Continuum fluid dynamics

This chapter is organized into two main parts. To give the reader an appreciation of real fluids, and the kinds of behaviors that it is hoped can be captured by CA models, the first part provides a mostly physical discussion of continuum fluid dynamics. The basic equations of fluid dynamics, the so-called Navier-Stokes equations, are derived, the Reynolds Number is defined and the different routes to turbulence are described. Part I also includes an important discussion of the role that conservation laws play in the kinetic theory approach to fluid dynamics, a role that will be exploited by the CA models introduced in Part II. 

A mathematical description of continuum fluid dynamics can proceed from two fundamentally different points of view (1) A macroscopic, or top-down ([hass87], [hass88]), approach, in which the equations of motions are derived as the most 

We begin our discus.sion with the top-down approach. Let F be a two or three dimensional region filled with a fluid, and let v x,t) be the velocity of a particle of fluid moving through the point x = ( r, y, z) at time t. Note that v x, t) is a vector-valued field on F, and is to be identified with a macroscopic fluid cell. The fact that we can make this so-called continuum assumption - namely that we can simultaneously speak of a velocity of a particle of fluid and think of a particle of fluid as a macroscopic cell - is not at all obvious, of course, and deserves some attention. 

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Diffusion in flowing fluids can be orders of magnitude faster than in nonfiowing fluids. This is generally estimated from continuum fluid dynamics simulations. 

Chapter 9 provides an introductory discussion of a research area that is rapidly growing in importance lattice gases. Lattice gases, which are discretized models of continuous fluids, represent an early success of CA modeling techniques. The chapter begins with a short primer on continuum fluid dynamics and proceeds with a discussion of CA lattice gas models. One of the most important results is the observation that, under certain constraints, the macroscopic behavior of CA models exactly reproduces that predicted by the Navier-Stokes equations. 

Continuum Fluid Dynamics can be traced as a function of TZ alone. 

Although the terminal settling velocities for nanoparticles are extremely small and of little consequence, it is interesting to note that they are also much faster than predicted from continuum fluid dynamics. As with nanoparticle diffusion, this is a result of particle slip between gas molecules. The terminal settling velocity for particles (Dp < 20 4,m) in air is given by the following equation (Seinfeld and Pandis 1998) ... 

The equations of motion for granular flows have been derived by adopting the kinetic theory of dense gases. This approach involves a statistical-mechanical treatment of transport phenomena rather than the kinematic treatment more commonly employed to derive these relationships for fluids. The motivation for going to the formal approach (i.e., dense gas theory) is that the stress field consists of static, translational, and collisional components and the net effect of these can be better handled by statistical mechanics because of its capability for keeping track of collisional trajectories. However, when the static and collisional contributions are removed, the equations of motion derived from dense gas theory should (and do) reduce to the same form as the continuity and momentum equations derived using the traditional continuum fluid dynamics approach. In fact, the difference between the derivation of the granular flow equations by the kinetic approach described above and the conventional approach via the Navier Stokes equations is that, in the latter, the material properties, such as viscosity, are determined by experiment while in the former the fluid properties are mathematically deduced by statistical mechanics of interparticle collision. 

One of the assumptions of continuum fluid dynamics is that the velocity of the fluid at the surface of the particle surface is zero. However, as the size of the particles approaches molecular dimensions, this assumption becomes increasingly unreasonable. The first slip correction factors were developed by Cunningham to take this into account. The correction factor Cc is given by... 

A tenet of textbook continuum fluid dynamics is the no-slip boundary condition, which means that the ensemble average of the velocity of fluid molecules directly at the surface of a solid is the same as the velocity of the solid. A possible slip was discussed only in the mainstream literature for complex liquids, for example, polymer melts [659,660]. Recent experiments, however, indicated that simple liquids might also slip past smooth surfaces [661-666]. 

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