A study of bubble coalescence on solid substrates

Direct numerical simulations have been employed to study coalescence of two bubbles on a flat solid substrate at a range of Ohnesorge number ( italic Oh <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:mi>O</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> </mml:math> $ \textit{Oh}$ ) between 0.019 and 0.858 aiming at a detailed understanding of the process as the viscous effects become increasingly significant. Bubble diameter ratio is varied between 1 and 1.5 and the substrate is assumed perfectly hydrophilic. Results show the onset of transition away from an inertio-capillary velocity scaling to be at an italic Oh <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:mi>O</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> </mml:math> $ \textit{Oh}$ of between 0.05 <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:mn>0.05</mml:mn> </mml:math> $0.05$ and 0.1. It is also observed that the merged bubbles stop jumping off the substrate in this range. When the inertio-capillary scaling holds, we report a maximum bubble velocity of 0.19–0.25 u Subscript i c <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:mi>u</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>c</mml:mi> </mml:mrow> </mml:msub> </mml:math> $u_{ic}$ depending on the value of diameter ratio, where u Subscript i c <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:mi>u</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>c</mml:mi> </mml:mrow> </mml:msub> </mml:math> $u_{ic}$ is the inertio-capillary velocity scale. Above italic Oh <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:mi>O</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> </mml:math> $ \textit{Oh}$ = 0.5, a perfect scaling with visco-capillary velocity and overdamping of bubble oscillations are observed. Furthermore, we observe that the total viscous dissipation rate peaks around the moment where the merged bubble assumes a `lemon-like shape’, and a slow down in the propagation of surface deformation after this moment. Spatially, the viscous dissipation rate is highly local and assumes its largest values around the apexes of the bubble interface at the moment of lemon-like shape. Different contributions to the total energy of the system until after the merged bubble jumps off the substrate are furthermore discussed in detail.