  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.
