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 $\mathrm{\Gamma }$ or $\alpha$ and is defined by

$$\mathrm{\Gamma }=\frac{c^4}{2{\upsilon }^3}{\left(\frac{{\partial }^2\upsilon }{\partial p^2}\right)}_s$$

where

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

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

$$\begin{array}{cc}\mathrm{\Gamma }& =\frac{{\upsilon }^3}{2c^2}{\left(\frac{{\partial }^{2}p}{\partial {\upsilon }^2}\right)}_s=\frac{1}{c}{\left(\frac{\partial \rho c}{\partial \rho }\right)}_s=1+\frac{c}{\upsilon }{\left(\frac{\partial c}{\partial p}\right)}_s\\ & =1+\frac{c}{\upsilon }{\left(\frac{\partial c}{\partial p}\right)}_T+\frac{cT}{\upsilon c_p}{\left(\frac{\partial \upsilon }{\partial T}\right)}_p{\left(\frac{\partial c}{\partial T}\right)}_p.\end{array}$$

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

$$\mathrm{\Gamma }=\frac{1}{2}(\gamma +1),$$

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

See also

• Landau damping

References

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