Origin of the reduced lubrication-scaling prefactor

Determine the origin of the saturated numerical prefactor c approximately equal to 0.11 for water droplets sliding beneath an inclined surface immersed in viscous oil, which is substantially smaller than the values reported for bubble experiments and predicted theoretically for creeping droplets on gently inclined surfaces.

Background

At sufficiently large inclination angles, the experimental capillary-number data collapse onto the pancake scaling law, and the fitted numerical prefactor saturates at approximately c = 0.11. This value is about three times smaller than the coefficient c approximately equal to 0.34 inferred from prior bubble experiments and more than four times smaller than the theoretical prediction c approximately equal to 0.474 for creeping droplets on gently inclined surfaces.

The paper does not resolve why the prefactor is reduced. It suggests that differences in the detailed lubrication-film structure, interfacial mobility, finite deformation of the moving droplet, or additional viscous-dissipation mechanisms could account for the discrepancy. Direct measurements of the lubrication-film thickness and contact geometry are identified as a means of testing these explanations.

References

A further feature of the data is the saturation of the numerical prefactor $c$ at approximately 0.11 for sufficiently large inclination angles, as shown in Fig. 4(c). The emergence of a common prefactor accompanies the collapse of the high-inclination data onto the pancake scaling law, suggesting that the dominant dissipation mechanism becomes relatively insensitive to further increases in inclination once this regime is reached. However, the magnitude of the prefactor remains significantly smaller than previously reported values. Specifically, the saturated value $c\simeq0.11$ is approximately three times smaller than the coefficient $c\simeq0.34$ inferred from the bubble experiments and more than four times smaller than the theoretical prediction $c\simeq0.474$ derived for creeping droplets on gently inclined surfaces .

The origin of this discrepancy remains unclear.

— Inclination-Induced Crossover in the Velocity Scaling of Lubrication-Mediated Droplet Motion  (2608.28203 - Hitomi et al., 28 Aug 2026) in Discussion section, paragraph beginning “A further feature of the data”