kinetic studies of many reactions involving anions.46,47,5° 51 In many cases the solvents are aprotic but not truly apolar, in the sense that their molecules have large dipole moments, for example, (CH3)2S=0, CH3CON(CH3)2. Derivatives of polyethylenimine can be made that have substituents mimicking these in chemical structure. For example, acylation of the polymer will produce CH3CO—N=C loci on the macromolecule. Such modified polymers should manifest substantial catalytic effects. 

Maruoka and coworkers also investigated the substantial reactivity enhancement of N-spiro chiral quaternary ammonium salt and simplification of its structure, the aim being to establish a truly practical method for the asymmetric synthesis of a-amino acids and their derivatives. As ultrasonic irradiation produces homogenization (i.e., very fine emulsions), it greatly increases the reactive interfacial area, which may in turn deliver a substantial rate acceleration in the liquid-liquid phase-transfer reactions. Indeed, sonication of the reaction mixture of 2, methyl iodide and (S,S)-lc (1 mol%) in toluene-50% KOH aqueous solution at 0 °C for 1 h gave rise to the corresponding alkylation product in 63% yield with 88% ee. Hence, the reaction was speeded up markedly, and the chemical yield and enantioselectivity were comparable with those of the reaction with simple stirring (0°C for 8h 64%, 90% ee) (Scheme 5.5) [10]. 

The Basset force can be substantial when the particle is accelerated at a high rate. The total force on a particle in acceleration can be many times that in a steady state [Hughes and Gilliland, 1952]. In a simple model with constant acceleration, the ratio of the Basset force to the Stokes drag, / gs> was derived [Wallis, 1969] and rearranged to [Rudinger, 1980]... 

---