Landau derivative explained

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

\Gamma or \alpha and is defined by^{[3] [4] [5]} \Gamma = (c ^{4})/(\left( 2 \upsilon ^{3}) (\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 = (\upsilon ^{3})/(\left( 2 c ^{2}) (\partial ^{2} p)/(\partial \upsilon ^{2}) \right) _{s} = (1)/(\left( c) (\partial \rho c)/(\partial \rho) \right) _{s=} 1 + (c)/(\left( \upsilon) (\partial c)/(\partial p) \right) _{s} = 1 + (c)/(\left( \upsilon) (\partial c)/(\partial p) \right) _{T} + (cT)/(\left( \upsilon c _{p}) (\partial\upsilon)/(\partial T) \right) _{p} \left( (\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 = (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

Notes and 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.