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 Γ {\displaystyle \Gamma } or α {\displaystyle \alpha } and is defined by
Γ = c 4 2 υ 3 ( ∂ 2 υ ∂ p 2 ) s {\displaystyle \Gamma ={\frac {c^{4}}{2\upsilon ^{3}}}\left({\frac {\partial ^{2}\upsilon }{\partial p^{2}}}\right)_{s}}
where
c {\displaystyle c} is the sound speed,
υ = 1 / ρ {\displaystyle \upsilon =1/\rho } is the specific volume,
ρ {\displaystyle \rho } is the density,
p {\displaystyle p} is the pressure, and
s {\displaystyle s} is the specific entropy. Alternate representations of Γ {\displaystyle \Gamma } include
Γ = υ 3 2 c 2 ( ∂ 2 p ∂ υ 2 ) s = 1 c ( ∂ ρ c ∂ ρ ) s = 1 + c υ ( ∂ c ∂ p ) s = 1 + c υ ( ∂ c ∂ p ) T + c T υ c p ( ∂ υ ∂ T ) p ( ∂ c ∂ T ) p . {\displaystyle {\begin{aligned}\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}\\[2ex]&=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}.\end{aligned}}}
For most common gases, Γ > 0 {\displaystyle \Gamma >0} , whereas abnormal substances such as the BZT fluids exhibit Γ < 0 {\displaystyle \Gamma <0} . In an isentropic process, the sound speed increases with pressure when Γ > 1 {\displaystyle \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
Γ = 1 2 ( γ + 1 ) , {\displaystyle \Gamma ={\tfrac {1}{2}}(\gamma +1),}
where γ > 1 {\displaystyle \gamma >1} is the specific heat ratio. Some non-ideal gases falls in the range 0 < Γ < 1 {\displaystyle 0<\Gamma <1} , for which the sound speed decreases with pressure during an isentropic transformation.
See also Landau damping
References