Magnus equation

There is, however, little exact data for its application. Lindemann and Magnus found that C, could be fairly well represented by adding to Einstein s equation an arbitrary term... 

Machines, 25, 52 Macro-differential, 39 Magnetic energy, 27 Magnus s vapour-pressure equation, 180... 

One therefore has to decide here which components of the phase interaction force (drag, virtual mass, Saffman lift, Magnus, history, stress gradients) are relevant and should be incorporated in the two sets of NS equations. The reader is referred to more specific literature, such as Oey et al. (2003), for reports on the effects of ignoring certain components of the interaction force in the two-fluid approach. The question how to model in the two-fluid formulation (lateral) dispersion of bubbles, drops, and particles in swarms is relevant... 

W. Magnus and S. Winkler, Hills Equation, Wiley-Interscience, New York, 1966. 

Magnus and coworkers have published full details55 on the direct a- or /J-azido functionalization of triisopropylsilyl (TIPS) enol ethers using an iodosylbenzene-TMS-azide combination (equation 13) the w-pathway, favoured at —78 °C, is an azide radical addition process, whereas the -pathway, favoured at —15 to — 20 °C, involves ionic dehydrogenation. Attempts to extend the /3-functionalization to other TMSX derivatives failed. 

For general particle motion formulation, additional forces such as the Saffman force, Magnus force, and electrostatic force should be included. Assuming that all forces applied on the moving particle are additive, the equation of motion of a particle in an arbitrary flow can be expressed by... 

The reagent has been used by Magnus in studies directed towards a synthesis of bachrachotoxin (71). A cir-decalin was oxidized selectively at the 7-position without the acetal or triple bond moieties present being affected (equation 37). The reagent has also found use in an approach to forskolin ° cyclohexadienone (72) was prepared by exclusive oxidation at the 7-position of (73 equation 38). 

A significant advance in the use of Friedel-Crafts acylation of alkenes to prepare divinyl ketones was the employment of vinylsilanes to control the site of electrophilic substitution. Two groups have developed this approach to cyclopentenone annulation using slightly different strategies. In the method described by Magnus the reagent vinyltrimethylsilane (80) is used primarily as an ethylene equivalent (equation 44). The construction of bicyclic systems followed readily as Nazarov cyclization proceeded under the reaction conditions. Tin(lV) chloride was found to be the most effective promoter of the overall transformation. As expected the position of the double bond is thermodynamically controlled. 

Magnus was the first to develop extensive synthetic applications of the Pauson-Khand preparation of the bicyclo[3.3.0]oct-l-en-3-one system. His efforts amply demonstrate the degree to which the high level of functionality in the Pauson-Khand products can be directly utilized in building more complex structures. A formal synthesis of the antitumor sesquiterpene coriolin illustrates a very efficient sequence for construction of the third ring in the linearly fused triquinane series in the presence of considerable functionality (Schemes 10 and 18). A synthesis of the related triquinane hirsutic acid utilizes the observation that the proper stereochemical relationship between the substituents at C-7 and the ring-fusion carbon (C-5) of the bicyclo[3.3.0]oct-l-en-3-one system, while not controllable in the cycloaddition reaction itself, may be readily established by acid- or base-catalyzed equilibration (equation 54 and Scheme 19). ... 

The ionic hydrogenation of indoles to indolines using EtsSiH/TFA has been described by Fames et al. and utilized by McKenzie, Magnus,Hlasta and Ward, and their coworkers in several different situations (equations 85-89). For the most part, the chemoselectivity and stereoselectivity that are observed with this reagent combination parallel that which was observed with NaBH4/TFA vide supra). 

In particular applications alternative relations for the slip velocity (3.428) can be derived introducing suitable simplifying assumptions about the dispersed phase momentum equations comparing the relative importance of the pressure gradient, the drag force, the added mass force, the Basset force, the Magnus force and the Saffman lift force [125, 119, 58]. For gas-liquid flows it is frequently assumed that the last four effects are negligible [201, 19[. 

W. Magnus, On the Exponential Solution of Differential Equations for a Linear Operator, Commun. Pure Appl. Math., VII, 649 (1954). 

Once W is known, P can always be found. Hence, the minimization in Equation (3.8) is only over W. The components T are restricted to be in the column-space of X. This restriction is inactive (see Appendix 3.A), that is, solving for T directly in Equation (3.8) automatically gives components in the column-space of X. Problem (3.8) is also known as the one-mode component problem [Magnus Neudecker 1988] because components are sought to summarize the columns of X (which is only one of the modes of X). Hence, the purpose is to reduce one mode of X. A more detailed treatment of the Pearson approach is given in Appendix 3.A. 

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