gaz parfait polytropique

\(\frac{P}{\rho^k}=cte=\frac{P_0}{\rho_0^k}\) avec \(P_0=P(x_3=0)\) et \(\rho_0=\rho(x_3=0)\)

donc \(\rho = \rho_0\left( \frac{P}{P_0}\right)^{1/k}\)

\(\rho\overrightarrow{g} -\nabla P = \vec 0\Leftrightarrow\left\{ \begin{array}{l c l} \frac{\partial P}{\partial x_1} = 0 &\Rightarrow & P(x_2,x_3) \\ \frac{\partial P}{\partial x_2} = 0 & \Rightarrow & P(x_3) \\ -\frac{\partial P}{\partial x_3}-\rho g = 0 &  &  \end{array}\right.\)

Donc \(\frac{\partial P}{\partial x_3}=\frac{d P}{d x_3}=-\rho g\)

\(\frac{d P}{d x_3}=-\rho_0\left( \frac{P}{P_0}\right)^{\frac 1k}g\)

\(\displaystyle \int P^{-\frac 1k} \left( \frac{d P}{dx_3}\right)dx_3 = \int-\rho_0 P_0^{-\frac 1k}gdx_3\)

\(\frac{1}{-\frac 1k+1}P^{- \frac 1k +1}=-\rho_0P_0^{-\frac 1k}gdx_3+C\)

avec \(P(x_3=0)=P_0\Rightarrow \frac{1}{-\frac 1k+1}P_0^{- \frac 1k +1}=C\)

donc \(\frac{k}{k-1}P^{- \frac 1k +1}=-\rho_0P_0^{-\frac 1k}gx_3+\frac{k}{k-1}P_0^{- \frac 1k +1}\)

\(P^{-\frac 1k +1}=P_0^{-\frac 1k +1} - \frac{k-1}{k}\rho_0gx_3P_0^{-\frac 1k +1 -1}\)

\(P^{\frac{k-1}{k}}=P_0^{\frac{k-1}{k}}\left( 1-\frac{k-1}{k}\frac{\rho_0}{P_0}gx_3\right)\)

\(\frac{P}{P_0}=\left( 1-\frac{k-1}{k}\frac{\rho_0}{P_0}gx_3\right)^{\frac 1k -1}\)