Competition evolution of Rayleigh-Taylor bubbles

Material mixing induced by a Rayleigh-Taylor instability occurs ubiquitously in either nature or engineering when a light fluid pushes against a heavy fluid, accompanying with the formation and evolution of chaotic bubbles. Its general evolution involves two mechanisms: bubble-merge and bubble-competition. The former obeys a universa1 evolution law and has been well-studied, while the latter depends on many factors and has not been well-recognized. In this paper, we establish a theory for the latter to clarify and quantify the longstanding open question: the dependence of bubbles evolution on the dominant factors of arbitrary density ratio, broadband initial perturbations and various material properties (e.g., viscosity, miscibility, surface tensor). Evolution of the most important characteristic quantities, i.e., the diameter of dominant bubble $D$ and the height of bubble zone $h$, is derived: (i) the $D$ expands self-similarly with steady aspect ratio $\beta \equiv D/h \thickapprox (1{\rm{ + }}A)/4$, depending only on dimensionless density ratio $A$, and (ii) the $h$ grows quadratically with constant growth coefficient $\alpha \equiv h/(Ag{t^2}) \thickapprox [2\phi/{\ln}(2{\eta _{\rm{0}}})]^2$, depending on both dimensionless initial perturbation amplitude ${\eta _{\rm{0}}}$ and material-property-associated linear growth rate ratio $\phi\equiv\Gamma_{actual}/\Gamma_{ideal}\leqslant1$. The theory successfully explains the continued puzzle about the widely varying $\alpha\in (0.02,0.12)$ in experiments and simulations, conducted at all value of $A \in (0,1)$ and widely varying value of ${\eta _{\rm{0}}} \in [{10^{ - 7}},{10^{ - 2}}]$ with different materials. The good agreement between theory and experiments implies that majority of actual mixing depends on initial perturbations and material properties, to which more attention should be paid in either natural or engineering problems.