Navier—Stokes momentum equations

Computational fluid dynamics (CFD) emerged in the 1980s as a significant tool for fluid dynamics both in research and in practice, enabled by rapid development in computer hardware and software. Commercial CFD software is widely available. Computational fluid dynamics is the numerical solution of the equations or continuity and momentum (Navier-Stokes equations for incompressible Newtonian fluids) along with additional conseiwation equations for energy and material species in order to solve problems of nonisothermal flow, mixing, and chemical reaction. 

The first term on the right-hand side of the identity above involves the divergence of the stress tensor, which also appears in the vector form of the momentum (Navier-Stokes) equations, Eq. 3.53. The momentum equation can be easily rearranged as... 

Atmospheric models for research and forecasting of weather, climate, and air quality are all based on numerical integration of the basic equations governing atmospheric behavior. These equations are the gas law, the equation of continuity (mass), the first law of thermodynamics (heat), the conservation equations for momentum (Navier-Stokes equations), and usually equations expressing the conservation of moisture and air pollutants. At one extreme, atmospheric models deal with the world s climate and climate change at the other extreme, they may account for the behavior of local flows at coasts, in mountain-valley areas, or even deal with individual clouds. This all depends on the selected horizontal scale and the available computing resources ... 

Other parameters, including the lattice sound speed Cs and weight factor fj, are lattice structure dependent. For example, for a typical D2Q9 (two dimensions and nine lattice velocities see Fig. 1) lattice structure, we have tQ = 4/9, ii 4 = 1/9, f5 8 = 1/36, and = A /3Afi, where Ax is the spatial distance between two nearest lattice nodes. Through the Chapman-Enskog expansion, one can recover the macroscopic continuity and momentum (Navier-Stokes) equations from the above-defined LBM dynamics ... 

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. 

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... 

Dissipative particle dynamics (DPD) is a technique for simulating the motion of mesoscale beads. The technique is superficially similar to a Brownian dynamics simulation in that it incorporates equations of motion, a dissipative (random) force, and a viscous drag between moving beads. However, the simulation uses a modified velocity Verlet algorithm to ensure that total momentum and force symmetries are conserved. This results in a simulation that obeys the Navier-Stokes equations and can thus predict flow. In order to set up these equations, there must be parameters to describe the interaction between beads, dissipative force, and drag. 

Dynamic meteorological models, much like air pollution models, strive to describe the physics and thermodynamics of atmospheric motions as accurately as is feasible. Besides being used in conjunction with air quaHty models, they ate also used for weather forecasting. Like air quaHty models, dynamic meteorological models solve a set of partial differential equations (also called primitive equations). This set of equations, which ate fundamental to the fluid mechanics of the atmosphere, ate referred to as the Navier-Stokes equations, and describe the conservation of mass and momentum. They ate combined with equations describing energy conservation and thermodynamics in a moving fluid (72) ... 

Cauchy Momentum and Navier-Stokes Equations The differential equations for conservation of momentum are called the Cauchy momentum equations. These may be found in general form in most fliiid mechanics texts (e.g., Slatteiy [ibid.] Denu Whitaker and Schlichting). For the important special case of an incompressible Newtonian fluid with constant viscosity, substitution of Eqs. (6-22) and (6-24) lead to the Navier-Stokes equations, whose three Cartesian components are... 

When the continmty equation and the Navier-Stokes equations for incompressible flow are time averaged, equations for the time-averaged velocities and pressures are obtained which appear identical to the original equations (6-18 through 6-28), except for the appearance of additional terms in the Navier-Stokes equations. Called Reynolds stress terms, they result from the nonlinear effects of momentum transport by the velocity fluctuations. In each i-component (i = X, y, z) Navier-Stokes equation, the following additional terms appear on the right-hand side ... 

Assuming laminar flow for a linear momentum equation in the a direction (an approximation from the Navier-Stokes equations) gives... 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

The basic model equations for a description of hydrodynamical flow are the Navier-Stokes equations, representing momentum conservation in the fluid... 

In this section we show how the fundamental equations of hydrodynamics — namely, the continuity equation (equation 9.3), Euler s equation (equation 9.7) and the Navier-Stokes equation (equation 9.16) - can all be recovered from the Boltzman equation by exploiting the fact that in any microscopic collision there are dynamical quantities that are always conserved namely (for spinless particles), mass, momentum and energy. The derivations in this section follow mostly [huangk63]. 

Recall that in the continuum case we derived the Navier-Stokes equations (9.16) by allowing for the possibility of having nonzero viscous terms in the form for the momentum tensor appearing in Euler s equation (9.9). While their LG analogs may be derived in essentially the same manner, however, the lack of Galilean invariance tend to make the calculations more involved. We will outline the procedure below. 

Isotropy of the Momentum Flux Density Tensor If we trace back our derivation of the macroscopic LG Euler s and Navier-Stokes equations, we see that the only place where the geometry of the underlying lattice really enters is through the form for the momentum flux density tensor, fwhere cp = x ) + y ), k = 1,..., V... 

General equations of momentum and energy balance for dispersed two-phase flow were derived by Van Deemter and Van Der Laan (V2) by integration over a volume containing a large number of elements of the dispersed phase. A complete system of solutions of linearized Navier-Stokes equations... 

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

A standard approach to modeling transport phenomena in the field of chemical engineering is based on convection-diffusion equations. Equations of that type describe the transport of a certain field quantity, for example momentum or enthalpy, as the sum of a convective and a diffusive term. A well-known example is the Navier-Stokes equation, which in the case of compressible media is given as... 

In the TFM, both the gas phase and the solid phase are described as fully interpenetrating continua using a generalized form of the Navier-Stokes equations for interacting fluids. The continuity and momentum equations for the gas phase are given by expressions identical to Eqs. (40) and (41), except for the gas solid interaction term ... 

Rather than setting up a force-momentum balance for a particular flow problem as was done in Chapter 1, general equations, known as the Navier-Stokes equations, may be formulated. Before discussing the Navier-Stokes equations, it is necessary to consider some related matters. 

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. 

The momentum equation (the Navier-Stokes equation) for fluid flow (De Groot and Mazur, 1962) is complicated and difficult to solve. It is the subject of fluid mechanics and dynamics and is not covered in this book. When fluid flow is discussed in this book, the focus is on the effect of the flow (such as a flow of constant velocity, or boundary flow) on mass transfer, not the dynamics of the flow itself. 

The interrelationships are embodied in variations of the Navier-Stokes equations, which describe mass and momentum balances in fluid systems (23). 

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