252 The UMAP Journal 23.3 (2002) Model of the water jet Ne model the spray from the fountain as a particle system. As water droplets spew forth from the nozzle, they are subjected to forces(gravity, air drag, turbulence, etc. ) We formulate a simplified differential equation governing the motion and then numerically integrate to find the trajectory for each droplet This equation is based on a physically realistic model of small droplets(around 1 mm radius)and we scale it up to an effective model for larger clumps of water(up to 10 cm across) because the physics of turbulence and viscosity at he larger scale cannot be computed accurately We need the following assumptions The drag force is proportional to the square of the speed and to the square f the radius [ Nasa 2002 Droplets break into smaller droplets when subjected to wind. Breakup rate is proportional to relative wind speed and surface area [INobauer 1999 When a droplet breaks turbulence causes the new droplet fragments to move slightly away from their initial trajectory. Modeling a Single Droplet We formulate the motion of a water droplet a m dt=-mgi+mlwul-ir where vis the velocity w is the wind velocity relative to the motion of the droplet (wind vector minus velocity vector), m and r are the droplet's mass and radius, and n is a constant of proportionality. According to the Virtual Science Center Project Team[2002 a raindrop with radius 1 mm falls at a terminal velocity of 7 m/s; so we determine that n=0.855 kg/m. Large drops fall quickly; very tiny drops fall very slowly, mimicking a fine mist that hangs in the air for a long time We assume droplet breakup is a modified Poisson process, with rate λoulr If the breakup rate did not depend on variable parameters w and r2, the process would be a standard Poisson process. We l determine Ao by fitting the water ream of our fountain to the streams of two real fountains: the jet D'Eau of Geneva, Switzerland, and the Five Rivers Fountain of Lights in Miami, Florida When a breakup occurs, we split the droplet into two new droplets and divide the mass randomly, using a uniform distribution. Air turbulence tends to impart to the two new droplets a small velocity component perpendicular to the relative wind direction w. This effect causes a tight stream of water to spread252 The UMAP Journal 23.3 (2002) Model of the Water Jet We model the spray from the fountain as a particle system. As water droplets spew forth from the nozzle, they are subjected to forces (gravity, air drag, turbulence, etc.). We formulate a simplified differential equation governing the motion and then numerically integrate to find the trajectory for each droplet. This equation is based on a physically realistic model of small droplets (around 1 mm radius) and we scale it up to an effective model for larger clumps of water (up to 10 cm across) because the physics of turbulence and viscosity at the larger scale cannot be computed accurately. We need the following assumptions: • The drag force is proportional to the square of the speed and to the square of the radius [Nasa 2002]. • Droplets break into smaller droplets when subjected to wind. Breakup rate is proportional to relative wind speed and surface area [Nobauer 1999]. • When a droplet breaks, turbulence causes the new droplet fragments to move slightly away from their initial trajectory. Modeling a Single Droplet We formulate the motion of a water droplet as m dv dt = −mgzˆ + η|w| 2wrˆ 2, wherev is the velocity, w is the wind velocity relative to the motion of the droplet (wind vector minus velocity vector), m and r are the droplet’s mass and radius, and η is a constant of proportionality. According to the Virtual Science Center Project Team [2002], a raindrop with radius 1 mm falls at a terminal velocity of 7 m/s; so we determine that η = 0.855 kg/m3. Large drops fall quickly; very tiny drops fall very slowly, mimicking a fine mist that hangs in the air for a long time. We assume droplet breakup is a modified Poisson process, with rate λbreakup = λ0|w|r2. If the breakup rate did not depend on variable parameters |w| andr2, the process would be a standard Poisson process. We l determine λ0 by fitting the water stream of our fountain to the streams of two real fountains: the Jet D’Eau of Geneva, Switzerland, and the Five Rivers Fountain of Lights in Miami, Florida. When a breakup occurs, we split the droplet into two new droplets and divide the mass randomly, using a uniform distribution. Air turbulence tends to impart to the two new droplets a small velocity component perpendicular to the relative wind direction w. This effect causes a tight stream of water to spread