Page 295 - Introduction to Computational Fluid Dynamics

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APPENDIX A. DERIVATION OF TRANSPORT EQUATIONS
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a pipe) of the flow. When Kn is very small (<10 ), the continuum approach is
considered valid. In engineering and environmental flows, therefore, the continuum
approach is adopted.
Control Volume
The notion of a controlvolume (CV) is very important in the continuum approach.
The CV may be defined as a region in space across the boundaries of which matter,
energy, and momentum may flow; it is a region within which source or sink of the
same quantities may prevail. Further, it is a region on which external forces may
act.
In general, a CV may be large or infinitesimally small. However, consistent with
the idea of a differential in a continuum, an infinitesimally small CV is considered.
Thus, when the laws are to be expressed through differential equations, the CV is
located within a moving fluid. Again, two approaches are possible:

1. a Lagrangian approach or
2. a Eulerian approach.
In the Lagrangian approach, the CV is considered to be moving with the fluid
as a whole. In the Eulerian approach, in contrast, the CV is assumed fixed in space
and the fluid is assumed to flow through and past the CV. Except when dealing
with certain types of unsteady flows (waves, for example), the Eulerian approach is
generallyusedforitsnotionalsimplicity.Also,measurementsmadeusingstationary
instruments can be directly compared with the solutions of differential equations
obtained using the Eulerian approach.
Finally, it is important to note that the fundamental laws define total flows of
mass, momentum, and energy not only in terms of magnitude but also in terms of
direction. In a general problem of convection, neither magnitude nor direction is
known a priori at different positions in the flowing fluid. The problem of ignorance
of direction is circumvented by resolving velocity, force, and scalar fluxes in three
directions that define the space.
In the derivations to follow, the three chosen directions will be along Cartesian
coordinates. The derivations are carried out using the continuum approach within a
Eulerian specification of the CV. Figure A.1 shows the considered CV of dimensions
x 1 , x 2 , and x 3 located at (x 1 , x 2 , x 3 ) from a fixed origin.

A.2 Mass Conservation – Fluid Mixture

The law of conservation of mass states that
˙
˙
Rate of accumulation of mass (M ac ) = Rate of mass in (M in )
˙
− Rate of mass out (M out ).