Raindrops-Model

This project aims to model rain and the dynamics of the raindrops once they land on a pond. I thought it was an interesting physical phenomena to model, where I could combine ideas from Stochastic Processes and differential equations. I made a model which simulates the random location and landing frequency of water drops as well as their effect on the creation of ripples on the surface.

How is the rain modeled?

I let $N(t)$ be the number of droplets that have fallen after $t$ units of time and $\{N(t),t>=0\}$ be a Poisson process with rate $\lambda$ droplets per unit of time. From this, it is known that $N(T) \sim POISSON(\lambda T)$. Another important result I used is that if $N(T) = n$ ,then the conditional distribution of the waiting times for each of the drops $S_1,...,S_n$ is like the distribution of the $n$-order statistics of $n$ iid $U(0,T)$ rvs. More precisely : $(S_1,...,S_n)|(N(T) = n)$ has the same distribution as $(Y^1,...,Y^n)$ where $Y_i \sim U(0,T)$ In addition, the location of an arbitrary drop is assumed to be a random vector: $(X,Y) \sim U(-l,l)*U(-l,l)$ where $2l$ is the length of the square grid where the raining simulation takes place.

Deterministic properties of the model:

Once a drop hits the water, how do the nearby water particles move?

The idea is that once a droplet hits the pond, the water particles that are at a distance $\epsilon$ away from the hitting position will be accelerated from rest with the maximum force right at that instant. From then on, these guys will all move in different directions away at a common velocity,whose rate of change (acceleration) will stay positive on the beginning but decreasing due to the short resistance offered by the time exponential contribution of the initial impact and counter force exerted in the opposite direction due to the water resistance.For this reason, at some later instant there will be no curvature and the particle will be travelling at its maximum velocity. From then, the acceleration will be negative until the velocity goes to 0 from above, and the particle will be barely moving , approaching a limit displacement. To be more exact, the distance from the hitting point to a nearby particle at a given time obeys the equationp: $R''(t) = Aexp(-t) -\mu R'(t)$ with $R(0) = \epsilon$ and $R'(0) = 0$. Here $A > 0$ is the maximum acceleration which happens at $t = 0$ and $\mu >0$ is the water drag coefficient. This is a first order linear drag model with a time exponential forcing, for the radial velocity $R'$. I also explored the dynamics of the model using other equations (same initial conditions) with very similar qualitative and quantitative results such as the nonlinear equation $R''(t) = 1/R(t) -\mu R'(t)$. However, they gave no more significant differences in the simulations and they suffer from the flaw of having no closed form solution, which meant more time consuming for the computer to previously approximate their solutions nummerically. The advantage of the linear drag model is that I could solve it analytically. The solution of the linear equation problem is then: $R(t) = \epsilon + \dfrac{A}{\mu}+\dfrac{A}{1-\mu}*(exp(-t)-\dfrac{exp(-\mu t)}{\mu})$ An interactive plot of the position, velocity and acceleration based on different parameters can be found here : https://www.desmos.com/calculator/shnfoqc6hr

Example: INTENSE RAIN

The following results uses $\lambda = 15$ drops per unit of time for the stochastic raining process , $A =1, \mu = 1.8$ It could serve as a decent way to simulate an intense rain during a storm.

Intense_video.1.mp4

Example: STARTING RAIN

The following case uses $\lambda = 3$ droplets per unit of time for the stochastic raining process, $A = 1, \mu = 1.8$ It could immitate the dynamics of a just started rain, where some few drops start hitting the ground with some longer pauses in between different to the above case, which could be the later behaviour

StartingRain.mp4.mp4