Velocity expansion

Note that the velocity expansion is multiplied by the Mach number [303]. The purpose of this multiplication is to re-scale the velocity to an order-one variable. Since the nondimen-sional velocity is scaled by the sound speed, V has inherently small values for low-speed (low Mach number) flows. In other words, for the flows of interest, the leading-order velocity should be approximately an order-one variable,... 

It is significant that, throughout the reaction zone, the velocity of transport of an acoustic perturbation w + c exceeds the detonation velocity expansion of one layer, caused by release of reaction heat, is conveyed to another and beyond to the original mixture. The mechanism of detonation... 

Step 1 is purely hydrodynamic and relates the perturbation Q to the velocity near the wall which is the only relevant quantity for the mass transfer response. at are either wall velocity gradients or coefficients involved in the velocity expansion near the wall. This step requires the use of Navier-Stokes equations and will be treated in Chapter 2. 

The approaches made to obtain such a solution differ in the types of assumptions made concerning the velocity expansions. 

The corrections to the current density were obtained by accounting for additiorud terms in the velocity expansion given as equation (11.77). 

The influence of the accuracy of the velocity expansion is illustrated in Figure 11.8. The characteristic length for mass transfer to a disk electrode is given by... 

In 2000, Filippov introduced the multipole velocity expansion (MVE) for the calculation of the velocity field u in and around an aggregate (or cluster of particles) which moves with defined velocity through a viscous medium. Once this velocity field is known, the hydrodynamic forces can be calculated straightforwardly. The approach is based on the following expansion of the velocity perturbation y, around a particle j (cf. Happel and Brenner 1983, p. 260) ... 

In conclusion, the multipole velocity expansion (MVE) is a fairly rigorous calculation scheme for the hydrodynamic behaviour of arbitrary clusters or... 

Fig. 4.20 Ratio of x t to Xg for DLCA Q t) and RLCA right) aggregates ealeulated with multipole velocity expansion (MVE), Kirkwood-Riseman theory (Eq. (4.84)) and its solution by Hess (Eq. (4.86)), as well as with porous sphere approaches from Gmachowski (1996) and Vanni (2000), and based on EDM results by Coelho et al. (1997 Eq. (4.96))...

There are many algorithms for integrating the equations of motion using finite difference methods, several of which are commonly used in molecular dynamics calculations. All algorithms assume that the positions and dynamic properties (velocities, accelerations, etc.) can be approximated as Taylor series expansions ... 

Let H and L be two characteristic lengths associated with the channel height and the lateral dimensions of the flow domain, respectively. To obtain a uniformly valid approximation for the flow equations, in the limit of small channel thickness, the ratio of characteristic height to lateral dimensions is defined as e = (H/L) 0. Coordinate scale factors h, as well as dynamic variables are represented by a power series in e. It is expected that the scale factor h-, in the direction normal to the layer, is 0(e) while hi and /12, are 0(L). It is also anticipated that the leading terms in the expansion of h, are independent of the coordinate x. Similai ly, the physical velocity components, vi and V2, ai e 0(11), whei e U is a characteristic layer wise velocity, while V3, the component perpendicular to the layer, is 0(eU). Therefore we have... 

Aluminum-containing propellants deflver less than the calculated impulse because of two-phase flow losses in the nozzle caused by aluminum oxide particles. Combustion of the aluminum must occur in the residence time in the chamber to meet impulse expectations. As the residence time increases, the unbumed metal decreases, and the specific impulse increases. The soHd reaction products also show a velocity lag during nozzle expansion, and may fail to attain thermal equiUbrium with the gas exhaust. An overall efficiency loss of 5 to 8% from theoretical may result from these phenomena. However, these losses are more than offset by the increase in energy produced by metal oxidation (85—87). 

Because bubbles occupy space in a bubbling fluid bed, the expansion of the bed becomes a function of both the bubble velocity and the volume of the gas entering the bed ... 

A. A. Avidan, Bed Expansion and S olids Mining in Eligh-Velocity Fluidi dBeds, Ph.D. dissertation. City University of New York, 1980. 

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows ... 

In practice, the loss term AF is usually not deterrnined by detailed examination of the flow field. Instead, the momentum and mass balances are employed to determine the pressure and velocity changes these are substituted into the mechanical energy equation and AFis deterrnined by difference. Eor the sudden expansion of a turbulent fluid depicted in Eigure 21b, which deflvers no work to the surroundings, appHcation of equations 49, 60, and 68 yields... 

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