Butter on a hot pan: self-regulating dynamics of melt-lubricated sliding

Abstract

When solids melt while sliding down heated inclines, their motion is governed by a complex coupling between heat transfer, phase change, gravity and viscous dissipation. Despite relevance across a variety of domains, like kitchen physics, geophysics, tribology, and manufacturing, this coupled problem lacks understanding and quantitative experimental validation. Here we report experiments with ice and paraffin wax on a temperature-controlled ramp that achieve terminal velocities from 0.01 m/s to 2 m/s across wide parameter ranges. We develop a theoretical model that captures the self-regulating feedback between melt-layer thickness, sliding velocity, and heat transfer. Without any adjustable parameters, our model collapses all measurements, validating the fundamental mechanism and enabling predictions for analogous systems.

Figures (5)

• Figure 1: (a) A block of butter slides down an inclined, heated cooking pan due to the lubricating effect of the liquid melt layer beneath it. (b) Schematic of the problem, with the relevant physical quantities. A cubic block of solid with sides $L=H$ slides down a heated ramp, which is inclined at an angle $\theta$, with a constant velocity $U$. The enlargement is tilted to be aligned with the ramp and is in the frame of reference of the block. $u(x,y)$ indicates the flow velocity; $T_\text{amb}$ is the ambient temperature; $T_\text{ramp}$ is the ramp temperature; $T_\text{melt}$ is the melting temperature; $j$ is the melt rate (with units of velocity). The symbol $\ell$ bounds the region $[\ell,L]$ where capillarity effects are dominant. (c) Superposed frames of a paraffin block. The block is moving from left to right. The time between the frames is 0.5, the angle of the ramp is 2.5° and the excess temperature is 5.
• Figure 2: Terminal velocity of ice (hollow squares) and paraffin (disks) blocks sliding down an inclined heated ramp as a function of excess temperature $\Delta T=T_\text{ramp}-T_\text{melt}$ (a) and inclination angle $\theta$ (b). In both plots, the variable that is not in abscissa is shown as the markers' colour. As the blocks are at terminal velocity, the errors in the measured velocities are tiny (less than 1%), and no error bars are shown.
• Figure 3: Schematic of the coupling between the effects of shear, transit time, and mass and heat balances.
• Figure 4: In panel (a): blocks' measured velocity $U_\text{exp}$ against the velocity from our analytical model $U_\text{2D}$; paraffin wax data are plotted as disks, while ice data are plotted as squares; the colour of the points indicate the excess temperature $\Delta T=T_\text{ramp}-T_\text{melt}$, while the size indicates the angle $\theta$, as in the legend (markers are 2.5°, 7.5°, 22.5°, 45°). The error bars on $U_\text{2D}$ were calculated with the bootstrapping method, but are not plotted as they never exceeded a marker's width. The rescaling does not involve any adjustable parameters; the solid line corresponds to $y=\frac{2}{7}x$. In panel (b), for synthetic data with 25 blocks of wax at angles in 0.5° $\leq \theta \leq$90°, and excess temperatures in 3$\leq \Delta T \leq$32, we calculate: the melt rate $j_\text{syn}$, the velocity $U_\text{syn}$, and the layer height $h_\text{syn}$, and we represent them as a 3D surface with iso-$\theta$ contours marked in red and iso-$\Delta T$ contours marked in blue.
• Figure S1: (a) Photograph of the experimental setup and (b) internal rendering of the ramp.