Fluid mechanics, equations continuity

The tliree conservation laws of mass, momentum and energy play a central role in the hydrodynamic description. For a one-component system, these are the only hydrodynamic variables. The mass density has an interesting feature in the associated continuity equation the mass current (flux) is the momentum density and thus itself is conserved, in the absence of external forces. The mass density p(r,0 satisfies a continuity equation which can be expressed in the fonn (see, for example, the book on fluid mechanics by Landau and Lifshitz, cited in the Furtlier Reading)... 

The evaluation of the parameters for this flow regime requires the calculation of the Reynolds number and hydraulic diameter for each continuous phase. The hydraulic diameter can be determined only if the holdup of each phase is known. This again illustrates the importance of understanding the fluid mechanics of two phase systems. Once the hydraulic diameter is known, the Reynolds number can be evaluated with the knowledge of the in situ phase velocity, and the parameters of the model equations can be evaluated. 

This may not seem like much help, because we have expanded three terms into six. However, if the flow is assumed to be incompressible, a derivation given in fluid mechanics texts (the continuity equation) is... 

The notion of fluid strains and stresses and how they relate to the velocity field is one of the fundamental underpinnings of the fluid equations of motion—the Navier-Stokes equations. While there is some overlap with solid mechanics, the fact that fluids deform continuously under even the smallest stress also leads to some fundamental differences. Unlike solid mechanics, where strain (displacement per unit length) is a fundamental concept, strain itself makes little practical sense in fluid mechanics. This is because fluids can strain indefinitely under the smallest of stresses—they do not come to a finite-strain equilibrium under the influence of a particular stress. However, there can be an equilibrium relationship between stress and strain rate. Therefore, in fluid mechanics, it is appropriate to use the concept of strain rate rather than strain. It is the relationship between stress and strain rate that serves as the backbone principle in viscous fluid mechanics. 

The Navier-Stokes equations express the conservation of momentum. Together with the continuity equation, which expresses conservation of mass, these equations are the fundamental underpinning of fluid mechanics. They are nonlinear partial differential equations that in general cannot be solved by analytical means. Nevertheless, there are a number of geometries and flow situations that permit considerable simplification and solution. We will explore many of these and their solution, usually by computational techniques. While... 

Stagnation flows represent a very important class of flow configurations wherein the steady-state Navier-Stokes equations, together with thermal-energy and species-continuity equations, reduce to systems of ordinary-differential-equation boundary-value problems. Some of these flows have great practical value in applications, such as chemical-vapor-deposition reactors for electronic thin-film growth. They are also widely used in combustion research to study the effects of fluid-mechanical strain on flame behavior. 

The general energy equation Eq. (10.5) and the continuity equation are two important keys to the solution of many problems in fluid mechanics. For compressible fluids it is necessary to have a third equation, which is the equation of state, i.e., the relation between pressure, temperature, and specific volume (or specific weight or density). 

In the previous section, stability criteria were obtained for gas-hquid bubble columns, gas-solid fluidized beds, liquid-sohd fluidized beds, and three-phase fluidized beds. Before we begin the review of previous work, let us summarize the parameters that are important for the fluid mechanical description of multiphase systems. The first and foremost is the dispersion coefficient. During the derivation of equations of continuity and motion for multiphase turbulent dispersions, correlation terms such as esv appeared [Eqs. (3) and (10)]. These terms were modeled according to the Boussinesq hypothesis [Eq. (4)], and thus the dispersion coefficients for the sohd phase and hquid phase appear in the final forms of equation of continuity and motion [Eqs. (5), (6), (14), and (15)]. However, for the creeping flow regime, the dispersion term is obviously not important. 

From fluid mechanics, the critical depth occurs at the minimum specific energy. Thus, the previous equation may be differentiated for E with respect to y and equated to zero. Convert V in terms of the flow Q and cross-sectional area of flow A using the equation of continuity, then differentiate and equate to zero. This will produce... 

FORTRAN computer program that predicts the species, temperature, and velocity profiles in two-dimensional (planar or axisymmetric) channels. The model uses the boundary layer approximations for the fluid flow equations, coupled to gas-phase and surface species continuity equations. The program runs in conjunction with CHEMKIN preprocessors (CHEMKIN, SURFACE CHEMKIN, and TRAN-FIT) for the gas-phase and surface chemical reaction mechanisms and transport properties. The finite difference representation of the defining equations forms a set of differential algebraic equations which are solved using the computer program DASSL (dassal.f, L. R. Petzold, Sandia National Laboratories Report, SAND 82-8637, 1982). 

While an understanding of the molecular processes at the fuel cell electrodes requires a quantum mechanical description, the flows through the inlet channels, the gas diffusion layer and across the electrolyte can be described by classical physical theories such as fluid mechanics and diffusion theory. The equivalent of Newton s equations for continuous media is an Eulerian transport equation of the form... 

The steady-state fluid mechanics problem is solved using the Fluent Euler-Euler multiphase model in the fluid domains. Mass, momentum and energy balances, the general forms of which are given by eqn. (4), (5), and (6), are solved for both the liquid and the gas phases. In solid zones the energy equation reduces to the simple heat conduction problem with heat source. By convention, / =1 designates the H2S04 continuous liquid phase whereas H2 bubbles constitute the dispersed phase 0 =2). 

The equations of fluid mechanics originate from the momentum and mass conservation principles. The overall mass conservation or continuity equation for laminar flows is... 

In Section 4.2.4, the governing equations of fluid mechanics for a turbulent flow are derived. Similarly, the governing equations for heat transfer and mass transfer can be derived from the principles of energy and mass conservation. In fact, the species conservation equation is an extension of the overall mass conservation (or the continuity) equation. For species i, it has the following form ... 

The traditional modeling framework describing reactive flow systems is presented in the following. The basic conservation equations applied in most reactor model analyzes are developed from the concept that the fluid is a continuum. This means that a fluid is considered to be a matter which exhibits no finer structure [168]. This model makes it possible to treat fluid properties at a point in space and mathematically as continuous functions of space and time. From the continuum viewpoint, fluid mechanics and solid mechanics have much in common and the subject of both these sciences are traditionally called continuum mechanics. 

Although there is no immediately useful information that we can glean from (2-56), we shall see that it provides a constraint on allowable constitutive relationships for T and q. In this sense, it plays a similar role to Newton s second law for angular momentum, which led to the constraint (2 41) that T be symmetric in the absence of body couples. In solving fluid mechanics problems, assuming that the fluid is isothermal, we will use the equation of continuity, (2-5) or (2-20), and the Cauchy equation of motion, (2-32), to determine the velocity field, but the angular momentum principle and the second law of thermodynamics will appear only indirectly as constraints on allowable constitutive forms for T. Similarly, for nonisothermal conditions, we will use (2-5) or (2-20), (2-32), and either (2-51) or... 

We are now in a position to begin to consider the solution of heat transfer and fluid mechanics problems by using the equations of motion, continuity, and thermal energy, plus the boundary conditions that were given in the preceding chapter. Before embarking on this task, it is worthwhile to examine the nature of the mathematical problems that are inherent in these equations. For this purpose, it is sufficient to consider the case of an incompressible Newtonian fluid, in which the equations simplify to the forms (2 20), (2-88) with the last term set equal to zero, and (2-93). 

Even with these simplifications, however, it is rarely possible to obtain analytic solutions for fluid mechanics or heat transfer problems. The Navier Stokes equation for an isothermal fluid is still nonlinear, as can be seen by examination of either (2 89) or (2 91). The Bousi-nesq equations involve a coupling between u and 6, introducing additional nonlinearities. It will be noted, however, that, provided the density can be taken as constant in the body-force term (thus neglecting any natural convection), the fluid mechanics problem is decoupled from the thermal problem in the sense that the equations of motion, (2 89) or (2-91), and continuity, (2-20), do not involve the temperature 0. The thermal energy equation, (2-93), is actually a linear equation in the unknown 6, once the Boussinesq approximation has been introduced. In that case, the only nonlinear term is dissipation, but this involves the product E E and can be treated simply as a source term that will be known once Eqs. (2-89) or (2 91) and (2 20) have been solved to determine the velocity. In spite of being linear, however, the velocity u appears as a coefficient (in the convective derivative term). Even when the form of u is known (either exactly or approximately), it is normally quite a complicated function, and this makes it extremely difficult to obtain analytic solutions for 0 even though the governing equation is linear. 

The assumption of a ID radial flow leads to a great simplification of the fluid mechanics problem. Indeed, the continuity equation, corresponding to (4-193), takes the very simple form... 

Nonstandard finite-difference schemes are, primarily, a generalization of the customary models of differential equations [ 1-4]. While the latter cannot satisfactorily model cumbersome physical phenomena, the particular forms maintain the underlying properties of their continuous analogs and evade elementary instabilities. In this manner, they have gained a noteworthy popularity in several scientific fields [5], like fluid mechanics, electromagnetics, photoconductivity, nonlinear optics, and mathematical biology to name a few. 

The principles of physics most useful in the applications of fluid mechanics are the mass-balance, or continuity, equations the linear- and angular-raomentum-balance equations and the mechanical-energy balance. They may be written in differential form, showing conditions at a point within a volume element of fluid, or in integrated form applicable to a finite volume or mass of fluid. Only the integral equations are discussed in this chapter. 

In this chapter we continue the quantitative development of thermodynamics by deriving the energy balance, the second of the three balance equations that will be used in the thermodynamic description of physical, chemical, and (later) biochemical processes. The mass and energy balance equations (and the third balance equation, to be developed in the following chapter), together with experimental data and information about the process, will then be used to relate the change in a system s properties to a change in its thermodynamic state. In this context, physics, fluid mechanics, thermodynamics, and other physical sciences are all similar, in that the tools of each are the same a set of balance equations, a collection of experimental observ ations (equation-of-state data in thermodynamics, viscosity data in fluid mechanics, etc.), and the initial and boundary conditions for each problem. The real distinction between these different subject areas is the class of problems, and in some cases the portion of a particular problem, that each deals with. 

Physical properties for all flows are inputted. The user must specify certain parameters such as mixing models, kinetic rates, turbulence, and others as required. The CFD model is generally full-scale with complete similitude. The governing differential equations that solve all aspects of mixing, heat transfer, chemistry, turbulence, fluid mechanics, species, and continuity are iterated across the entire model until a converged solution is obtained for all cells and boundary conditions. 

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