Equations of Fluid Mechanics

Aris, R., 1989. Vectors, Tensors and the Basic Equations of Fluid Mechanics, Dover Publications, New York. 

One-dimensional Flow Many flows of great practical importance, such as those in pipes and channels, are treated as onedimensional flows. There is a single direction called the flow direction velocity components perpendicmar to this direction are either zero or considered unimportant. Variations of quantities such as velocity, pressure, density, and temperature are considered only in the flow direction. The fundamental consei vation equations of fluid mechanics are greatly simphfied for one-dimensional flows. A broader categoiy of one-dimensional flow is one where there is only one nonzero velocity component, which depends on only one coordinate direction, and this coordinate direction may or may not be the same as the flow direction. 

We attempt here to describe the fundamental equations of fluid mechanics and heat transfer. The main emphasis, however, is on understanding the physical principles and on application of the theory to realistic problems. The state of the art in high-heat flux management schemes, pressure and temperature measurement, pressure drop and heat transfer in single-phase and two-phase micro-channels, design and fabrication of micro-channel heat sinks are discussed. 

The constitutive relations along with the conservation equations give the basic equations of fluid mechanics, which are a set of five nonlinear partial differential equations involving the seven variables, p, g,e, P, and T. Because five equations [Eqs. (1), (2), (3), (5), and (6)] cannot determine seven quantities, the equations are closed by expressing any two variables of the set (p,e,P,T) in terms of the other two remaining variables. This is done by using the assumption of local equilibrium and thermodynamic equations of state. 

A general, tensorially based description of the basis of the governing equations of fluid mechanics soon appeared in the Prentice-Hall series of texts for chemical engineers [4], and this material was later incorporated in several widely used undergraduate textbooks for chemical engineers [see 5, 6]. 

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 ... 

A brief survey of the different classes of ODE methods that are commonly applied solving the governing equations of fluid mechanics, is given in the following. The elementary notation and the basic properties of these ODE discretizations are briefly mentioned. Analysis of these methods can be found in numerous textbooks on the numerical solution of ordinary differential equations and will not be repeated here [156, 66, 170, 158, 134, 93, 28]. 

It is more difficult to explain the motion of particles that are larger than the mean free path. The explanation Is based on the tangential slip velocity that develops at the surface of a particle in a temperature gradient (Kennard, 1938). This creep velocity is directed toward the high-temperature side, propelling the particle in the direction of lower temperature. An expression for the thermophoretic velocity based on the continuum equations of fluid mechanics with slip-corrected boundary conditions was derived by Brock (1962). Talbot et al. (1980) proposed an interpolation formula for the thermophoretic velocity... 

To describe the theoretical dynamical and thermal behavior of the atmosphere, the fundamental equations of fluid mechanics must be employed. In this section these equations are presented in a relatively simple form. A more conceptual view will be presented in Section 3.6. The circulation of the Earth s atmosphere is governed by three basic principles Newton s laws of motion, the conservation of energy, and the conservation of mass. Newton s second law describes the response of a fluid to external forces. In a frame of reference which rotates with the Earth, the first fundamental equation, called the momentum equation, is given by ... 

Hence this chapter is devoted to an investigation of hydrodynamic fluctuation theory. Much of what we use in this chapter is developed more formally in Chapter 11. In order to use hydrodynamic fluctuation theory it is necessary to discuss the derivation of the usual equations of fluid mechanics. Unfortunately this task would require the writing of a separate monograph. Space does not permit us to present a detailed account of the equations of fluid dynamics. We refer the reader to the excellent monograph on this subject by Landau and Lifshitz (1960), and will only highlight the important points here. 

In Section 10.2 we saw that the macroscopic relaxation equations can be used to determine correlation functions. In this section we summarize the traditional methods for deducing the macroscopic relaxation equations of fluid mechanics. In subsequent sections these equations are used to determine the Rayleigh-Brillouin spectrum. The first step in the derivation of the relaxation equation involves a discussion of conservation laws. 

In molecular fluids, rotational and vibrational relaxation can effect the density fluctuation. It is then necessary to supplement the equations of fluid mechanics with equations describing the molecular relaxation. We shall consider this momentarily. The whole picture developed here must be modified in the neighborhood of the critical point or near a phase transition. The long range correlations discussed in Sections 10.1 and 10.7 then affect the whole structure of the theory. See, for example, the review of Stanley, et al. (1971) and particularly references to the work of Kawasaki cited therein. Some aspects of scattering in the critical region are considered in Sec. (10.7). 

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