Page:THEORY OF SHOCK WAVES AND INTRODUCTION TO GAS DYNAMICS.pdf/13

Density, pressure and temperature of a fluid are connected by &n equation known as the equation of state. If we know the thermal capacity, we can connect temperature with energy. To determine the connection between density and pressure, we must set up another equation - the equation of energy of a fluid in motion. In the absence of dissipative forces (viscosity and thermal conductivity) we have

$dE=-pdv;\quad {\frac {dE}{dt}}=-p{\frac {dv}{dt}}=-p{\frac {d(1/\rho )}{dt}}={\frac {p}{\rho ^{2}}}{\frac {d\rho }{dt}},$

(I-4)

where $v$ is specific volume, a quantity inverse to density $\rho$.

The energy of any element of matter under investigation can only change on account of the work of compression that is being performed on it by the surrounding volumes of the fluid (gas).

Bearing in mind the fundamental thermodynamics equation

$dE=TdS-pdv,$

(I-5)

from the energy equation we readily obtain for the studied case of the absence of dissipative forces the natural conclusion

$TdS=0;\quad {\frac {dS}{dt}}=0.$

(I-6)

In other words, the state of matter changes according to the adiabatic curve, it changes with constant entropy.

As is known, for an ideal gas with constant thermal capacity, the adiabatic equation is

$p=A\rho ^{k},$

(I-7)

where $k=c_{p}/c_{v}$, $k=$const. It can also be found without considering entropy, and it was found that way in 1818 by Poisson who integrated Eq. (1-4), in which for an ideal gas we substitute Clapeyron's law

$E=c_{v}T={\frac {c_{v}}{R}}RT={\frac {c_{v}}{R}}pv;\quad dE={\frac {c_{v}}{R}}pdv+{\frac {c_{v}}{R}}vdp.$

(I-8)

Which are the conditions of applicability of the above equations$^{3}$ in which the effect of viscosity and thermal conductivity was disregarded? It is obvious, in the first place, that in order to apply these equations the Reynolds and Peclet numbers must be high.