Untitled - Aerobib - Universidad Politécnica de Madrid

60 CHAPTER 3. GENERAL EQUATIONS
1) The need to establish a separate balance for each one of the species by means of
partial continuity equations for each different species. In each of these equations
the effects of the chemical reactions and of diffusion must be included.
2) The modifications due to diffusion of some of the transport coefficients of the
mixture. As previously seen in chapter 2, diffusion does not change the values
of the viscosity coefficients but modifies the heat flux, mainly due to the transport
of the enthalpy of the various species through diffusion.
3) The presence of a new form of energy: the chemical energy released or absorbed
in the chemical reactions.
n
q
f
Σ
V
F
Figure 3.1: Schematic diagram of the actions upon a isolated gas element.
In Gas Dynamics the equations are established by applying the laws of the conservation
of mass, momentum and energy to an ideally isolated gas element following
its motion. 1
The material element thus isolated (Fig. 3.1) is bounded by a fluid surface,
that is to say by a surface Σ, in which the velocity at each point is the velocity
¯v of the gas particle at the same point. This element evolves in accordance with the
aforementioned laws of conservation and under the action of the surrounding mass,
which acts upon its boundary Σ. This action result in the surface forces and in the
heat transport ¯q throughout each of its surface elements. Furthermore, the action of
the possible force fields acting upon the element must be included; for example, the
action ¯F of the gravity field. In this manner 2 we obtain:
a) The continuity equation, which expresses the laws of the conservation of mass.
b) The equation of motion which expresses that Newton’s Second Law of Mechanics
is satisfied.
c) The energy equation which expresses that the First Law of Thermodynamics is
satisfied.
1 See, i.e., Ref. [1], pp. 337 and following.
2 The same equations could be established by starting from the molecular structure of the gas and applying
the laws of the Kinetic Theory of gases. See, i.e., Ref. [2], p. 25.