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Abstract

In the preparation of café latte, spectacular layer formation can occur between the espresso
shot in a glass of milk and the milk itself. Xue et al. (Nat. Commun., vol. 8, 2017, pp. 1–6)
showed that the injection velocity of espresso determines the depth of coffee–milk mixture.
After a while, when a stable stratification forms in the mixture, the layering process can
be modelled as a double diffusive convection system with a stably stratified coffee–milk
mixture cooled from the side. More specifically, we perform (two-dimensional) direct
numerical simulations of laterally cooled double diffusive convection for a wide parameter
range, where the convective flow is driven by a lateral temperature gradient while
stabilized by a vertical concentration gradient. Depending on the strength of stabilization
as compared to the thermal driving, the system exhibits different flow regimes. When
the thermal driving force dominates over the stabilizing force, the flow behaves like
vertical convection in which a large-scale circulation develops. However, with increasing
strength of the stabilizing force, a meta-stable layered regime emerges. Initially, several
vertically-stacked convection rolls develop, and these well-mixed layers are separated by
sharp interfaces with large concentration gradients. The initial thickness of these emerging
layers can be estimated by balancing the work exerted by thermal driving and the required
potential energy to bring fluid out of its equilibrium position in the stably stratified fluid.
In the layered regime, we further observe successive layer merging, and eventually only
a single convection roll remains. We elucidate the following merging mechanism: as
weakened circulation leads to accumulation of hot fluid adjacent to the hot sidewall, larger buoyancy forces associated with hotter fluid eventually break the layer interface. Then
two layers merge into a larger layer, and circulation establishes again within the merged structure.