Page 23 - Introduction to Computational Fluid Dynamics

P2: IWV
P1: JYD/GKJ
0 521 85326 5
12:20
May 20, 2005
CB908/Date
0521853265c01
2
INTRODUCTION
required considerable experience as well as ingenuity, for it is not an easy matter
to simultaneously maintain the three aforementioned similarities. The activity had
to, therefore, be supported by flow-visualization studies and by simple (typically,
one-dimensional) analytical solutions to equations governing the phenomenon un-
der consideration. Ultimately, experience permitted judicious compromises. Being
very expensive to generate, such information is often of a proprietary kind. In more
recent times, of course, scaling difficulties are encountered in the opposite direction.
This is because electronic equipment is considerably miniaturised and, in mate-
rials processing, for example, the more relevant phenomena occur at microscales
(even molecular or atomic scales where the continuum assumption breaks down).
Similarly, small-scale processes occur in biocells.
Clearly, designers need a design tool that is scale neutral. The tool must be
scientific and must also be economical to use. An individual designer can rarely, if
at all, acquire or assimilate this scale neutrality. Fortunately, the fundamental laws of
mass, momentum, and energy, in fact, do embody such scale-neutral information.
The key is to solve the differential equations describing these laws and then to
interpret the solutions for practical design.
The potential of fundamental laws (in association with some further empiri-
cal laws) for generating widely applicable and scale-neutral information has been
known almost ever since they were invented nearly 200 years ago. The realisation
of this potential (meaning the ability to solve the relevant differential equations),
however, has been made possible only with the availability of computers. The past
five decades have witnessed almost exponential growth in the speed with which
arithmetic operations can be performed on a computer.
By way of reminder, we note that the three laws governing transport are the
following:

1. the law of conservation of mass (transport of mass),
2. Newton’s second law of motion (transport of momentum), and
3. the first law of thermodynamics. (transport of energy).

1.2 Transport Equations

The aforementioned laws are applied to an infinitesimally small control volume
located in a moving fluid. This application results in partial differential equations
(PDEs) of mass, momentum and energy transfer. The derivation of PDEs is given in
1
Appendix A. Here, it will suffice to mention that the law of conservation of mass is
written for a single-component fluid or for a mixture of several species. When ap-
plied to a single species of the mixture, the law yields the equation of mass transfer
when an empirical law, namely, Fick’s law of mass diffusion (m =−ρ D ∂ω/∂x i ),

i
1 The reader is strongly advised to read Appendix A to grasp the main ideas and the process of
derivations.