  This work focuses on the interfacial dynamics with interfacial mass flux in
the presence of acceleration and surface tension. We employ the general matrix
method to find the fundamental solutions for the linearized boundary value
problem conserving mass, momentum and energy. We find that the dynamics can be
stable or unstable depending on the values of the acceleration, the surface
tension and the density ratio. In the stable regime, the flow has the
non-perturbed fields in the bulk, is shear-free at the interface, and has the
constant interface velocity. The dynamics is unstable only when it is
accelerated, and when the acceleration value exceeds a threshold combining
contributions of the inertial stabilization mechanism and the surface tension.
The properties of this instability unambiguously differentiate it from other
fluid instabilities. Particularly, its velocity field has potential and
vortical components in the bulk and is shear free at the interface. Its
dynamics describes the standing wave with the growing amplitude, and has the
growing interface velocity. For strong accelerations, this fluid instability of
the conservative dynamics has the fastest growth-rate and the largest
stabilizing surface tension value when compared to the classical Landau and
Rayleigh-Taylor dynamics. We find the values of the initial perturbation
wavelength at which the fluid instability can be stabilized and at which it has
the fastest growth. We identify theory benchmarks for experiments and
simulations in high energy density plasmas and its outcomes for application
problems in nature and technology.
