Laws Navier-Stokes equations

Key words convective dispersion, fluid mechanics, package dyeing geometry, modelling fluid flow in dyeing, Darcy s law, Navier-Stokes equations. 

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

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. 

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. 

The Navier-Stokes equations have a complex form due to the necessity of treating many of the terms as vector quantities. To understand these equations, however, one need only recognize that they are not mass balances but an elaboration of Newton s second law of motion for a flowing fluid. Recall that Newton s second law states that the vector sum of all the forces acting on an object ( F) will be equal to the product of the object s mass (m) and its acceleration (a), or XF = ma. Now consider the first of the three Navier-Stokes equations listed above, Eq. (10). The object in this case is a differential fluid element, that is, a small cube of fluid with volume dx dy dz and mass p(dx dy dz). The left-hand side of the equation is essentially the product of mass and acceleration for this fluid element (ma), while the right-hand side represents the sum of the forces... 

Navier-Stokes equations A series of differential equations derived from Newton s second law of motion (XF = raa) that describe the relationship between fluid velocity and applied forces in a moving fluid. See Eqs. (10)—(12). 

The momentum balance is a version of Newton s Second Law of mechanics, which students first encounter in introductory physics as F = mo, with F the force, m the mass, and a the acceleration. For an element of fluid the force becomes the stress tensor acting on the fluid, and the resulting equations are called the Navier-Stokes equations. 

Application of Newton s second law of motion to an infinitesimal element of an incompressible Newtonian fluid of density p and constant viscosity p, acted upon by gravity as the only body force, leads to the Navier-Stokes equation of motion ... 

At the phenomenological level, there are enough further relations between the 14 variables to reduce the number to 5 and make the problem determinate. These further relations are the thermodynamic ones and Stokes and Newton s laws of viscosity and heat flow. These lead from the transport equations to the Navier-Stokes equations. It is noted that these are irreversible. 

Conservation Law for a System Conservation laws (e.g., Newton s second law or the conservation of energy) are most conveniently written for a system, which, by definition, is an identified mass of material. In fluid mechanics, however, since the fluid is free to deform and mix as it moves, a specific system is difficult to follow. The conservation of momentum, leading to the Navier-Stokes equations, is stated generally as... 

The energy equation is a statement of the first law of thermodynamics, just as the Navier-Stokes equations are a statement of Newton s second law, F = ma. For a system, the first law states that the rate of change of stored energy equals the rate of heat... 

The second is the law of conservation of momentum which, for a fluid of constant density and viscosity, is the Navier—Stokes equation... 

Keywords Navier-Stokes equation, diffuse double-layer, Darcy s law... 

The basic hydrodynamic equations are the Navier-Stokes equations [51]. These equations are listed in their general form in Appendix C. The combination of these equations, for example, with Darcy s law, the fluid flow in crossflow filtration in tubular or capillary membranes can be described [52]. In most cases of enzyme or microbial membrane reactors where enzymes are immobilized within the membrane matrix or in a thin layer at the matrix/shell interface or the live cells are inoculated into the shell, a cake layer is not formed on the membrane surface. The concentration-polarization layer can exist but this layer does not alter the value of the convective velocity. Several studies have modeled the convective-flow profiles in a hollow-fiber and/or flat-sheet membranes [11, 35, 44, 53-56]. Bruining [44] gives a general description of flows and pressures for enzyme membrane reactor. Three main modes... 

In the derivation of Stokes law, the assumption of a perfectly viscous medium means that no inertial forces are considered. This was done to linearize the Navier-Stokes equation. If these inertial effects are included in a first-order approximation, it is possible to extend the applicability of Stokes law up to a Reynolds number of about 5. Then the resisting force can be expressed as... 

Now we can see how the differential form of the property conservation law can generate the equations of the velocity distribution for a flowing fluid (Navier-Stokes equations), the temperature or the enthalpy distribution (Fourier second law) and the species concentration distribution inside the fluid (second Fick s law). 

The equation for the momentum transport in vectorial form, gives (by particularization) the famous Navier-Stokes equation. This equation is obtained considering the conservation law of the property of movement quantity in the differential form P = mw. At the same time, if we consider the expression of the transport vector Jt = f + w(pw) and that the molecular momentum generation rate is given with the help of one external force F, which is active in the balance point, the par-d(pw)... 

Koelman and Hoogerbrugge (1993) have developed a particle-based method that combines features from molecular dynamics (MD) and lattice-gas automata (LGA) to simulate the dynamics of hard sphere suspensions. A similar approach has been followed by Ge and Li (1996) who used a pseudo-particle approach to study the hydrodynamics of gas-solid two-phase flow. In both studies, instead of the Navier-Stokes equations, fictitious gas particles were used to represent and model the flow behavior of the interstial fluid while collisional particle-particle interactions were also accounted for. The power of these approaches is given by the fact that both particle-particle interactions (i.e., collisions) and hydrodynamic interactions in the particle assembly are taken into account. Moreover, these modeling approaches do not require the specification of closure laws for the interphase momentum transfer between the particles and the interstitial fluid. Although these types of models cannot yet be applied to macroscopic systems of interest to the chemical engineer they can provide detailed information which can subsequently be used in (continuum) models which are suited for simulation of macroscopic systems. In this context improved rheological models and boundary condition descriptions can be mentioned as examples. 

Conseruatton of momentum leads to Newton s second law, derived in sec. 1.6. lb, see [1.6.1.14], and variants of it. including the Navier-Stokes equation. For the present case it is written in the following general form... 

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