Motion quasi-steady

Turbulent mass transfer near a wall can be represented by various physical models. In one such model the turbulent flow is assumed to be composed of a succession of short, steady, laminar motions along a plate. The length scale of the laminar path is denoted by x0 and the velocity of the liquid element just arrived at the wall by u0. Along each path of length x0, the motion is approximated by the quasi-steady laminar flow of a semiinfinite fluid along a plate. This implies that the hydrodynamic and diffusion boundary layers which develop in each of the paths are assumed to be smaller than the thickness of the fluid elements brought to the wall by turbulent fluctuations. Since the diffusion coefficient is small in liquids, the depth of penetration by diffusion in the liquid element is also small. Therefore one can use the first terms in the Taylor expansion of the Blasius expressions for the velocity components. The rate of mass transfer in the laminar microstructure can be obtained by solving the equation... 

Neglect any effects due to the climb motion of the dislocations and assume quasi-steady-state diffusion. The vacancy isoconcentration contours around the dislocations in the boundary will then appear approximately as illustrated in Fig. 13.20. 

Statement 3. If the concentrations are fixed, iVa = const, Nb = const, the set of kinetic equations (8.2.17), (8.2.22) and (8.2.23) as functions of the control parameter demonstrates two kinds of motions for k k the stationary (quasi-steady-state) solution holds, whereas for k < k a regular (quasi-regular) oscillations in the correlation functions like standing waves... 

This statement comes from analytical and topological studies [4], Unlike the Lotka-Volterra model where due to the dependence of the reaction rate K(t) on concentrations NA and NB, the nature of the critical point varied, in the Lotka model the concentration motion is always decaying. Autowave regimes in the Lotka model can arise under quite rigid conditions. It is easy to show that not any time dependence of K(t) emerging due to the correlation motion is able to lead to the principally new results. For example, the reaction rate of the A + B -> 0 reaction considered in Chapter 6 was also time dependent, K(t) oc t1 d/4 but its monotonous change accompanied by a strong decay in the concentration motion has resulted only in a monotonous variation of the quasi-steady solutions of (8.3.20) and (8.3.21) jVa(t) (3/K(t) and N, (t) p/f3 = const. 

Numerical simulations that combine the details of the thermal-capillary models described previously with the calculation of convection in the melt should be able to predict heat transfer in the CZ system. Sackinger et al. (175) have added the calculation of steady-state, axisymmetric convection in the melt to the thermal-capillary model for quasi steady-state growth of a long cylindrical crystal. The calculations include melt motion driven by buoyancy, surface tension, and crucible and crystal rotation. Figure 24 shows sample calculations for growth of a 3-in. (7.6-cm)-diameter silicon crystal as a function of the depth of the melt in the crucible. 

Smoluchowski (16) has treated the kinetics of coagulation assuming that the particles which collide during their Brownian motion coagulate. The rate of collision was computed assuming quasi-steady dif-... 

Equation (1) holds for a particle or a cell of any shape, even when the shape changes with the distance x, provided that it is reaching the surface mainly by a translational motion. Equation (1) also assumes a quasi-steady state of the motion of particles over the potential barrier. This approximation can be made because the region over which the potential acts is very thin and consequently the flux of particles through it can be considered practically constant with respect to the distance a at a given time t. 

The analysis can be significantly simplified by reahzing that the rate with which the vorticity diffuses inwards, and hence establishes the fluid motion, is represented by the kinematic viscosity coeflBcient, which is of the order of 10 cmVsec and is at least one order of magnitude greater than the droplet surface regression rate. Hence quasi-steadiness for both the gas and liquid motion, with a stationary droplet surface and constant interfacial heat and mass flux, can be assumed. Once the fluid mechanical aspect of the problem is solved, the transient liquid-phase heat and mass transfer analyses, with a regressing droplet surface, can be performed. 

Statement 2. A set (8.3.22) to (8.3.24) with fixed concentrations = P/K and N = VIP has two kinds of motions dependent on the value of parameter n. As k kq, the stationary (quasi-steady-state) solution occurs, whereas for /c < kq the correlation functions demonstrate the regular (quasiregular) oscillations of the standing wave type. The marginal magnitude is Ko = o(p,/3)-... 

The objective is to find the steady-state escape rate k out of the potential well. Before presenting the Kramers solution it is important to note that for such a (quasi) steady state to be established, a clear separation of time scales has to exist, whereupon the escape occurs on a time scale much longer than all time scales associated with the motion inside the well. In particular this implies that the well should be deep enough (see below). 

In the quasi-steady-state approximation, which is also known as the step method [9], it is assumed that the rate of variation in the WP shape, that is, the anodic dissolution rate, is small compared with the rates of transfer processes in the gap therefore, for calculating the distribution of the current density, the WP surface can be considered as being immobile. This approximation can be used at not very high current densities. At very high current densities, ignoring the WP surface motion during anodic dissolution and the hydrodynamic flow induced by this motion causes a considerable error in the calculated distribution of current density [33]. 

The equations governing the fluid motion and heat transfer in these quasi-steady regimes are (if the property values and boundary conditions are the same) identical to those for steady-state convection in the same geometry with a uniform internal generation of energy [73]. The heat transfer equations from one situation can therefore be readily transferred to the other by replacing the constant pcp dT/dt by the internal generation rate q " (in W/m3 or Btu/h-ft3). 

As Nj 0, corresponding to quasi-steady motion, the growth coefficient should be reduced by a factor of In 2, as in Eq. (102), in agreement with the numerical solution. 

These also result even if the motion is unsteady, providing that ajxV and the other dimensionless terms remain finite in the limit R = 0 Equations (7) and (8) are then referred to as the quasi-static or quasi-steady Stokes equations. In this case the time variable enters the equations of motion only in an implicit form. The precise relationship between the solutions of Eqs. (7) and (8) and the asymptotic solutions of the Navier-Stokes equations at small Reynolds numbers is discussed in Section III. 

In this section we treat the steady and quasi-steady motions of rigid, three-dimensional particles in a fluid at rest at infinity on the basis of Stokes... 

Complete characterization of the hydrodynamic resistance of a solid particle in quasi-steady Stokes motion generally requires knowledge of 21... 

By quasi-steady motion here, we mean an unsteady motion for which the time-dependent terms in the equations of motion and boundary conditions are of higher order in R than 0(.R In R). 

The two-phase motion problem is very stiff, with a wide separation of timescales and a transport matrix which becomes singular as the solution relaxes to its quasi-steady state. The asymptotic analysis presented eliminates the stiffness that is the bane of numerical simulations, affording computational speed-up of 3-4 orders of magnitude over the full system. Building this model into a unit cell simulation code promises huge reductions in computational cost and admits the possibility of performing either full stack-based calculations or doing extensive inverse calculations and parameter estimation. 

Transient molecular deformation and orientation in the systems subjected to flow deformation results in transient and orientation dependent crystal nucleation. Quasi steady-state kinetic theory of crystal nucleation is proposed for the polymer systems exhibiting transient molecular deformation controlled by the chain relaxation time. Access time of individual kinetic elements taking part in the nucleation process is much shorter than the chain relaxation time, and a quasi steady-state distribution of clusters is considered. TVansient term of the continuity equations for the distribution of the clusters scales with much shorter characteristic time of an individual segment motion, and the distribution approaches quasi steady state at any moment of the time scaled with the chain relaxation time. Quasi steady-state kinetic theory of nucleation in transient polymer systems can be used for elongation rates in a wide range 0 < esT C N. ... 

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