The Governing Equations of Fluid Dynamics

In this paragraph the conservation equations of fluid dynamic are derived from the Boltzmann equation. 

The equation of change, (2.199), becomes particularly simple if tp refers to a quantity which is conserved in collisions between molecules so that J Aip = 0) = 0. Therefore, for conservative quantities (2.199) reduces to  

By letting tp in (2.200) be m, me, and mc, respectively, we obtain the three fundamental conservation equations that are all satisfied by the gas. 

This relation corresponds to the Cauchy equation of motion in fluid dynamics. 

This is the total energy equation, for which the potential energy term is expressed in terms of the external force F. By use of the momentum equation we can derive a transport equation for the mean kinetic energy, and thereafter extract the mean kinetic energy part from the equation (i.e, the same procedure was used manipulating the continuum model counterpart in chap. 1, sect. 1.2.4). The result is  

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More or less as a spin-off result of the foregoing analysis determining the transport coefficients, a rigorous procedure deriving the governing equations of fluid dynamics from first principles was established. It is stressed that in classical fluid dynamics the continuum hypothesis is used so that the governing... 

The model derivation given above using the Liouville theorem is in many ways equivalent to the Lagrangian balance formulation [83]. Of course, a consistent Eulerian balance formulation would give the same result, but includes some more manipulations of the terms in the number balance. However, the Eulerian formulation is of special interest as we have adopted this framework in the preceding discussion of the governing equations of classical fluid dynamics, chap 1. 

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 ... 

There are several approaches to developing the correct scaling relationships. Probably the most straightforward is the nondimensional-ization of the governing equations. If we can write the proper equations governing the fluid and particle dynamic behavior, we can develop the proper scaling relationships even if we can t solve the equations (at present we can t). In essence, if a model is designed which follows the... 

All fluid dynamical systems are continuous system with infinite degrees of freedom and the governing equations depend continuously upon both space and time. While for any system, the time-dependent signal cannot move back in time, the space dependent signal can propagate in all directions with respect to the location of the source. This therefore requires that we develop a theory based on bilateral Laplace transform - a topic described in great details in Papoulis (1962) and Van der Pol Bremmer (1959). 

To analyze this problem, we need to go back to a statement of the problem in general fluid dynamical terms. Thus we begin by restating the governing equations and boundary conditions for an oscillating bubble in a quiescent, incompressible fluid. These are the Navier-Stokes and continuity equations the three boundary conditions... 

We begin by considering the flow within a shallow, horizontal (a = 0) cavity as sketched in Fig. 6-7a. We assume that the ratio, d/L, is asymptotically small. We seek only the leading-order approximation within the shallow cavity. Hence the starting point for analysis is the thin-film equations, (6—1)—(6—3). In the present case of a 2D cavity, we can use a Cartesian coordinate system, and, for the present problem, we assume that the fluid is isothermal, so that the body-force term in (6-3) can be incorporated into the dynamic pressure, and hence plays no role in the fluid s motion. In this case, the governing equations become... 

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