From hyperbolic regularization to exact hydrodynamics for linearized Grad’s equations

Inspired by a recent hyperbolic regularization of Burnett’s hydrodynamic equations [A. Bobylev, J. Stat. Phys. 124, 371 (2006)], we introduce a method to derive hyperbolic equations of linear hydrodynamics to any desired accuracy in Knudsen number. The approach is based on a dynamic invariance principle which derives exact constitutive relations for the stress tensor and heat flux, and a transformation which renders the exact equations of hydrodynamics hyperbolic and stable. The method is described in detail for a simple kinetic model—a 13 moment Grad system.

Figure 1

Dispersion relation. Acoustic mode $\mathrm{Re}\left({\omega }_{\mathrm{ac}}\right)$ for Navier-Stokes and Burnett hydrodynamics.

Figure 2

(Color online) Dispersion relation for the linearized 1D13M Grad system (1). The unique solution of hydrodynamical modes obtained from Eq. (13) with Eq. (17) coincides with the real parts of the modes of the original Grad system (the plot also shows when pairs of conjugate complex roots appear), the solution of the original system (1) features five $\omega$’s, while the exact solution of Eq. (13) with Eq. (17) has three $\omega$’s for each $k$ and degenerated over the hydrodynamic branches at $k\ge k_c$.

Figure 3

(Color online) Imaginary parts of coefficients $A$ to $Z$ solving Eq. (17). Shown is the unique solution leading to hydrodynamic branches, cf. Fig. 2, which is symmetric about the real axis.

Figure 4

(Color online) Real parts of complex-valued functions $A,\dots ,Z$ solving Eq. (17). It is clearly visible that the solution matches the initial condition (7).

Figure 5

Dispersion relations $\omega \left(k\right)$ for acoustic and diffusive modes obtained via Newton iteration. In the plots, shown is also the approximation obtained through Bobylev’s hyperbolic regularization (HR) [13]. Newton iterations fail for $k\ge k_c=0.3023$.