Lateral self-propulsion of Marangoni droplets in stratified fluids

Stable vertical stratification of liquids significantly influences the dynamics of both rigid and deformable bodies immersed in them. Recent research has shown that the interplay between buoyancy and Marangoni convection induced by surface tension gradients gives rise to periodic vertical bouncing of droplets (e.g. of silicone oil) immersed in a stable concentration gradient (e.g. of ethanol in water). Here, we report that such droplets also exhibit lateral self-propulsion, which may or may not be accompanied by bouncing, when the radius of the droplet is large enough. We investigate experimentally and theoretically how the droplet radius and the magnitude of the ethanol concentration gradient influence the onset of self-propulsion and the steady lateral self-propulsion speed far beyond onset. We find that lateral self-propulsion occurs when the droplet Marangoni number italic Ma <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Ma</mml:mtext> </mml:mrow> </mml:math> $\textit{Ma}$ exceeds a critical value ( italic Ma Subscript italic crit Baseline tilde 10 Superscript 5 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Ma</mml:mtext> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">crit</mml:mtext> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>∼</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>5</mml:mn> </mml:msup> </mml:math> $\textit{Ma}_{\textit{crit}}\sim 10^5$ ), at which point spontaneous symmetry breaking of the initially axisymmetric flow field occurs. The condition italic Ma equals italic Ma Subscript italic crit Baseline <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Ma</mml:mtext> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Ma</mml:mtext> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">crit</mml:mtext> </mml:mrow> </mml:mrow> </mml:msub> </mml:math> $\textit{Ma} = \textit{Ma}_{\textit{crit}}$ implies that the critical droplet radius for lateral self-propulsion scales as the negative 1 divided by 2 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:math> $-1/2$ power of the magnitude of the solute concentration gradient, a result that is consistent with experiment. Far beyond onset, the self-propulsion speed scales with the theoretical Marangoni velocity for infinite diffusivity. These findings offer insights for diverse applications such as modelling biological systems and developing microrobots.