Falling Raindrop - Evaporation and - Diff Eq project

[Mobilizer]

I'm working on a project where I'm trying to find the differential equation of a falling raindrop. The drop I'm considering will be under 10mm to eliminate the drops breaking apart and I've found that for that size, the drops will be spherical and traveling 6-8m/s. I've determined my air resistance to be F = =bv[tex]^{2}[/tex] because of the size of raindrops being molded.
So far my equation is simple with ma + bv[tex]^{2}[/tex] = mg

Next I was trying to figure out the part for evaporation. The raindrop will be decreasing in size due to evaporation but and also it's velocity will be changing too due to a smaller size; and here's where I got stuck.

What would be the best way to go about this?

 

[gato_]

the problem will be harder the more detail you get into, but roughly, both mass and friction coefficients will vary in time because of the change in volume/geometry of the drop. You need to model too the process of evaporation. The rate is constant in time? does it depend on velocity? (on account of the drop heating), and add a new equation for the rate of volume/friction area loss rate.
The simplest approximation I can think about (may be pretty unaccurate) is: consider the friction to be given, over the rate of velocities considered, by
[tex]F_{D}=1/2\rho A C_{D}v^{2}[/tex]
where A is the cross area of the drop, and Cd a drag coefficient. Let us consider it constant, and the drop, to keep a spherical shape (again, unaccurate) of volume [tex]4/3\pi r(t)^{3}[/tex] and cross section [tex]A=4\pi r(t)^{2}[/tex]. You still need to specify the rate at which r changes. Let's imagine it loses mass at a constant rate q:
[tex]\dot{m}=\rho 4/3\pi \dot{r^3}=q [/tex]
you would now have two equations for variables (r(t),v(t)).

 

[Mute]

Also note that the net force starting point should be

[tex]\frac{dp}{dt} = \frac{d(m(t)v(t))}{dt} = \sum F[/tex]

as the mass is changing. (Or alternatively you could still say the net force is ma, but there is now an additional force on the drop due to the changing mass).

 

[gato_]

True. Also, I think it is better to consider evaporation rate to be proportional to the surface area rather than constant, the reason being only molecules in the interface can evaporate. That results in a linearly varying radius with time.

 

[blue_raver22]

I find your project very interesting and relevant to understanding the dynamics of raindrops. The differential equation approach is a great way to understand the behavior of falling raindrops.

Regarding the evaporation aspect, you are correct in considering the decrease in size of the raindrop due to evaporation. This will affect both the mass and velocity of the raindrop. One approach could be to incorporate the rate of evaporation into the equation, considering it as a negative acceleration term. This would result in a more complex differential equation, but it would better reflect the real-world behavior of a falling raindrop.

Another approach could be to consider the rate of evaporation as a function of the raindrop's surface area, which decreases as the raindrop shrinks. This could be incorporated into the air resistance term, as a decrease in the drag force on the raindrop as it decreases in size.

It is also important to consider the environmental factors that could affect evaporation, such as humidity and temperature. Including these variables in your equation could improve its accuracy.

Overall, I would suggest experimenting with different approaches and considering all the relevant factors to create a comprehensive and accurate differential equation for a falling raindrop. Good luck with your project!