Gravity can lead to multiple peaks in the early stages of coffee ring formation

We consider the role of gravity in solute transport when a thin droplet evaporates. Under the physically-relevant assumptions that the contact line is pinned and the solutal P\'{e}clet number, $\mbox{Pe}$ is large, we identify two fundamental regimes that depend on the size of the Bond number, $\mbox{Bo}$. When $\mbox{Bo} = O(1)$, the asymptotic structure of solute transport follows directly from the surface tension-dominated regime, whereby advection drives solute towards the contact line, only to be countered by local diffusive effects, leading to the formation of the famous ``coffee ring". For larger Bond numbers, we identify the distinguished limit in which $\mbox{Bo}^{-1/2}\mbox{Pe}^{2/3} = O(1)$, where the diffusive boundary layer is comparable to the surface tension boundary layer. In each regime, we perform a systematic asymptotic analysis of the solute transport and compare our predictions to numerical simulations of the full model. Our analysis identifies the effect of gravity on the nascent coffee ring, providing quantitative predictions of the size, location and shape of the solute mass profile. Furthermore, we reveal that, for certain values of $\mbox{Bo}$, $\mbox{Pe}$ and the evaporation time, a secondary peak may exist inside the classical coffee ring. We find that the onset of this secondary peak is linked to the change in behaviour of the critical point in the droplet centre. Both the onset and the peak characteristics are shown to be independent of $\mbox{Pe}$, but solutal diffusion may act to remove the secondary peak when the classical coffee ring becomes so large as to subsume it.