Rain Drops falling down in a uniform cloud

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SUMMARY

The discussion focuses on the dynamics of a rain drop falling through a uniform cloud of density ρ1, with the drop's density being ρ0. The key equations involve establishing a relationship between the drop's radius change rate (\dot{r}), its velocity, and the two densities. Additionally, the participants explore the rain drop's acceleration under the assumption of negligible air resistance, although they consider the potential impact of linear air resistance on the solution of the non-linear differential equation governing the drop's motion.

PREREQUISITES
  • Understanding of fluid dynamics principles
  • Familiarity with non-linear differential equations
  • Knowledge of spherical geometry and volume calculations
  • Basic concepts of density and its implications in fluid motion
NEXT STEPS
  • Study the derivation of non-linear differential equations in fluid dynamics
  • Research the effects of air resistance on falling objects
  • Learn about the Navier-Stokes equations and their applications
  • Explore the relationship between density and buoyancy in fluids
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Students and professionals in physics, particularly those focusing on fluid dynamics, as well as engineers and researchers interested in the behavior of particles in a fluid medium.

Erfan
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Homework Statement


Consider a uniforms cloud of density ρ1 which consists of many little drops that are approximately still.Consider a special rain drop that is a little more bigger than the others so that it starts to fall down.Water drops stays spherical all the way.But the radius gets higher as it falls down.Also water's density is ρ0.
(a) Find a relation between \dot{r}, velocity of the drop, and the two densities.
(b) Find rain drop's acceleration.

Homework Equations





The Attempt at a Solution


The differential equation is non-linear.
 
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Any thoughts on how to go about this? Any logical assumptions?
 
We have no air resistance force.But it might be possible to find a solution when the air exerts linear resistance force.
 

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