Rectilinear vortex sheets of inviscid liquid-gas two-phase flow: Linear stability

Abstract

The vortex sheet solutions are considered for the inviscid liquid-gas two-phase flow. In particular, the linear stability of rectilinear vortex sheets in two spatial dimensions is established for both constant and variable coefficients. The linearized problem of vortex sheet solutions with constant coefficients is studied by means of Fourier analysis, normal mode analysis, and Kreiss symmetrizer, while the linear stability with variable coefficients is obtained by Bony-Meyer paradifferential calculus theory. The linear stability is crucial to the existence of vortex sheet solutions of the nonlinear problem. A novel symmetrization and some weighted Sobolev norms are introduced to study the hyperbolic linearized problem with characteristic boundary.

Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow

“…In particular, the matrices C(U r,l , ∇U r,l , ∇Φ r,l ) are C ∞ function of (U r,l , ∇U r,l , ∇Φ r,l ) vanishing when (U r,l , ∇U r,l , ∇Φ r,l ) = 0. In view of the previous results in [34], we neglect the zeroth order term in Ψ ± in the linearized equation (3.9) and thus we consider the effective linear operators:…”

“…It is important to obtain energy estimates and define the dual problem. From previous analysis [34], we know that the effective linearized equation (3.9) form a symmetrizable hyperbolic system. The Friedrichs symmetrizer for the operator L ′ e (U r,l , Φ r,l ) is…”

“…In this paper, we establish the nonlinear stability and local existence of compressible vortex sheet solutions for the two-phase flow. Strongly motivated by the paper of Coulombel and Secchi [10], we extend the linear stability result in Ruan-Wang-Weng-Zhu [34] to the nonlinear stability using the estimates in anisotropic Sobolev spaces. The difficulty for the two-phase flow, compared with the Euler flow, is that the difference of the densities of the liquid and gas makes the analysis more complicated and much harder.…”

Elementary symmetrization of inviscid two-fluid flow equations giving a number of instant results

“…We can rewrite the speeds c ± in (28) in terms of the original unknowns m and n: c ± = c(m ± , n ± ), c(m, n) = γ(γ − 1)(m + n) γ n = 1 + m n P n , where P n = ∂P (m, n)/∂n. Now c ± are written exactly in the same form as in [30] where the linear stability of 2D vortex sheets for equations (1) was studied. Introducing the entropy-like function S = m/n and just referring to [26,27], we easily recover not only the linear stability result in [30] but also a recent structural stability result in [13].…”

Shock waves and characteristic discontinuities in ideal compressible two-fluid MHD

“…It is still unclear whether neutrally stable nonplanar Alfvén discontinuities do exist locally in time. However, the linear results in [7] are automatically carried over two-fluid MHD because the free boundary problem (20), (21) has absolutely the same form as that for Alfvén discontinuities in classical MHD.…”