Extended symmetry analysis of an isothermal no-slip drift flux model

Abstract

We perform extended group analysis for a system of differential equations modeling an isothermal no-slip drift flux. The maximal Lie invariance algebra of this system is proved to be infinite-dimensional. We also find the complete point symmetry group of this system, including discrete symmetries, using the megaideal-based version of the algebraic method. Optimal lists of one-and two-dimensional subalgebras of the maximal Lie invariance algebra in question are constructed and employed for obtaining reductions of the system under study. Since this system contains a subsystem of two equations that involves only two of three dependent variables, we also perform group analysis of this subsystem. The latter can be linearized by a composition of a fiber-preserving point transformation with a two-dimensional hodograph transformation to the Klein-Gordon equation. We also employ both the linearization and the generalized hodograph method for constructing the general solution of the entire system under study. We find inter alia genuinely generalized symmetries for this system and present the connection between them and the Lie symmetries of the subsystem we mentioned earlier. Hydrodynamic conservation laws and their generalizations are also constructed.

Generalized symmetries, conservation laws and Hamiltonian structures of an isothermal no-slip drift flux model

“…x . As was noted in [27,Remark 19], this subspace is closed with respect to the Lie bracket of generalized vector fields, and thus we can call it an algebra. The indicated property is shared by all strictly hyperbolic diagonalizable hydrodynamic-type systems.…”

“…Following the procedure analogous to that in [27], we find the complete set of local solutions of the system (1) via the linearization of the subsystem (1a)-(1b).…”

“…Alternatively, to get the subfamily of regular solutions with nonconstant parameter function W , one can employ the generalized hodograph transformation [39], see details in [27,Section 9].…”

“…In [27] we carried out the extended classical symmetry analysis of the system S. In particular, for this system we constructed the general solution via the generalized hodograph transformation and the entire set of local solutions via the linearization of the subsystem S 0 , the maximal Lie invariance algebra g and the algebra of generalized symmetries of order not greater than one, the complete point symmetry group and group-invariant solutions. Thus, the algebra g is spanned by the vector fields…”

“…where (τ, ξ) is a tuple of smooth functions of (r 1 , r 2 ), running through the solution set of the system ξ r 1 = V 2 τ r 1 , ξ r 2 = V 1 τ r 2 . In [27], for the system S we also found the zeroth-order local conservation laws using the direct method and, following [4], constructed the entire space of first-order (t, x)-translation-invariant conservation laws of and a subspace of (t, x)-translationinvariant conservation laws of arbitrarily high order. Generalizing a subalgebra of generalized symmetries of order not greater than one, we obtain an infinite-dimensional subspace of generalized symmetries of arbitrarily high order for S. (In the present paper we show that this subspace is an ideal in the entire algebra of generalized symmetries of the system S.) At the same time, the system S possesses two properties that allow us to exhaustively describe the entire spaces of generalized symmetries, cosymmetries and local conservation laws.…”