Computational fluid mechanics separation

Materials in the macroscopic sense follow laws of continuum models in which the nanoscale phenomenon is accounted for by statistical averages. Continuum models and analysis separate materials into solids (structures) and fluids. Computational solid mechanics and structural mechanics emphasize the analysis of solid materials and its structural design. Computational fluid mechanics treats material behaviors that involve the equilibrium and motion of liquid and gases. A relative new area, called multiphysics, includes materials systems that contain interacting fluids and structures such as phase changes (solidification, melting), or interaction of control, mechanical and electromagnetic (MEMS, sensors, etc.). 

The above set of equations is similar to that used in traditional computational fluid dynamics (CFD) (except for the electron energy balance and the EM equations), and advances made in that field can be used to benefit the plasma reactor simulation problem. As an example, the sheath near the wall can be thought of as similar to a boundary layer in fluid flow (chemically reacting or not) [130]. Separating the flow into bulk (inviscid) and boundary layer (viscous) and then patching the two solutions (asymptotic analysis) has long been practiced in fluid mechanics and may also be applied to the plasma problem [102, 103, 151]. 

While the Navier-Stokes equation is a fundamental, general law, the boundary conditions are not at all dear. In fluid mechanics, one usually relies on the assumption that when liquid flows over a solid surface, the liquid molecules adjacent to the solid are stationary relative to the solid and that the viscosity is equal to the bulk viscosity. We applied this no-slip boundary condition in Eq. (6.18). Although this might be a good assumption for macroscopic systems, it is questionable at molecular dimensions. Measurements with the SFA [644—647] and computer simulations [648-650] showed that the viscosity of simple liquids can increase many orders of magnitude or even undergo a liquid to solid transition when confined between solid walls separated by only few molecular diameters water seems to be an exception [651, 652]. Several experiments indicated that isolated solid surfaces also induce a layering in an adjacent liquid and that the mechanical properties of the first molecular layers are different from the bulk properties [653-655]. An increase in the viscosity can be characterized by the position of the plane of shear. Simple liquids often show a shear plane that is typically 3-6 molecular diameters away from the solid-liquid interface [629, 644, 656-658]. 

The attenuation of the reflected shock wave over 12 cycles of reflection within cylindrical and spherical vessels has been examined. Computations without added dissipation simulate the qualitative features of the measured pressure histories, but the shock amplitudes and decay rates are incorrect. Computations using turbulent channel flow dissipation models have been compared with measurements in a cylindrical vessel. These comparisons indicate that the nonideal aspects of the experiments result in a much more rapid decay of the shock wave than predicted by the simple channel flow model. Dissipation mechanisms not directly accounted for in the present model include multidimensional flow associated with transverse shock waves (originating in detonation or shock instability) separated flow due to shock wave-boundary layer interactions the influence of flow in the initiator tube arrangement and real gas (dissociation and ionization) effects and fluid dynamic instabilities near the shock focus in cylindrical and spherical geometries. 

Before proceeding to such types of analysis and computations in the sections that follow, we begin with a statement of the full problem with as much of the physics represented as possible. Our approach is to work with macroscopic models of the interface separating the fluid phases. This approach represents the interface by a sharp dynamic surface embedded in three-dimensional space, across which flow and concentration variables can jump in a manner specified by physical boundary conditions. The alternative microscopic approach seeks to describe the three-dimensional thin transition layer between the two phases using statistical or continuum mechanical methods. The reader is referred to Chapters 15-18 of the text by Edwards, Brenner and Wasan as well as the many references therein. 

---