  For an immiscible oil drop immersed in a stably stratified ethanol-water
mixture, a downwards solutal Marangoni flow is generated on the surface of the
drop, owing to the concentration gradient, and the resulting propulsion
competes against the downwards gravitational acceleration of the heavy drop. In
prior work of Li et al. (Phys. Rev. Lett., vol. 126, issue 12, 2021, 124502),
we found that for drops of low viscosity, an oscillatory instability of the
Marangoni flow is triggered once the Marangoni advection is too strong for
diffusion to restore the stratified concentration field around the drop. Here
we experimentally explore the parameter space of the concentration gradient and
drop radius for large oil viscosities and find a different and new mechanism
for triggering the oscillatory instability in which diffusion is no longer the
limiting factor. For such drops of higher viscosities, the instability is
triggered when the gravitational effect is too strong so that the viscous
stress cannot maintain a stable Marangoni flow. This leads to a critical drop
radius above which the equilibrium is always unstable. Subsequently, a unifying
scaling theory that includes both the mechanisms for low and for high
viscosities of the oil drops is developed. The transition between the two
mechanisms is found to be controlled by two length scales: The drop radius $R$
and the boundary layer thickness $\delta$ of the Marangoni flow around the
drop. The instability is dominated by diffusion for $\delta<R$ and by viscosity
for $R<\delta$. The experimental results for various drops of different
viscosities can well be described with this unifying scaling theory. Our
theoretical description thus provides a unifying view of physicochemical
hydrodynamic problems in which the Marangoni stress is competing with a stable
stratification.
