Landau derivative

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]

where

is the sound speed;
is the specific volume;
is the density;
is the pressure;
is the specific entropy.

Alternate representations of include

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

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.

  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.