# Splash Dynamics of Paint on Dry, Wet, and Cooled Surfaces

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Section

#### Materials and Methods

## 3. Results

## 4. Fractal Dimension

## 5. Statistical Analysis

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Tables of Statistical Data

Estimate | Std. Error | t Value | Pr(>|t|) | |
---|---|---|---|---|

(Intercept) | 5.5634 | 0.2156 | 25.81 | 0.0000 |

factor(Temp)2 | 0.1926 | 0.1931 | 1.00 | 0.3188 |

factor(Temp)3 | –1.8528 | 0.1841 | –10.06 | 0.0000 |

factor(Height)2 | 0.2908 | 0.1398 | 2.08 | 0.0378 |

Time | 0.1579 | 0.0088 | 17.89 | 0.0000 |

factor(Viscosity)2 | –2.3478 | 0.1580 | –14.86 | 0.0000 |

factor(Viscosity)3 | 0.7302 | 0.2721 | 2.68 | 0.0074 |

factor(Paper)2 | 0.8153 | 0.1941 | 4.20 | 0.0000 |

factor(Paper)3 | –0.4384 | 0.2769 | –1.58 | 0.1138 |

factor(Volume)2 | 0.5852 | 0.1895 | 3.09 | 0.0021 |

Estimate | Std. Error | z Value | Pr(>|z|) | |
---|---|---|---|---|

(Intercept) | –26.6447 | 1457.5012 | –0.02 | 0.9854 |

factor(Temp)2 | 20.1488 | 1457.5009 | 0.01 | 0.9890 |

factor(Temp)3 | 24.0557 | 1457.5010 | 0.02 | 0.9868 |

factor(Height)2 | 1.0877 | 0.3406 | 3.19 | 0.0014 |

Time | 0.3074 | 0.0365 | 8.42 | 0.0000 |

factor(Viscosity)2 | –28.0664 | 1122.4010 | –0.03 | 0.9801 |

factor(Viscosity)3 | –20.9046 | 2602.5289 | –0.01 | 0.9936 |

factor(Paper)2 | 2.2046 | 0.4948 | 4.46 | 0.0000 |

factor(Paper)3 | 0.5124 | 0.4571 | 1.12 | 0.2623 |

factor(Volume)2 | –2.8894 | 0.5487 | –5.27 | 0.0000 |

Estimate | Std. Error | z Value | Pr(>|z|) | |
---|---|---|---|---|

(Intercept) | –0.8532 | 0.2482 | –3.44 | 0.0006 |

factor(Temp)2 | 0.0944 | 0.2309 | 0.41 | 0.6827 |

factor(Temp)3 | 1.3492 | 0.2220 | 6.08 | 0.0000 |

factor(Height)2 | –0.2046 | 0.1654 | –1.24 | 0.2161 |

Time | 0.0035 | 0.0101 | 0.35 | 0.7289 |

factor(Viscosity)2 | 1.3398 | 0.1847 | 7.25 | 0.0000 |

factor(Viscosity)3 | 3.6557 | 0.5041 | 7.25 | 0.0000 |

factor(Paper)2 | –1.2459 | 0.2277 | –5.47 | 0.0000 |

factor(Paper)3 | –1.2640 | 0.3389 | –3.73 | 0.0002 |

factor(Volume)2 | 0.8644 | 0.2167 | 3.99 | 0.0001 |

Estimate | Std. Error | t Value | Pr(>|t|) | |
---|---|---|---|---|

(Intercept) | 1.8432 | 0.0537 | 34.35 | 0.0000 |

factor(Temp)2 | –0.0329 | 0.0297 | –1.11 | 0.2762 |

factor(Temp)3 | –0.0720 | 0.0303 | –2.38 | 0.0232 |

factor(Height)2 | –0.0126 | 0.0240 | –0.53 | 0.6027 |

Time | 0.0004 | 0.0021 | 0.18 | 0.8553 |

factor(Viscosity)2 | –0.0582 | 0.0269 | –2.16 | 0.0377 |

factor(Viscosity)3 | –0.1247 | 0.0483 | –2.58 | 0.0144 |

factor(Paper)2 | 0.0035 | 0.0414 | 0.08 | 0.9338 |

factor(Paper)3 | 0.0774 | 0.0476 | 1.63 | 0.1134 |

factor(Volume)2 | 0.0359 | 0.0337 | 1.06 | 0.2947 |

## References

- Lane, R. Images from the Floating World, The Japanese Print; Oxford University Press: Oxford, UK, 1978. [Google Scholar]
- Taylor, R.; Micolich, A.P.; Jonas, D. The construction of Jackson Pollock’s fractal drip paintings. Leonardo
**2002**, 203–207. [Google Scholar] [CrossRef] - Taylor, R.P.; Micolich, A.P.; Jonas, D. Fractal Analysis of Pollock Drip Paintings. Nature
**1999**, 399, 422. [Google Scholar] [CrossRef] - Hercynzki, A.; Cernuschi, C.; Mahadevan, L. Painting with drops, jets and sheets. Phys. Today
**2011**, 2011, 31–36. [Google Scholar] - Worthington, A.M. A Study of Splashes; Longmans: London, UK, 1908; p. 129. [Google Scholar]
- Josserand, C.; Thoroddsen, S.T. Drop Impact on a Solid Surface. Annu. Rev. Fluid Mech.
**2016**, 48, 365–391. [Google Scholar] [CrossRef] - Sefiane, K. Patterns from drying drops. Adv. Colloid Interface Sci.
**2014**, 206, 372–381. [Google Scholar] [CrossRef] [PubMed] - Yarin, A.L. Drop Impact Dynamics: Splashing, Spreading, Receding, Bouncing. Ann. Rev. Fluid Mech.
**2006**, 38, 159–192. [Google Scholar] [CrossRef] - Rioboo, R.; Tropea, C.; Marengo, M. Outcomes from a drop impact on solid surfaces. At. Sprays
**2001**, 11, 155–165. [Google Scholar] - Bartolo, D.; Narcy, G.; Boudaoud, A.; Bonn, D. Dynamics of Non-Newtonian Droplets. Phys. Rev. Lett.
**2007**, 99, 174502. [Google Scholar] [CrossRef] [PubMed] - Li-Hua, L.; Forterre, Y. Drop impact of yield-stress fluids. J. Fluid Mech.
**2009**, 632, 301–327. [Google Scholar] - Marston, J.O.; Mansoor, M.M.; Thoroddsen, S.T. Impact of granular drops. Phys. Rev. E
**2013**, 88, 010201. [Google Scholar] [CrossRef] [PubMed] - Nicolas, M. Spreading of a drop of neutrally buoyant suspension. J. Fluid Mech.
**2005**, 545, 271–280. [Google Scholar] [CrossRef] - Peters, I.R.; Xu, Q.; Jaeger, H.M. Splashing onset in dense suspensions. Phys. Rev. Lett.
**2013**, 111, 028301. [Google Scholar] [CrossRef] [PubMed] - Guemas, M.; Marin, A.G.; Lohse, D. Drop impact experiments of non-Newtonian liquids on micro-structured surfaces. Soft Matter
**2012**, 8, 10725–10731. [Google Scholar] [CrossRef] - Mysels, K.J. Visual Art: The role of capillarity and rheological properties in painting. Leonardo
**1981**, 13, 22–27. [Google Scholar] [CrossRef] - Lagubeau, G.; Fontelos, M.A.; Josserand, C.; Maurel, A.; Pagneux, V.; Petitjeans, P. Spreading dynamics of drop impacts. J. Fluid Mech.
**2012**, 713, 50–60. [Google Scholar] [CrossRef] - How Watercolor Paints Are Made. Available online: http://www.handprint.com/HP/WCL/pigmt1.html (accessed on 15 December 2015).
- Blair, G.W.S. Rheology and Painting. Leonardo
**1969**, 2, 51–53. [Google Scholar] [CrossRef] - Li, X.; Zhang, H.; Fang, Y.; Al-Assaf, S.; Phillips, G.O.; Nishinari, K. Rheological Properties of Gum Arabic Solution: The Effect of Arabinogalactan Protein Complex (AGP). In Gum Arabic; Kennedy, J.F., Phillips, G.O., Williams, P.A., Eds.; Royal Society of Chemistry: London, UK, 2011. [Google Scholar]
- Vernon-Carter, E.J.; Sherman, P. Rheological properties and applications of mesquite tree (Prosopis juliflora) gum. 1. Rheological properties of aqueous mesquite gum solutions. J. Text. Stud.
**1980**, 11, 339–349. [Google Scholar] [CrossRef] - Lopez-Franco, Y.L.; Gooycolea, F.M.; Lizardi-Mendoza, J. Gum of Prosopis/Acacia Species; Ramawat, K.G., Merillon, J.-M., Eds.; Springer International Publishing: Cham, Switzerland, 2015. [Google Scholar]
- De Marsily, G. Quantitative Hydrogeology; Academic Press Inc.: Orlando, FL, USA, 1986. [Google Scholar]
- Marmanis, H.; Thoroddsen, S.T. Scaling of the fingering pattern of an impacting drop. Phys. Fluids
**1996**, 8, 1344–1346. [Google Scholar] [CrossRef] - Rioboo, R.; Marengo, M.; Tropea, C. Time evolution of a liquid drop impact onto solid, dry surfaces. Expts Fluids
**2002**, 33, 112–124. [Google Scholar] - Yarin, A.L.; Weiss, D.A. Impact of drops on solid surfaces: self-similar capillary waves, and splashing as a new type of kinematic discontinuity. J. Fluid Mech.
**1995**, 283, 141–173. [Google Scholar] [CrossRef] - Cossali, G.E.; Coghe, A.; Marengo, M. The impact of a single drop on a wetted solid surface. Exp. Fluids
**1997**, 22, 463–472. [Google Scholar] [CrossRef] - Sivakumar, D.; Tropea, C. Splashing impact of a spray onto a liquid film. Phys. Fluids
**2002**, 14, L85–L88. [Google Scholar] [CrossRef] - Taylor, R.P.; Guzmana, R.; Martina, T.P.; Halla, G.D.R.; Micolichb, A.P.; Jonasb, D.; Scannella, B.C.; Fairbanksa, M.S.; Marlowa, C.A. Authenticating Pollock Paintings Using Fractal Geometry. Pattern Recognit. Lett.
**2007**, 28, 695. [Google Scholar] [CrossRef] - Jones-Smith, K.; Mathur, H.; Krauss, L.M. Drip paintings and fractal analysis. Phys. Rev. E
**2009**, 79. [Google Scholar] [CrossRef] [PubMed] - Joye, Y. Some reflections on the relevance of fractals for art therapy. Arts Psychother.
**2006**, 33, 143–147. [Google Scholar] [CrossRef] - Lesmior-Gordon, N. Introducing Fractal Geometry; Icon Books Ltd.: Duxford, UK, 2006; p. 176. [Google Scholar]
- Howison, S.D.; Moriarty, J.A.; Terrill, E.L.; Wilson, S.K. A mathematical model for drying paint layers. J. Eng. Math.
**1997**, 32, 377–394. [Google Scholar] [CrossRef] - Kim, H.Y.; Chun, J.H. The recoiling of liquid droplets upon collision with solid surfaces. Phys. Fluids
**2001**, 13, 643–659. [Google Scholar] [CrossRef]

**Figure 2.**Temperature of the canvas as a function of time for all three cases. The dashed lines point to the times when the canvas surface appears to be in molten liquid state.

**Figure 3.**Closeup view of the three canvases used in this study showing their surface features and porous nature. The pictures were taken with a Leica CME microscope (Meyer Instruments, Houston, TX, USA) at magnification of 100×. Images (

**a**)–(

**c**) correspond to 90 lb, 140 lb and 140 lb cold pressed canvas, respectively. The image (

**d**) is of stained, uncoated printing paper at the same magnification.

**Figure 5.**The effective radius of the splash as a function of time showing the effect of (

**a**) height of droplet; (

**b**) the temperature of the canvas; (

**c**) the viscosity of paint; and (

**d**) the type of canvas used.

**Figure 6.**Examples of the four distinct splash patterns seen in our experiments. A single splash can contain one or more of these patterns. The images are organized in pairs, with the first image (X1, where $X=A,B,C,D$), immediately upon impact and the final image (X2) are the images at the final times, at equilibrium.

**Figure 7.**The graph shows the evolution of the effective radius of the splash with time. A power law fit to the data is also shown, with an exponent value of 0.13, which is much smaller than those seen in other kinds of splashes.

**Figure 8.**Number of fingers as a function of height of release or impact velocity. Figure (

**a**) shows the results for Permanent Rose and figure (

**b**) shows the count for Prussian Blue. The x-axis represents different experimental cases where fingering was observed.

**Figure 11.**Residual against fitted value plot for multiple regression model of fractal dimension. Normal Q-Q plot for multiple regression model of fractal dimension.

**Figure 12.**Cross sectional image of canvas showing absorption and penetration of paint through canvas around the droplet impact point.

Viscosity | Permanent Rose | Prussian Blue | Sepia |
---|---|---|---|

Average Viscosity (dPa-s) | 0.0257 | 0.0132 | 0.0098 |

Std. Dev. | 0.0085 | 0.008 | 0.0071 |

**Table 2.**Repeated patterns observed in our experiments are analyzed for their qualitative dependence upon canvas roughness, paint viscosity, height(or impact speed) and temperature of canvas.

Property | Symmetric/Circular | Visible Inner Stamp | Radial Fingering | Satellites |
---|---|---|---|---|

Increase in... | ||||

Height | ↓ | - | ↑ | ↑ |

Paper Roughness | ↓ | ↓ | ↓ | |

Paint Viscosity | - | ↑ | ↓ | - |

Freezing | ↑ | ↑ | - | - |

Temperature | Permanent Rose | Prussian Blue | Sepia |
---|---|---|---|

Unfrozen (6in) | 1.637 | 1.832 | 1.826 |

Frozen 30 min (6in) | 1.975 | 1.570 | 1.707 |

Unfrozen (12in) | 1.822 | 1.820 | 1.738 |

Frozen 30 min (12in) | 1.931 | 1.722 | 1.650 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Baron, D.; Su, H.; Vaidya, A.
Splash Dynamics of Paint on Dry, Wet, and Cooled Surfaces. *Fluids* **2016**, *1*, 12.
https://doi.org/10.3390/fluids1020012

**AMA Style**

Baron D, Su H, Vaidya A.
Splash Dynamics of Paint on Dry, Wet, and Cooled Surfaces. *Fluids*. 2016; 1(2):12.
https://doi.org/10.3390/fluids1020012

**Chicago/Turabian Style**

Baron, David, Haiyan Su, and Ashwin Vaidya.
2016. "Splash Dynamics of Paint on Dry, Wet, and Cooled Surfaces" *Fluids* 1, no. 2: 12.
https://doi.org/10.3390/fluids1020012