II. Governing Equations
In aerodynamics, or fluid mechanics, there are six properties of the flow an
engineer is usually interested in – pressure p, density
ρ
, the three velocity components
(u,v,w), and the temperature T. For gases and mixtures of gases (e.g. air) the equation of
state links p,
ρ
and T by:
Equation of State:
p =
ρ
RT
(2.1)
We need to come up five additional equations linking the 6 properties. These five
equations are PDEs and turn out to be:
a)
Conservation of Mass or Continuity
b)
Conservation of u momentum
c)
Conservation of vmomentum
d)
Conservation of wmomentum
e)
Conservation of energy
These equations may be derived using a Lagrangean approach, or an Eulerian
approach.
In the Lagrangean approach, we follow a fixed set of fluid particles (e.g. a cloud,
a tornado, tip vortices from an aircraft) and write down equations governing their motion.
This is somewhat like tracking satellites and missiles in space, using equations to
describe their position in space and the forces acting on them.
In the Eulerian approach, we look at a (usually) fixed or (sometimes) moving
volume in space surrounded by permeable boundaries.
We develop equations describing
what happens to the fluid inside the control volume as new fluid enters and old fluid
particles leave. Eulerian approach is the preferred approach in most fluid dynamics
applications. This is what we will follow in our derivations.
Conservation of Mass (Continuity):
The conservation of mass stems from the principle that mass can not be created or
destroyed
inside the control volume. Obviously, we are situations (e.g. nuclear reactions)
involving the conversion of mass into energy.
Let V be a control volume, a balloon like shape in space. We will assume that it, and its
surface S remain fixed in space. The surface is permeable so that fluid can freely enter in
and leave. The continuity equation says
The time rate of change of Mass within the control volume V =
Rate at which mass enters V through the boundary S
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 Fall '08
 Yeung
 Fluid Dynamics, Fluid Mechanics, Derivative, Force, Eulerian

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