  We present mesoscale numerical simulations of Rayleigh-B\'enard (RB)
convection in a two-dimensional model emulsion. The systems under study are
constituted of finite-size droplets, whose concentration Phi_0 is
systematically varied from small (Newtonian emulsions) to large values
(non-Newtonian emulsions). We focus on the characterisation of the heat
transfer properties close to the transition from conductive to convective
states, where it is known that a homogeneous Newtonian system exhibits a steady
flow and a time-independent heat flux. In marked contrast, emulsions exhibit a
non-steady dynamics with fluctuations in the heat flux. In this paper, we aim
at the characterisation of such non-steady dynamics via detailed studies on the
time-averaged heat flux and its fluctuations. To understand the time-averaged
heat flux, we propose a side-by-side comparison between the emulsion system and
a single-phase (SP) system, whose viscosity is constructed from the shear
rheology of the emulsion. We show that such local closure works well only when
a suitable degree of coarse-graining (at the droplet scale) is introduced in
the local viscosity. To delve deeper into the fluctuations in the heat flux, we
propose a side-by-side comparison between a Newtonian emulsion and a
non-Newtonian emulsion, at fixed time-averaged heat flux. This comparison
elucidates that finite-size droplets and the non-Newtonian rheology cooperate
to trigger enhanced heat-flux fluctuations at the droplet scales. These
enhanced fluctuations are rooted in the emergence of space correlations among
distant droplets, which we highlight via direct measurements of the droplets
displacement and the characterisation of the associated correlation function.
The observed findings offer insights on heat transfer properties for confined
systems possessing finite-size constituents.

Fluid Dynamics;Soft Condensed Matter;Statistical Mechanics