Landau derivative

In gas dynamics, the Landau derivative or fundamental derivative of gas dynamics, named after Lev Landau who introduced it in 1942, ^{[1] [2]} 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 ${\displaystyle \Gamma }$ or ${\displaystyle \alpha }$ and is defined by ^{[3] [4] [5]}

$${\displaystyle \Gamma ={\frac {c^{4}}{2\upsilon ^{3}}}\left({\frac {\partial ^{2}\upsilon }{\partial p^{2}}}\right)_{s}}$$

where

• ${\displaystyle c}$ is the sound speed,
• ${\displaystyle \upsilon =1/\rho }$ is the specific volume,
• ${\displaystyle \rho }$ is the density,
• ${\displaystyle p}$ is the pressure, and
• ${\displaystyle s}$ is the specific entropy.

Alternate representations of ${\displaystyle \Gamma }$ include

$${\displaystyle {\begin{aligned}\Gamma &={\frac {\upsilon ^{3}}{2c^{2}}}\left({\frac {\partial ^{2}p}{\partial \upsilon ^{2}}}\right)_{s}={\frac {1}{c}}\left({\frac {\partial \rho c}{\partial \rho }}\right)_{s}=1+{\frac {c}{\upsilon }}\left({\frac {\partial c}{\partial p}}\right)_{s}\\[2ex]&=1+{\frac {c}{\upsilon }}\left({\frac {\partial c}{\partial p}}\right)_{T}+{\frac {cT}{\upsilon c_{p}}}\left({\frac {\partial \upsilon }{\partial T}}\right)_{p}\left({\frac {\partial c}{\partial T}}\right)_{p}.\end{aligned}}}$$

For most common gases, ${\displaystyle \Gamma >0}$, whereas abnormal substances such as the BZT fluids exhibit ${\displaystyle \Gamma <0}$. In an isentropic process, the sound speed increases with pressure when ${\displaystyle \Gamma >1}$; 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

$${\displaystyle \Gamma ={\tfrac {1}{2}}(\gamma +1),}$$

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

See also

• Landau damping

References

1. 1942, Landau, L.D. "On shock waves" J. Phys. USSR 6 229-230.
2. Thompson, P. A. (1971). A fundamental derivative in gasdynamics. The Physics of Fluids, 14(9), 1843-1849.
3. Landau, L. D., & Lifshitz, E. M. (2013). Fluid mechanics: Landau And Lifshitz: course of theoretical physics, Volume 6 (Vol. 6). Elsevier.
4. W. D. Hayes, in Fundamentals of Gasdynamics, edited by H. W. Emmons (Princeton University Press, Princeton, N.J., 1958), p. 426.
5. Lambrakis, K. C., & Thompson, P. A. (1972). Existence of real fluids with a negative fundamental derivative Γ. Physics of Fluids, 15(5), 933-935.