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
α
and is defined by[3][4][5]
υ
υ
where Alternate representations of
include
υ
{\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,
, 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
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.
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