Landau derivative

In gas dynamics, the Landau derivative or fundamental derivative of gas dynamics, named after Lev Landau who introduced it in 1942, refers to a dimensionless physical quantity characterizing the curvature of the isentrope drawn on the specific volume versus pressure plane. Specifically, the Landau derivative is a second derivative of specific volume with respect to pressure. The derivative is denoted commonly using the symbol $$\Gamma$$ or $$\alpha$$ and is defined by


 * $$\Gamma = \frac{c^4}{2\upsilon^3}\left(\frac{\partial^2\upsilon}{\partial p^2}\right)_s$$

where

Alternate representations of $$\Gamma$$ include


 * $$\Gamma = \frac{\upsilon^3}{2c^2} \left(\frac{\partial^2 p}{\partial \upsilon^2}\right)_s = \frac{1}{c} \left(\frac{\partial \rho c}{\partial \rho}\right)_s= 1 + \frac{c}{\upsilon} \left(\frac{\partial c}{\partial p}\right)_s = 1 + \frac{c}{\upsilon} \left(\frac{\partial c}{\partial p}\right)_T + \frac{cT}{\upsilon c_p}\left(\frac{\partial\upsilon}{\partial T}\right)_p \left(\frac{\partial c}{\partial T}\right)_p.$$

For most common gases, $$\Gamma>0$$, whereas abnormal substances such as the BZT fluids exhibit $$\Gamma<0$$. In an isentropic process, the sound speed increases with pressure when $$\Gamma>1$$; this is the case for ideal gases. Specifically for polytropic gases (ideal gas with constant specific heats), the Landau derivative is a constant and given by


 * $$\Gamma = \frac{1}{2}(\gamma+1),$$

where $$\gamma>1$$ is the specific heat ratio. Some non-ideal gases falls in the range $$0<\Gamma<1$$, for which the sound speed decreases with pressure during an isentropic transformation.