Exact solutions to drift-flux multiphase flow models through Lie group symmetry analysis

Abstract

In the present paper, Lie group symmetry method is used to obtain some exact solutions for a hyperbolic system of partial differential equations (PDEs), which governs an isothermal no-slip drift-flux model for multiphase flow problem. Those symmetries are used for the governing system of equations to obtain infinitesimal transformations, which consequently reduces the governing system of PDEs to a system of ODEs. Further, the solutions of the system of ODEs which in turn produces some exact solutions for the PDEs are presented. Finally, the evolutionary behavior of weak discontinuity is discussed.

Similarity solutions for two-phase fluids models

“…We found that in the general scenario where the polytropic exponents γ 1 , γ 2 of the two fluids are arbitrary the admitted Lie symmetries from a algebra of dimension fourth, while when in the special case where the polytropic exponents are equal, that is, γ 1 = γ 2 the admitted Lie symmetries form a sixth dimensional Lie algebra. That result is different from that previously found in the literature where it was found that a sixth dimensional Lie algebra it is admitted only when γ 1 , γ 2 are equal with one, that is, γ 1 = γ 2 = 1 [41].…”

“…Another important system of hyperbolic equations where Lie point symmetries have been used for the determination of new solutions is the two-phase flow model [41,42]. The two-phase fluids model describes the evolution of two fluids with different phases, such as liquid and gas, in a tube.…”

“…In [41,42] Lie's theory applied for the simplest two-phase flow model where there is not any mass transfer from the one fluid to the other while the pressure and the energy density of the two fluids is given by an polytropic equation of state as it is given by Lane-Emden equation. In this work we revise the results of [41,42] for a non-flip drift flux model of multi-phase flow defined [53]. More specifically, we find new solutions which have not been presented before, for linear and nonlinear equation of state parameters for the two fluids.…”

Similarity solutions for two‐phase fluids models

“…We found that in the general scenario where the polytropic exponents γ 1 , γ 2 of the two fluids are arbitrary the admitted Lie symmetries from a algebra of dimension fourth, while when in the special case where the polytropic exponents are equal, that is, the admitted Lie symmetries form a sixth dimensional Lie algebra. That result is different from that previously found in the literature where it was found that a sixth dimensional Lie algebra is admitted only when γ 1 , γ 2 are equal to 1, that is, 41 …”

“…In Bira and Sekhar 41 and Bira et al, 42 Lie's theory was applied for the simplest two‐phase flow model where there is not any mass transfer from the one fluid to the other while the pressure and the energy density of the two fluids is given by a polytropic equation of state as it is given by Lane–Emden equation. In this work we revise the results of Bira and Sekhar 41 and Bira et al, 42 for a non‐flip drift–flux model of multi‐phase flow defined 53 . More specifically, we find new solutions which have not been presented before, for linear and nonlinear equation of state parameters for the two fluids.…”