TY - JOUR

T1 - Dispersion and Group Analysis of Dusty Burgers Equations

AU - Stoyanovskaya, O. P.

AU - Turova, G. D.

AU - Yudina, N. M.

N1 - The dispersion analysis (paragraph 2) was done by O.P. Stoyanovskaya (funded by the RSF grant 23-11-00142), the group analysis (paragraph 3) was done by N.M. Yudina as a part of her bachelor thesis at Novosibirsk State University, invariant solutions (paragraph 4) were derived by G.D. Turova (funded by the RSF grant 22-21-20063).

PY - 2024/1

Y1 - 2024/1

N2 - We investigate the system of non-stationary one-dimensional equations consisting of a parabolic Burgers equation for the velocity of a viscous gas and a hyperbolic Hopf equation for the velocity of solid particles. The Burgers and Hopf equations are connected into a system due to relaxation terms simulating the momentum transfer between the carrier phase (gas) and the dispersed phase (particles). The momentum transfer intensity is inversely proportional to the relaxation time of the particle velocity to the gas velocity (stopping time). A dispersion relation is constructed for this system. A particular solution corresponding to the damping of a low-amplitude sound wave is found. For an infinitely short velocity relaxation time, the effective viscosity of the gas-dust medium is derived, which is determined by the viscosity of the gas and the mass fraction of particles in the mixture. The Lie algebra of symmetries of Burgers–Hopf system is found. Invariant submodels with respect to the basis operators of the symmetry algebra are derived. These submodels are explicitly integrated, except for one that defines stationary motion. For this submodel, a code has been developed for the numerical generation of particular solutions of the system. It is shown that the invariant solution determined by this submodel, in the asymptotic case of infinitely short velocity relaxation time also makes it possible to obtain the effective viscosity of the gas-dust mixture. Moreover, this effective viscosity coincides with the viscosity value determined from the dispersion relation.

AB - We investigate the system of non-stationary one-dimensional equations consisting of a parabolic Burgers equation for the velocity of a viscous gas and a hyperbolic Hopf equation for the velocity of solid particles. The Burgers and Hopf equations are connected into a system due to relaxation terms simulating the momentum transfer between the carrier phase (gas) and the dispersed phase (particles). The momentum transfer intensity is inversely proportional to the relaxation time of the particle velocity to the gas velocity (stopping time). A dispersion relation is constructed for this system. A particular solution corresponding to the damping of a low-amplitude sound wave is found. For an infinitely short velocity relaxation time, the effective viscosity of the gas-dust medium is derived, which is determined by the viscosity of the gas and the mass fraction of particles in the mixture. The Lie algebra of symmetries of Burgers–Hopf system is found. Invariant submodels with respect to the basis operators of the symmetry algebra are derived. These submodels are explicitly integrated, except for one that defines stationary motion. For this submodel, a code has been developed for the numerical generation of particular solutions of the system. It is shown that the invariant solution determined by this submodel, in the asymptotic case of infinitely short velocity relaxation time also makes it possible to obtain the effective viscosity of the gas-dust mixture. Moreover, this effective viscosity coincides with the viscosity value determined from the dispersion relation.

KW - Burgers equation

KW - dispersion analysis

KW - dispersion relation

KW - dusty gas

KW - effective viscosity

KW - group analysis

KW - momentum exchange

KW - partial differential equation

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85192965840&origin=inward&txGid=c501414c3f1514ca360000d7d3c46a98

UR - https://www.elibrary.ru/item.asp?id=67311894

UR - https://www.mendeley.com/catalogue/b99c34f5-a37e-3d2a-9b92-64d9f47c8e00/

U2 - 10.1134/S1995080224010505

DO - 10.1134/S1995080224010505

M3 - Article

VL - 45

SP - 108

EP - 118

JO - Lobachevskii Journal of Mathematics

JF - Lobachevskii Journal of Mathematics

SN - 1995-0802

IS - 1

M1 - 12

ER -