Navier-Stokes theory

This chapter is aimed at giving a concise presentation of the necessary tools. The first section of this chapter is devoted to the basic principles of hydrodynamics. In the second section, a description of hydrodynamic interactions between moving particles in a fluid is presented. Limitation is made to the level of the Navier-Stokes theory commonly used in the theory of electrolyte solutions. 

The hydrodynamic theory for uniaxial nematic liquid crystals was developed around 1968 by Leslie [10, 11] and Ericksen [12, 13] (Leslie-Ericksen theory, LE theory). An introduction into this theory is presented by F. M. Leslie (see Chap. Ill, Sec. 1 of this Volume). In 1970 Parodi [14] showed that there are only five independent coefficients among the six coefficients of the original LE theory. This LEP theory has been tested in numerous experiments and has been proved to be valid between the same limits as the Navier-Stokes theory. An alternative derivation of the stress tensor was given by Vertogen [15]. 

Hadjiconstantinou NG The limits of Navier-Stokes theory and kinetic extensions for describing small-scale gaseous hydrodynamics, Phys Fluids 18 111301, 2006. 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

A proposal based on Onsager s theory was made by Landau and Lifshitz [27] for the fluctuations that should be added to the Navier-Stokes hydrodynamic equations. Fluctuating stress tensor and heat flux temis were postulated in analogy with the Onsager theory. Flowever, since this is a case where the variables are of mixed time reversal character, tlie derivation was not fiilly rigorous. This situation was remedied by tlie derivation by Fox and Ulilenbeck [13, H, 18] based on general stationary Gaussian-Markov processes [12]. The precise fomi of the Landau proposal is confimied by this approach [14]. 

Ocily n. - 1 of the n equations (4.1) are independent, since both sides vanish on suinming over r, so a further relation between the velocity vectors V is required. It is provided by the overall momentum balance for the mixture, and a well known result of dilute gas kinetic theory shows that this takes the form of the Navier-Stokes equation... 

Temam R. (1979) Navier-Stokes equations. Theory and numerical analysis. North-Holland, Amsterdam, New-York, Oxford. 

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. 

These conditions show us immediately that in the case of the four-neighbor HPP lattice (V = 4) f is noni.sotropic, and the macroscopic equations therefore cannot yield a Navier-Stokes equation. For the hexagonal FHP lattice, on the other hand, we have V = 6 and P[. is isotropic through order Wolfram [wolf86c] predicts what models are conducive to f lavier-Stokes-like dynamics by using group theory to analyze the symmetry of tensor structures for polygons and polyhedra in d-dimensions. 

New questions have arisen in micro-scale flow and heat transfer. The review by Gad-el-Hak (1999) focused on the physical aspect of the breakdown of the Navier-Stokes equations. Mehendale et al. (1999) concluded that since the heat transfer coefficients were based on the inlet and/or outlet fluid temperatures, rather than on the bulk temperatures in almost all studies, comparison of conventional correlations is problematic. Palm (2001) also suggested several possible explanations for the deviations of micro-scale single-phase heat transfer from convectional theory, including surface roughness and entrance effects. 

In Spite of the existence of numerous experimental and theoretical investigations, a number of principal problems related to micro-fluid hydrodynamics are not well-studied. There are contradictory data on the drag in micro-channels, transition from laminar to turbulent flow, etc. That leads to difficulties in understanding the essence of this phenomenon and is a basis for questionable discoveries of special microeffects (Duncan and Peterson 1994 Ho and Tai 1998 Plam 2000 Herwig 2000 Herwig and Hausner 2003 Gad-el-Hak 2003). The latter were revealed by comparison of experimental data with predictions of a conventional theory based on the Navier-Stokes equations. The discrepancy between these data was interpreted as a display of new effects of flow in micro-channels. It should be noted that actual conditions of several experiments were often not identical to conditions that were used in the theoretical models. For this reason, the analysis of sources of disparity between the theory and experiment is of significance. 

Historically, gas lubrication theory was developed from the classical liquid lubrication equation—Re5molds equation [4]. The first gas lubrication equation was derived by Harrison [5] in 1913, taking the compressibility of gases into account. Because the classical gas lubrication equation is based on the Navier-Stokes equation, it does not incorporate some gas flow characteristics rooted in the rarefaction effects of dilute gases. As early as 1959, Brunner s experiment [6] showed that the classical gas lubrication equation was... 

Alternative methods of analysis have been examined and evaluated. Shokoohi and Elrod[533] solved the Navier-Stokes equations numerically in the axisymmetric form. Bogy15271 used the Cosserat theory developed by Green.[534] Ibrahim and Linl535 conducted a weakly nonlinear instability analysis. The method of strained coordinates was also examined. In spite of the mathematical or computational elegance, all of these methods suffer from inherent complexity. Lee15361 developed a 1 -D, nonlinear direct-simulation technique that proved to be a simple and practical method for investigating the nonlinear instability of a liquid j et. Lee s direct-simulation approach formed the... 

However, one difference exists with classical theory in this latter case, the Navier-Stokes equation (443) and the incompressibility condition (444) are assumed to be valid for all distances rict. In this case, it is an easy matter to calculate explicitly the higher-order terms in Eq. (445), and the boundary condition at the B-particle (assumed to be spherical) imposes the condition... 

Stokes Postulates Stokes s postulates provide the theory to relate the strain-rate to the stress. As a result the forces may be related to the velocity field, leading to viscous-force terms in the Navier-Stokes equations that are functions of the velocity field. Working in the principal coordinates facilitates the development of the Stokes postulates. 

There are a very wide variety of theories and approaches to determine transport properties and to report their functional dependencies [178,332]. The brief discussion here serves only to establish the basic functional dependencies, and thus facilitate understanding the role of viscosity in the Navier-Stokes equations. 

In addition to the general treatments of wavy flow, a number of theories concerning the stability of film flow have been published in these the flow conditions under which waves can appear are determined. The general method of dealing with the problem is to set up the main equations of flow (usually the Navier-Stokes equations or the simplified Nusselt equations), on which small perturbations are imposed, leading to an equation of the Orr-Sommerfeld type, which is then solved by various approximate means to determine the conditions for stability to exist. The various treatments are lengthy, and only the briefest summaiy of the results can be given here. 

Since in hydrodynamic lubrication the friction force is completely determined by the viscous friction of the lubricant, the coefficient of friction can be calculated from hydrodynamics using the Navier-Stokes equations. This had already been done in 1886 when Reynolds published his classical theory of hydrodynamic lubrication [494], The friction force Fp between two parallel plates of area A separated by the distance d is given by ... 

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