Water flowing from one vertical cylinder to another

[Tellerath]

1. The problem statement, all variables, and given/known data
Two vertical cylinders rest on a flat surface, next to each other. Both have a hole on the side at the bottom so that the water can flow from one cylinder to the other without leakage. The diameter of the first cylinder is D₁=3 m, the second D₂ = 4 m, and the hole diameter is d=10 cm.
If the first cylinder is filled with water to the height of H=5m, and the other one is empty, find the time τ it takes for water levels to equalize.

Homework Equations

Continuity equation: Sv = const
Bernoulli equation:

The Attempt at a Solution

First, I determined the "equalized" height as follows:
(total volume) = (volume in the first cylinder after equalization) x (volume in the second cylinder after equalization)
Which gives:
D₁²H = D₁²h_e + D₂²h_e
So from here we get
h_e = 9H/25
This seemed like a simple problem but ended up being quite challenging (at least to me). First, let v₁, v₃ and v₂ be speeds of water (at any moment) in the first cylinder, through the opening, and in the second cylinder respectively.

(sorry for writing like this, but I was already comfortable with latex, so it seemed a faster approach )

I was very tempted to integrate here, but the problem is - h₁ and h₂ do not change in the same manner.
I cannot seem to be able to get past here.

 

[kuruman]

Probably a typo, but you give two different expressions for the equilibrium height. One has 26 in the denominator and the other 25.

More importantly, your final expression has three variables, h, h₁ and h₂. There should be only one, say h representing the height in the cylinder that was empty originally. Since the volume of the liquid is fixed, you can express the height of the liquid in the other cylinder as a function of h. Then get an expression for ##\frac{dt}{dh}##.