## Theses, Dissertations and Culminating Projects

8-2017

Thesis

#### Degree Name

Master of Science (MS)

#### College/School

College of Science and Mathematics

#### Department/Program

Mathematical Sciences

Ashwin Vaidya

Haiyan Su

Eric Forgoston

#### Abstract

In his classic study in 1908, A.M. Worthington gave a thorough account of splashes and their formation through visualization experiments. In more recent times, there has been renewed interest in this subject, and much of the underlying physics behind Worthington's experiments has now been clarified. One specific set of such recent studies, which motivates this thesis, concerns the fluid dynamics behind Jackson Pollock's drip paintings. The physical processes and the mathematical structures hidden in his works have received serious attention and have made the scientific pursuit of art a compelling area of exploration. Our current work explores the interaction of watercolors with watercolor paper. Specifically, we conduct experiments to analyze the settling patterns of droplets of watercolor paint on wet and frozen paper. Variations in paint viscosity, paper roughness, paper temperature, and the height of a released droplet are examined from time of impact, through its transient stages, until its final, dry state. Observable phenomena such as paint splashing, spreading, fingering, branching, rheological deposition, and fractal patterns are studied in detail and classified in terms of the control parameters. Using the one-dimensional (1-D) Saint-Venant differential equations, which are a simplification of the three-dimensional (3-D) Navier-Stokes equations from fluid dynamics, we created a computer-simulated, mathematical model of a droplet splash of watercolor paint onto a flat surface. The mathematical model is analyzed using a MATLAB code which considered changes in droplet height, radius, and velocity of dispersal over time. We also implemented a stochastic version of the Saint-Venant equations which captured the random fingering patterns of a droplet splash. Initial conditions for height, radius, and velocity of a radially spreading droplet were given at the onset of the simulation. Dynamic viscosity and fluid density were parameters incorporated into this system of differential equations, which could be easily adjusted in the MATLAB code for the paint type to be simulated. The stochastic nature of our model was designed to recreate the complex behavior of water splashes, the non-homogeneity of the watercolor paper, and the resulting patterns. We then computed the fractal dimension of each computer-generated droplet image to compare theoretical and experimental values. Analysis of the set of data consisting of over 10,000 trials was conducted to determine any significant statistical correlations among the spreading pattern, the number of fingers, viscosity, density and fractal dimension. Finally, we extended the system of differential equations based on the Saint-Venant equations to include the effects of temperature upon the paint-spreading pattern. In a similar manner, we compared the theoretical values of fractal dimensions generated by our MATLAB model to the experimental results for paint droplets on a frozen substrate.

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