The absolute instability of an inviscid compound jet

Abstract

This paper examines the emergence of the absolute instability from convectively unstable states of an inviscid compound jet. A compound jet is composed of a cylindrical jet of one fluid surrounded by a concentric annulus of a second, immiscible fluid. For all jet velocities $v$, there are two convectively unstable modes. As in the single-fluid jet, the compound jet becomes absolutely unstable below a critical dimensionless velocity or Weber number $V ({:=}\,\sqrt{v^2\,{\rho_1 R_1}/\sigma_1}$ where $\rho_{1}$, $R_{1}$ and $\sigma _{1}$ are the core density, radius and core–annular interfacial tension), which is a function of the annular/core ratios of densities $\beta$, surface tensions $\gamma$ and radii $a$. At $V\,{=}\,0$, the absolutely unstable modes and growth recover the fastest growing temporal waves. We focus specifically on the effect of $\gamma$ at $a\,{=}\,2$ and $\beta\,{=}\,1$ and find that when the outer tension is significantly less than the inner $(0.1\,{<}\,\gamma\,{<}\,0.3)$, the critical Weber number $V_{\hbox{\scriptsize{\it crit}}}$ decreases with <$\gamma$, whereas for higher ratios $(0.3\,{<}\,\gamma\,{<}\,3)$ it increases. The values (1.2–2.3) of $V_{\hbox{\scriptsize{\it crit}}}$ for the compound jet include the parameter-independent critical value of 1.77 for the single jet. Therefore, increasing the outer tension can access the absolute instability at higher dimensional velocities than for a single jet with the same radius and density as the core and a surface tension equal to the compound jet's liquid–liquid tension. We argue that this potentially facilitates distinguishing experimentally between absolute and convective instabilities because higher velocities and surface tension ratios higher than 1 extend the breakup length of the convective instability. In addition, for $0.3\,{<}\,\gamma\,{<}\,1.16$, the wavelength for the absolute instability is roughly half that of the fastest growing convectively unstable wave. Thus choosing $\gamma$ in this range exaggerates its distinction from the convective instability and further aids the potential observation of absolute instability.