Stability of a hydrodynamic discontinuity

Abstract

While looking from a far field at a discontinuous front separating incompressible ideal fluids of different densities, we identify two qualitatively different behaviors of the front-unstable and stable-depending upon whether the energy flux produced by the perturbed front is large or small compared to the flux of kinetic energy across the planar front. Landau's solution for the Landau-Darrieus instability is consistent with one of these cases.

Analysis of dynamics, stability, and flow fields' structure of an accelerated hydrodynamic discontinuity with interfacial mass flux by a general matrix method

“…32,33 For incompressible fluids, it has been recently found that the interface stability is highly sensitive to energy fluctuations produced by the perturbed interface. 24,34 These theories and models successfully expanded the classical framework 20,23 and explained experiments. [15][16][17][21][22][23][25][26][27][28]31 Some fundamental challenges still remain.…”

“…3 That is, the specific internal energy refers to energy per unit mass that is contained within a system, excluding the kinetic energy of motion of the system as a whole and the potential energy of the system as a whole due to the external force fields. 3,24 The inertial frame of reference refers to the frame of reference moving with a constant velocityṼ 0 ¼ ð0; 0;Ṽ 0 Þ. 3,34 We introduce a continuous local scalar function hðx; y; z; tÞ, whose derivatives _ h and rh exist (the dot marks a partial time-derivative), such that h ¼ 0 at the interface, and the heavy (light) fluid is located in the region h > 0 (h < 0); its fields are ðq; v; P; EÞ !…”

“…ðq; v; P; EÞ hðlÞ and are marked with subscript hðlÞ hereafter. 4,11,24,31 By using the Heaviside step-function HðhÞ, we represent in the fluid fields in the entire domain as ðq; v; P; EÞ ¼ ðq; v; P; EÞ h HðhÞ þðq; v; P; EÞ l HðÀhÞ. 24,31 At the interface, the balance of fluxes of mass and normal and tangential components of momentum and energy obey the boundary conditions 3,4,24,34…”

“…where ½… ¼ 0 denotes the jump of functions across the interface; n and s are the unit vectors normal and tangential at the interface with n ¼ rh=jrhj and ðn Á sÞ ¼ 0;j ¼ qðn _ h=jrhj þ vÞ is the mass flux across the moving interface; W ¼ e þ P=q is the specific enthalpy in its physics definition; particularly, the value W includes the enthalpy of formation. 3,24 Here, we provide (the first, to our knowledge) systematic discussion of the boundary conditions Eq. (2a), including their direct derivation from the governing equations Eq.…”

“…We consider only the first of these conditions here; other conditions can be derived likewise. 3,4,24,31,34 In particular, in the entire domain the fluid density is q ¼ q h HðhÞ þ q l HðÀhÞ, 4,24,34 leading to…”