Landau Derivative

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 $\Gamma$ or $\alpha$ and is defined byLandau, L. D., & Lifshitz, E. M. (2013). Fluid mechanics: Landau And Lifshitz: course of theoretical physics, Volume 6 (Vol. 6). Elsevier.Lambrakis, K. C., & Thompson, P. A. (1972). Existence of real fluids with a negative fundamental derivative Γ. Physics of Fluids, 15(5), 933-935.

$\Gamma = \frac\left(\frac\right)_s$

where
* $c$ is the sound speed,
* $\upsilon = 1/\rho$ is the specific volume,
* $\rho$ is the density,
* $p$ is the pressure, and
* $s$ is the specific entropy.

Alternate representations of $\Gamma$ include

\begin
\Gamma &= \frac \left(\frac\right)_s
= \frac \left(\frac\right)_s
= 1 + \frac \left(\frac\right)_s \\ ex&= 1 + \frac \left(\frac\right)_T + \frac\left(\frac\right)_p \left(\frac\right)_p.
\end

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

$\Gamma = \tfrac(\gamma+1),$

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

References

Fluid dynamics

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