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86 CHAPTER 3. GENERAL EQUATIONS
If τ is symmetric
∇ · (ā · τ) = ā · (∇ · τ) + (∇ā) : τ = ā · (∇ · τ) + 1 (∇ā + ∇ā) : τ
2
The Gauss formula:
In a domain V enclosed by surface S:
∫∫
∫∫∫
(¯n ◦ ϕ) dσ =
S
V
(∇ ◦ ϕ) dV
ϕ can be a scalar, vector or tensor. Symbol ◦ represents any of the several types of
products previously defined.
Derivative of an integral with changing boundary:
∫∫∫
∫∫∫
∫∫
d
∂ϕ (t, ¯x)
ϕ (t, ¯x) dV =
dV +
dt
∂t
V (t)
V (t)
Σ(t)
(¯n · ¯v) ϕ dσ
where ¯v is the velocity of Σ at dσ and ¯n is the outwards normal to Σ at dσ.
References
[1] Kuethe, A. N. and Schetzer, J. D.: Foundations of Aerodynamics. John Wiley
and Sons, Inc. New York. 1950.
[2] Green, R. S.: The Molecular Theory of Fluids. Interscience Publishers, Inc,
New York, 1952.
[3] Richardson, J. M. and Brinkley, S. R.: Mechanics of Reacting Continua. Combustion
Processes, Sec. F, Vol. II of High Speed Aerodynamics and Jet Propulsion,
Princeton University Press, 1956, p. 203.
[4] Hirschfelder, J. O., Curtiss, C. F. and Bird, R. B.: Molecular Theory of Liquid
and Gases. John Wiley and Sons, Inc., New York, 1954.
[5] De Groot, S. R.: Thermodynamics of Irreversible Processes. North Holland
Publishing Comp., Amsterdam, 1951.
[6] Prigogine, I. and Defay, R.: Chemical Thermodynamics. Longmans Green and
C., New York, 1954.
[7] von Kármán, Th.: The Theory of Shock Waves and the Second Law of Thermodynamics.
L’Aerotecnica, Febr. 15, 1951, pp. 82-3.
[8] Chambré, P. and Lin, C. C.: On the Steady Flow of a Gas through a Tube with
Heat Exchange or Chemical Reaction. The Journal of Aeronautical Sciences,
Oct. 1946, pp. 537-42.