Continuum fluids

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

StoKes-Einstein and Free-Volume Theories The starting point for many correlations is the Stokes-Einstein equation. This equation is derived from continuum fluid mechanics and classical thermodynamics for the motion of large spherical particles in a liqmd. 

While there are mairy variants of the basic, model, one can show that there is a well-defined minimal set of niles that define a lattice-gas system whose macroscopic behavior reproduces that predicted by the Navier-Stokes equations exactly. In other words, there is critical threshold of rule size and type that must be met before the continuum fluid l)cliavior is matched, and onec that threshold is reached the efficacy of the rule-set is no loner appreciably altered by additional rules respecting the required conservation laws and symmetries. 

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. 

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... 

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

In situations where the surrounding fluid behaves as a non-continuum fluid, for example at very high temperatures and/or at low pressures, it is possible for Nu to be less than 2. A gas begins to exhibit non-continuum behaviour when the mean free path between collisions of gas molecules or atoms with each other is greater than about 1/100 of the characteristic size of the surface considered. The molecules or atoms are then sufficiently far apart on average for the gas to begin to lose the character of a homogeneous or continuum fluid which is normally assumed in the majority of heat transfer or fluid... 

When the length scale approaches molecular dimensions, the inner spinning" of molecules will contribute to the lubrication performance. It should be borne in mind that it is not considered in the conventional theory of lubrication. The continuum fluid theories with microstructure were studied in the early 1960s by Stokes [22]. Two concepts were introduced couple stress and microstructure. The notion of couple stress stems from the assumption that the mechanical interaction between two parts of one body is composed of a force distribution and a moment distribution. And the microstructure is a kinematic one. The velocity field is no longer sufficient to determine the kinematic parameters the spin tensor and vorticity will appear. One simplified model of polar fluids is the micropolar theory, which assumes that the fluid particles are rigid and randomly ordered in viscous media. Thus, the viscous action, the effect of couple stress, and... 

In nonreacting, continuum fluid mechanics the fluid velocity normal to a solid wall is zero, which is a no-slip boundary condition. However, if there are chemical reactions at the wall, then the velocity can be nonzero. The so-called Stefan flow velocity occurs... 

The simple formula derived for viscosity in Eq. 12.49 predicts that /z should be independent of pressure and should increase as the square root of temperature. It is typically found that the viscosity of a gas is independent of pressure except at high and low pressure extremes. At very high pressure, molecular interactions become more important and the rigid-sphere approximation becomes inappropriate, leading to a breakdown in Eq. 12.49. At very low pressures, the gas no longer behaves like a continuum fluid, and the steady-state flow picture of Fig. 12.1 is no longer valid. Viscosity is usually found experimentally to increase with T faster than the n = 1 /2 power. Consideration of the interaction potential between molecules, as is discussed in the next section, is needed to more closely match the observed temperature dependence of /x. 

The transition to the continuum fluid may be mimicked by a discretization of the model choosing > 1. To this end, Panagiotopoulos and Kumar [292] performed simulations for several integer ratios 1 < < 5. For — 2 the tricritical point is shifted to very high density and was not exactly located. The absence of a liquid-vapor transition for = 1 and 2 appears to follow from solidification, before a liquid is formed. For > 3, ordinary liquid-vapor critical points were observed which were consistent with Ising-like behavior. Obviously, for finely discretisized lattice models the behavior approaches that of the continuum RPM. Already at = 4 the critical parameters of the lattice and continuum RPM agree closely. From the computational point of view, the exploitation of these discretization effects may open many possibilities for methodological improvements of simulations [292], From the fundamental point of view these discretization effects need to be explored in detail. 

Ex 2. A classical one-dimensional continuum fluid with next neighbor interaction7 [Pg.135]

Viscous flow of a pure fluid (Kn 1). The gas acts as a continuum fluid driven by a pressure gradient, and both molecule-molecule collisions and molecule-wall collisions are important. This is sometimes called convective or bulk flow. 

However, the hydrodynamic problem that Stokes solved to get F = 67cn/v pertains to a sphere moving in an incompressible continuum fluid. This is a far cry indeed from the actuality of an ion drifting inside a discontinuous electrolyte containing particles (solvent molecules, other ions, etc.) of about the same size as the ion. Furthermore, the ions considered may not be spherical. 

To reconcile this apparent contradiction the membrane skeleton fence and anchored transmembrane picket model was proposed (54). According to this model, transmembrane proteins anchored to and lined up along the membrane skeleton (fence) effectively act as a row of posts for the fence against the free diffusion of lipids (Fig. 11). This model is consistent with the observation that the hop rate of transmembrane proteins increases after the partial removal of the cytoplasmic domain of transmembrane proteins, but it is not affected by the removal of the major fraction of the extracellular domains of transmembrane proteins or extracellular matrix. Within the compartment borders, membrane molecules undergo simple Brownian diffusion. In a sense, the Singer-Nicolson model is adequate for dimensions of about 10 x lOnm, the special scale of the original cartoon depicted by the authors in 1972. However, beyond such distances simple extensions of the fluid mosaic model fail and a substantial paradigm shift is required from a two-dimensional continuum fluid to the compartmentalized fluid. 

Kusumi A, et al. Paradigm shift of the plasma membrane concept from the two-dimensional continuum fluid to the partitioned fluid high-speed single-molecule tracking of membrane molecules. Ann. Rev. Biophys. Biomolec. Struct. 2005 34 351-378. 

Using the theory developed by Chapman-Enskog (see Ref. 14), a hierarchy of continuum fluid mechanics formulations may be derived from the Boltzmann equation as perturbations to the Maxwellian velocity distribution function. The first three equation sets are well known (1) the Euler equations, in which the velocity distribution is exactly the Maxwellian form (2) the Navier-Stokes equations, which represent a small deviation from Maxwellian and rely on linear expressions for viscosity and thermal conductivity and (3) the Burnett equations, which include second order derivatives for viscosity and thermal conductivity. 

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