Preprint Article Version 2 This version is not peer-reviewed

# Scaling Law for Liquid Splashing inside a Container Drop Impact on a Solid Surface

Version 1 : Received: 22 October 2018 / Approved: 23 October 2018 / Online: 23 October 2018 (07:58:07 CEST)
Version 2 : Received: 14 November 2018 / Approved: 15 November 2018 / Online: 15 November 2018 (05:06:10 CET)

How to cite: Sun, B. Scaling Law for Liquid Splashing inside a Container Drop Impact on a Solid Surface. Preprints 2018, 2018100525 (doi: 10.20944/preprints201810.0525.v2). Sun, B. Scaling Law for Liquid Splashing inside a Container Drop Impact on a Solid Surface. Preprints 2018, 2018100525 (doi: 10.20944/preprints201810.0525.v2).

## Abstract

This letter attempts to find splashing height of liquid-filled container drop impact to a solid surface by dimensional analysis (DA). Two solutions were obtained by both traditional DA and directed DA without solving any governing equations. It is found that the directed DA can provide much more useful information than the traditional one. This study shows that the central controlling parameter is called splash number $\mathrm{Sp}=\mathrm{Ga} \mathrm{La}^\beta=(\frac{gR^3}{\nu^2})(\frac{\sigma R}{\rho \nu^2})^\beta$, which is the collective performance of each quantity. The splash height is given by $\frac{h}{H}=(\frac{\rho\nu^2}{\sigma R})^\alpha f[\frac{gR^3}{\nu^2}(\frac{R\sigma}{\rho\nu^2})^\beta]=\frac{1}{\mathrm{La}^\alpha}f(\mathrm{Ga}\cdot \mathrm{La}^\beta)$. From the physics of the splashing number, we can have a fair good picture on the physics of the liquid splashing as follows: the jets propagation will generate vortex streets from the container bottom due to sudden pressure increasing from drop impact (water-hammer effect), which will travel along the container sidewall to the centre of the container and subsequently excite a gravity wave on the liquid surface. The interaction between the gravitational force, surface force and viscous force is responsible for creating droplet splash at the liquid surface.

## Subject Areas

liquid splashing; dimensional analysis; directed dimensional analysis