Page 70 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 53

need to supplement Newton’s laws of motion with a constitutive law. For pure
homogeneous fluids (such as air and water) this constitutive law is provided by the
Newtonian fluid model, which as the name suggests also originated with Newton. In
simple terms the constitutive law for a Newtonian fluid states that:
Viscous stress cx Rate of strain
At a fundamental level these simple physical laws are, of course, merely theoretical
models. The principal theoretical assumption is that the fluid consists of continuous
matter - the so-called continurn model. At a deeper level we are, of course, aware
that the fluid is not a continuum, but is better considered as consisting of myriads of
individual molecules. In most engineering applications even a tiny volume of fluid
(measuring, say, 1 pm3) contains a large number of molecules. Equivalently, a typical
molecule travels on average a very short distance (known as the mean free path)
before colliding with another. In typical aerodynamics applications the m.f.p. is less
than lOOnm, which is very much smaller than any relevant scale characterizing
quantities of engineering significance. Owing to this disparity between the m.f.p.
and relevant length scales, we may expect the equations of fluid motion, based on the
continuum model, to be obeyed to great precision by the fluid flows found in almost
all engineering applications. This expectation is supported by experience. It also has
to be admitted that the continuum model also reflects our everyday experience of the
real world where air and water appear to our senses to be continuous substances.
There are exceptional applications in modern engineering where the continuum model
breaks down and ceases to be a good approximation. These may involve very small-
scale motions, e.g. nanotechnology and Micro-Electro-Mechanical Systems (MEMS)
technology,* where the relevant scales can be comparable to the m.f.p. Another
example is rarefied gas dynamics (e.g. re-entry vehicles) where there are so few mole-
cules present that the m.f.p. becomes comparable to the dimensions of the vehicle.
We first show in Section 2.2 how the principles of conservation of mass, momen-
tum and energy can be applied to one-dimensional flows to give the governing
equations of fluid motion. For this rather special case the flow variables, velocity
and pressure, only vary at most with one spatial coordinate. Real fluid flows are
invariably three-dimensional to a greater or lesser degree. Nevertheless, in order to
understand how the conservation principles lead to equations of motion in the form
of partial differential equations, it is sufficient to see how this is done for a two-
dimensional flow. So this is the approach we will take in Sections 2.4-2.8. It is usually
straightforward, although significantly more complicated, to extend the principles
and methods to three dimensions. However, for the most part, we will be content to
carry out any derivations in two dimensions and to merely quote the final result for
three-dimensional flows.

2.1.1 Air flow
Consider an aeroplane in steady flight. To an observer on the ground the aeroplane is
flying into air substantially at rest, assuming no wind, and any movement of the air is
caused directly by the motion of the aeroplane through it. The pilot of the aeroplane,
on the other hand, could consider that he is stationary, and that a stream of air is
flowing past him and that the aeroplane modifies the motion of the air. In fact both

* Recent reviews are given by M. Gad-el-Hak (1999) The fluid mechanics of microdevices - The Freeman
Scholar Lecture. J. Fluids Engineering, 121, 5-33; L. Lofdahl and M. Gad-el-Hak (1999) MEMS applica-
tions in turbulence and flow control. Prog. in Aerospace Sciences, 35, 101-203.