Linear Stability Analysis for a Korteweg Fluid

Linear stability analysis of the homogeneous equilibrium solution of Euler equations for an isothermal, inviscid, compressible fluid endowed with a Korteweg-type stress tensor, in a harmonic potential field, is performed. We show that, by perturbing the fluid at rest, the transition from stability to instability takes place via a marginal state exhibiting a stationary cellular pattern of motions. By analyzing the disturbances in normal modes we obtain threshold values of the harmonic angular frequency that, in correspondence of a given temperature, could trigger fluid's fragmentation. In the picture of initial stage of a volcano eruption, this model could describe the transition from the two-phase system magma-dissolved gas, at supersaturation pressure in the chamber, to the rising foam at conduit's base induced by an external stress of elastic-type.