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
or
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b8e15d565215ea2b92c04587d8dfaf1fac61197)
where
Alternate representations of
include
![{\displaystyle \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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c79b69ff51d05802583100bc046e88cfbb80e47)
For most common gases,
, whereas abnormal substances such as the BZT fluids exhibit
. In an isentropic process, the sound speed increases with pressure when
; 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 ={\frac {1}{2}}(\gamma +1),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb5fd293ae34111c25594fc223c8e4e090a1354)
where
is the specific heat ratio. Some non-ideal gases falls in the range
, for which the sound speed decreases with pressure during an isentropic transformation.
- ^ 1942, Landau, L.D. "On shock waves" J. Phys. USSR 6 229-230.
- ^ Thompson, P. A. (1971). A fundamental derivative in gasdynamics. The Physics of Fluids, 14(9), 1843-1849.
- ^ Landau, L. D., & Lifshitz, E. M. (2013). Fluid mechanics: Landau And Lifshitz: course of theoretical physics, Volume 6 (Vol. 6). Elsevier.
- ^ W. D. Hayes, in Fundamentals of Gasdynamics, edited by H. W. Emmons (Princeton University Press, Princeton, N.J., 1958), p. 426.
- ^ Lambrakis, K. C., & Thompson, P. A. (1972). Existence of real fluids with a negative fundamental derivative Γ. Physics of Fluids, 15(5), 933-935.