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Xu et al. proposed the following scaling parameter by comparing the destabilizing
stresses from the gas and the stabilizing stresses from the surface tension [168],
∑ ∑G / =
d 0 v 0
γM
L G p
4k B T
νL
σ , (2.9)
where γ is the adiabatic constant (1.4 for air), M is the relative molecular mass
(29 for air), p is the pressure, d 0 and v 0 are drop diameter and impact velocity
respectively, k B is the Boltzmann constant, T is temperature in Kelvin, ν is kinematic
viscosity, σ is surface tension. G denotes gas and the L denotes liquid. In
this equation, the air compressibility is considered based on the assumption that a
weak shock in the air presents at the early stages of drop spreading. However, the
importance of the shock wave is questionable at a low impact speed of 3.74m/s applied
in their experiment. This scaling method failed to provide a universal splash
threshold for either their own data or the data from Mishra [94, 166, 168].
Eq. 2.9 predicts that the liquid viscosity encourages splash. This prediction was
examined by the author with various liquids covering a wide range of kinematic
viscosity from 0.68 cSt to 18 cSt. It was found out that at lower viscosities from
0.68 cSt to 2.60 cSt, higher viscosity promoted the splash, while the effect was
reversed at higher viscosities (see Fig. 5. in [166]). This nonmonotonic influence
of the viscosity explains the disparities in other experiments. Cossali et al. [34]
and Rioboo et al. [114] found that the viscosity inhibited splashing, because they
had only one value in the low viscosity range, while all the other test points were
in the high viscosity range. Vander Wal et al. [156] found out that the viscosity
promoted the splash because their range of viscosity lay exclusively in the low
viscosity range. Range and Feuillebois had the same findings as Xu with a large
variety of liquids (see Fig. 7. in [115]), but the nonmonotonic effect was much
less prominent in the expression of We crit as a function of the Oh, which led to
their counterintuitive conclusion that the viscosity played a negligible role in the
splash threshold. Nonetheless the encouraging effect of viscosity on the splash
is surprising. Xu hypothesized that [166] for low viscosities, larger ν L causes a
thicker film which was easier to destabilize, while viscous drag became important
and helped to stabilized the spreading drop at higher viscosities.
All the experiments above were conducted on smooth surfaces with negligible
roughnesses. Decreasing the surrounding gas pressure to a level, where no corona
splash occurred, rough surface was applied for drop impact by Xu et al. [167], in
order to observe the prompt splash exclusively. It was found out that the roughness
20 2. Background