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

$c$	is the sound speed;
$\upsilon =1/\rho$	is the specific volume;
$\rho$	is the density;
$p$	is the pressure;
$s$	is the specific entropy.

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.

See also

• Landau damping