A fluid mechanics problem -- Shape of a falling water drop

[LordGfcd]

Homework Statement

A drop of water fall towards the ground with initial mass [m][/0] and radius [r][/0] (assume the initial shape of that water drop is sphere). the air resistance is F=½.ρ.A.[v][/2].C (C is the drag coefficient, A is the area that the air contact with the water drop and ρ is the specific weigh of the air). The bulk modulus of water is K, the atmospheric pressure is [p][/0], the gravitational acceleration is g. Assume the height was enough so the speed of the water drop can be constant at some point, find the formula that describe the shape of the water drop.

Homework Equations

I think this is the most general problem.

The Attempt at a Solution

I tried to consider a function in the Cartesian coordinate which the (0,0) point is the center of mass of that water drop. Apply the air resistant force on some dS area to the bulk modulus definition K=ρ.dP/dV but I can't somehow translate it to a differential equation form or even a dy/dx form to integrate. I don't think I did the math wrong. Is my way of approaching wrong ? My teacher said I can use another method using Larangian mechanic, the field I haven't studied throughout yet.
I will very appreciate if someone can tell me if my method is wrong there is a better approaching or if you attempted this problem before can you share with me your opinion. Thank you very much.

 

[LordGfcd]

Hi, I was wrong about the bulk modulus definition :v It was K=-v.dP/dV. My bad :(

 

[haruspex]

Hi, I was wrong about the bulk modulus definition :v It was K=-v.dP/dV. My bad :(

So is this solved now? If so, please click the MARK SOLVED button.

 

[LordGfcd]

Not yet, I'm still working on it but haven't got much progress.

 

[Chestermiller]

What is your assessment of the physical mechanisms involved in determining the shape of the water drop? Can you articulate these?

 

[LordGfcd]

Well, there is the atmospheric pressure and the bulk modulus that determine the size of the water drop. If the velocity is constant then the dragging force should be constant too but different with each part of the water drop. The force on the x-axis will be p0.dS and the force on the y-axis is ½ρ.dS.v^2 (v^2 could be found easily since we known the mass of of the water drop). The problem is if I only consider the shape of the water drop only in 2 dimensional (Cartesian coordinate) I can't consider the bulk modulus but if I do it in 3 axis the analysis of the forces acting on the water drop will be much more complicate.

 

[Chestermiller]

It seems to me that there is much more going on here. An important factor is going to be surface tension which surrounds the drop and helps maintain its shape. The compressibility effect is definitely going to be negligible. There is also the shear stress and normal stress distributions at the surface of the drop that are key, determined by the drag. These are balanced by the surface tension effect, via variations in surface curvature.

 

[haruspex]

It seems to me that there is much more going on here. An important factor is going to be surface tension which surrounds the drop and helps maintain its shape. The compressibility effect is definitely going to be negligible. There is also the shear stress and normal stress distributions at the surface of the drop that are key, determined by the drag. These are balanced by the surface tension effect, via variations in surface curvature.

I agree. In the given circumstances the water will be essentially incompressible. Surface tension is the significant player.
But since the bulk modulus is given and the surface tension is not, it seems that the author has chosen an inappropriate physical context for the intended exercise. The puzzle is, what model is intended?

 

[LordGfcd]

Oh, it's must be my fault, I totally forgot the surface tension. The original problem was just "find the mathematical description of a shape of a falling drop of water in the atmosphere" so I added some variables and data that I thought would be involved. I will take another attempt and see what it brings.