Hydrodynamics convection

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

We have so far assumed that the atoms deposited from the vapor phase or from dilute solution strike randomly and balHstically on the crystal surface. However, the material to be crystallized would normally be transported through another medium. Even if this is achieved by hydrodynamic convection, it must nevertheless overcome the last displacement for incorporation by a random diffusion process. Therefore, diffusion of material (as well as of heat) is the most important transport mechanism during crystal growth. An exception, to some extent, is molecular beam epitaxy (MBE) (see [3,12-14] and [15-19]) where the atoms may arrive non-thermalized at supersonic speeds on the crystal surface. But again, after their deposition, surface diffusion then comes into play. 

In many crystal growth phenomena, transportation of material occurs not only via diffusion but also via hydrodynamic convection. We assume... 

Here, D is the diffusion constant for heat or material and the kinematic viscosity of the liquid. A consequence of the existence of such a diffusive surface barrier is that the diffusion length = D/F is to be replaced by in all formulas, as soon as growth rate V the more important become the hydrodynamic convection effects. 

In distinction to the equilibrium state the dynamics of adsorption is characterised by time dependence and inhomogeneous distributions of surface and bulk concentration. This is accompanied by hydrodynamic convection and convective diffusion as transport process for the molecules between the bulk phase and the adsorption layer. 

Thermal limited. In the hydrodynamic convection and conduction limited cases, there is usually an amount of fluid entering the system that is in excess of that required to cool the hot surface as a result of evaporation. However, there will exist a thermal limit in which the heat generation in the system exceeds that which can be removed by the liquid being introduced. 

In many works on chemical patterns gels inactive regarding the chemistry involved have been used as reaction medium to suppress hydrodynamic convection while allowing the control of the system far from equilibrium. Instead, in this study, we consider the case of an active gel the behavior of which is influenced by some products of the reaction. We propose a simple model where the gel exhibits mechanical oscillations in response to a chemical reaction. 

Instantaneous adsorption. During the flow of polymer solutions through porous media, the hydrodynamic convection of macromolecules brings them into contact with solid surface in a mean time t which is of the order of a few seconds under our experimental conditions. Since the probability for a macromolecule to be adsorbed during the first contact with the pore surface free of adsorbed pol3mier is very high, the polymer adsorption can be considered as instantaneous. Indeed, this mean time is negligible compared to the residence time of macromolecules t inside the column. 

In cases where the mass transport conditions reach a steady state, e.g. with a microelectrode where spherical diffusion dominates (Section 11.2.4) or with a rotating disc electrode where hydrodynamic convection controls the diffusion layer thickness (Section 11.2.5), the current-voltage curve takes the form of a sigmoidal wave (Figure 11.7). The voltammogram is described by equation (11.2.49) when the kinetics are reversible. If however the... 

A. Joets, R. Ribotta Electro-hydrodynamical convective structures and transitions to chaos in liquid crystals, in J. E. Wesfreid, S. Zaleski (eds.) Cellular Structure in Instabilities, Springer, Berlin, p. 294 (1984)... 

The function / is called the orientation distribution function. Changes in this function are governed by a dynamic conservation equation. It takes into account contributions to the rotational flux from Brownian motion (jj) and from the hydrodynamic convection (j/,) (Brenner 1972 Hinch and Leal 1972, Schowalter 1978 Bird et al., 1987)... 

The hydrodynamic convection can be expressed in terms of the angular velocity Qa c, whereas the Brownian term is determined by the product of the rotary diffusion coefficient Dr and the gradient V/ ... 

Forced convection (hydrodynamic) generator - collector systems are commonly employed in rotating ring-disc or wall-jet geometries or in channel flow cells to improve collection efficiencies. For macroscopic interelectrode gap systems hydrodynamic agitation can be employed to improve feedback, but for diffusion - dominated nano-gap electrode systems hydrodynamic convection effects usually remain insignificant, whereas heating can be used to enhance the rate of diffusion processes and therefore to improve feedback currents. 

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