| Summary: | The dynamic of droplet spreading on a free-slip surface was studied experimentally and numerically, with particularly interest in the impacts under relatively small droplet inertias (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mi>e</mi><mo>≤</mo><mn>30</mn></mrow></semantics></math></inline-formula>). Our experimental results and numerical predictions of dimensionless droplet maximum spreading diameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>β</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> agree well with those of Wildeman et al.’s widely-used model at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mi>e</mi><mo>></mo><mn>30</mn></mrow></semantics></math></inline-formula>. The “1/2 rule” (i.e., approximately one half of the initial kinetic energy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi><mrow><mi>k</mi><mn>0</mn></mrow></msub></mrow></semantics></math></inline-formula> finally transferred into surface energy) was found to break down at small Weber numbers (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mi>e</mi><mo>≤</mo><mn>30</mn></mrow></semantics></math></inline-formula>) and droplet height is non-negligible when the energy conservation approach is employed to estimate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>β</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula>. As <i>We</i> increases, surface energy and kinetic energy alternately dominates the energy budget. When the initial kinetic energy is comparable to the initial surface energy, competition between surface energy and kinetic energy finally results in the non-monotonic energy budget. In this case, gas viscous dissipation contributes the majority of the dissipated energy under relatively large Reynolds numbers. A practical model for estimating <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>β</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow></semantics></math></inline-formula> under small Weber numbers (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mi>e</mi><mo>≤</mo><mn>30</mn></mrow></semantics></math></inline-formula>) was proposed by accounting for the influence of impact parameters on the energy budget and the droplet height. Good agreement was found between our model predictions and previous experiments.
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