The role of surface tension in splashing

Letters to the Editors The Role of Surface Tension in Splashing The instability of a rapidly decelerating interface and the subsequent formation of surface tension dominated waves is proposed as the basic mechanism of splashing. Surface tension plays a vital role in splashing but the detailed mechanism involved does not appear to be documented. In the following, a drop of fluid is considered to fall onto a dry surface. A "blot" results, with characteristic "fingers" extending radially from the edge of the blot, Fig. 1. Each of these fingers is the progenitor of a splash droplet. Evidence of this has been presented by Edgerton (1) using high-speed photographic techniques. The fingers are in fact the wave crests of an extremely unstable interface. The fluid motion is nonlinear, but linear stability analysis of the interface does give the wave length of the fingers. The interface, between the fluid of the original drop and the environment, is unstable because of its enormous deceleration. A dense liquid underlying a lighter one presents a stable interface unless the interface is accelerating downwards. The latter is an example of Rayleigh-Taylor instability. In the present case, the part of the interface in question is represented by the strong line shown in Fig. 2. I t is moving radially outwards, but the motion is heavily damped by viscous forces and the interface is decelerating. At this decelerating interface RayleighTaylor instability is exhibited, and interracial waves (dotted) develop. Ignoring the viscosity of both fluid and environment, it can be shown, see Chandrasekhar (2), that the wave amplitude A is given by A = A0 exp(nt).

[i]

surface tension T, which also involves the following: m and p,., the fluid densities on either side of the interface; gt.., the acceleration of the interface towards fluid 2; and k, the wave number of the interracial waves where k = 2,r/X, X being the wavelength. This relationship is n~

=

P:

--

Tk 3

pl

p2 + ,ol

glzk

P2 + pl

.

1-23

E2] with respect to k and putting

Differentiating Eq.

On/Ok = O, the value of k which maximizes the growth

rate n can be determined. It is reasonable that the waves which grow most rapidly are the ones most likely to appear and so the wave number of the interfacial waves is given by k~

=

ga2(p,.

pa)/3T,

--

[33

which depends critically upon surface tension T. For a water (or ink) blot in an air environment this gives k~ ~ 9 X lO-3/gt~.. Clearly the greater the acceleration, the smaller will be the wavelength. Referring again to Fig. 2 it is evident, in the model presented, that gl.. is not constant. However, from Eq. 1-2] it is seen than n increases as g1~. increases and so the wavelength most likely to occur is given by k s ~-~ 9 X 10-3/g,,

[4]

where g, is the maximum value of gl-. achieved. Making no allowance for air resistance, a drop failing from rest at height h will have an impact velocity v = (2gh)~, where g is acceleration due to gravity. Assuming that this velocity is turned through 90 ° on impact, and reduced to rest in a distance r,

In this expression A0 is constant, n is the exponential rate of growth of the interfacial waves, and t is time. There is a relationship, between n and the interfacial

( ~-)incider,t ~\~ " drc~ t

!

.......

, •

i I

C

1:}ran/.

(iuid 1

Fie. 1. An ink blot with fingers, h = 800 mm, r = 10mm, N = 3 5 .

FIG. 2. Single arrows denote velocities. Double arrows denote accelerations. 350

Journal of Colloid and Interface Science, Vol. 51, No. 2, May 1975

Copyright ~) 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.

LETTERS TO THE EDITORS the radius of the blot, then the average retardation, gay of the interface is given by v = (2g~vr)½. Hence gav = gh/r. This retardation is essentially due to the transient growth of a liquid boundary layer at the liquid/solid interface. At the initial moment of contact, between the incident drop and the solid surface, this retardation would be zero. The simplest model would then give the maximum deceleration g,~ as twice the average deceleration gay. Thus g. = 2gh/r giving ~* ,~ 9 X lO-~r/2gh for water in air. For the example depicted in Fig. 1 this gives g. = 160g a surprisingly high retardation and ~ = 2.4 mm. From Fig. 1 we see there are approximately N = 35 fingers. Thus k = 2 z r / N = 1.8 ram. The two values for k are sufficiently close in this case~ and in many others not shown, to verify the above concept of splash generation, particularly in view of the many gross assumptions made.

351 REFERENCES

1. EDGERTON, H. E., AND KILLIAN, J. R. "flash[ Seeing the unseen by ultra high-speed photography," p. 111, Charles T. Branford Company, Boston, MA 1954. 2. CHANDRASEKIIAR, S., "Hydrodynamic and Hydromagnetic Stability," p. 435, Oxford University Press, London, 1961. ROBERT FRANCIS ALIEN Department of Civil Engineering University of Nairobi P.O. Boa: 30197 Nairobi, Kenya Received October 21, 1974; accepted December 30, 1974

Journal of Colloid and Interface Science. Vol. Sl. No. 2. May 1975