Instability development of a viscous liquid drop impacting a smooth substrate

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Instability development of a viscous liquid drop impacting a smooth substrate

Lei Xu (徐磊)

Phys. Rev. E 82, 025303(R) – Published 30 August 2010

Abstract

We study the instability development during a viscous liquid drop impacting a smooth substrate, using high-speed photography. The onset time of the instability highly depends on the surrounding air pressure and the liquid viscosity: it decreases with air pressure with the power of minus two, and increases linearly with the liquid viscosity. From the real-time dynamics measurements, we construct a model which compares the destabilizing stress from air with the stabilizing stress from liquid viscosity. Under this model, our experimental results indicate that at the instability onset time, the two stresses balance each other. This model also illustrates the different mechanisms for the inviscid and viscous regimes previously observed: the inviscid regime is stabilized by the surface tension and the viscous regime is stabilized by the liquid viscosity.

Received 13 July 2010

DOI:https://doi.org/10.1103/PhysRevE.82.025303

Authors & Affiliations

Lei Xu (徐磊)

Department of Physics, The Chinese University of Hong Kong, Hong Kong, People’s Republic of China

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Issue

Vol. 82, Iss. 2 — August 2010

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Images

Figure 1
Instability development under different pressures. The liquid drop has diameter $d = 3.1 \pm 0.1 mm$ , dynamic viscosity $μ = 6.71 \pm 0.02 mPa s$ , and impact velocity $V_{0} = 4.03 \pm 0.05 m s^{- 1}$ . The left column shows an impact under the air pressure $P = 40 kPa$ . The impact is initially stable, but instability shows up from the third image. The right column shows an identical impact under higher pressure, $P = 63 kPa$ . The instability appears at a much earlier time from the second image.Reuse & Permissions
Figure 2
(Color online) The instability onset time, $t_{o n}$ , vs $P$ for liquids of different viscosities. From bottom to top, the four curves correspond to increasing viscosities: $μ = 4.65$ (●), 6.7 (○), 9.3(▲), and 13.2 (×) $mPa s$ . All the curves can be fitted by a universal functional form: $t_{o n} = A / P^{2} + t_{0}$ . $t_{0}$ ranges from 0.03 to 0.09 ms, much smaller than most $t_{o n}$ values. The prefactor $A$ increases with $μ$ , as demonstrated by the higher locations of the liquids with larger $μ$ . Limited by the experimental condition, each data set has only about one decade in $x$ and $y$ directions. But it is nonetheless impressive that one simple functional form can fit all the data sets well.Reuse & Permissions
Figure 3
Direct measurement of the thickness $d$ vs time $t$ . The impact is by a liquid drop of $μ = 4.65 mPa s$ and $V_{0} = 4.03 \pm 0.01 m / s$ . The inset shows a typical snapshot from which $d$ is measured: $d$ is the liquid film thickness measured at the edge. Main panel shows the measured $d (t)$ . Because $d$ is quite small, the four discrete values correspond to one, two, three, and four pixels of our camera. The fit is: $d = 1.9 \sqrt{ν t}$ , indicating that $d$ is determined by the boundary layer thickness: $\sqrt{ν t}$ .Reuse & Permissions
Figure 4
The prefactor, $A$ , vs liquid viscosity, $μ$ , for the curves shown in Fig. 2. The prefactors are obtained from the best fits in Fig. 2. $A$ varies linearly with $μ$ and intercepts the $x$ axis at $μ_{0} = 3.4 mPa S$ . $μ_{0}$ agrees with the threshold viscosity separating the inviscid and viscous regimes observed in previous experiment [9].Reuse & Permissions
Figure 5
(Color online) The ratio between the destabilizing and the stabilizing stresses, $Σ_{G} / Σ_{μ}$ , measured at $t = t_{o n}$ , for various pressures and viscosities. All experiments are done under almost identical impact conditions, with only the pressure being varied. Different symbols represent liquids of different viscosities: $μ = 4.65$ (●), 6.7 (○), 9.3 (▲), and 13.2 (×) $mPa s$ . The ratio, $Σ_{G} / Σ_{μ} \sim ρ_{G} C_{G} d / μ$ , is computed from direct measurements: $ρ_{G}$ is calculated from $P$ , and $d$ is from the best fit to the high-speed images at the time $t_{o n}$ . Without any fitting parameter, all the ratios are within the narrow range between 3 and 4, confirming that $Σ_{G}$ and $Σ_{μ}$ are comparable at the time $t_{o n}$ .Reuse & Permissions

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