# Help with fluid dynamics with differntial equations

Hi, I have been doing a simple experiment with a 2L bottle and a cut out slot near the bottom. I have gathered data regarding the time it takes the water level in the bottle to go from 13cm above the slot to the slot itself, versus the area of the slot. My data shows a perfect inverse relationship between the time for the water level to reach the slot with a constant of 0.0169. I am working in standard units and my functions looks as follows T=0.0169/A. Now I need to determine what the 0.0169 is so i can derive a formula that can determine the time it takes to empty the bottle depending on initial volume, bottle radius, and slot area.

I have been searching everytwhere for relevent information but I honestly cannot find anything I can use to help me. I am desperate to know because it would solve all my problems. The next part of my experiment is to derive a formula for the volume in the bottle after a certian time has elasped. The data for this experiment fits a parabola which I also cannot explain. If someone can please clear this up for me I would be so grateful, but please don't think I am asking someone to do my homework, I have doen the experiments and sat down since 5:00 (and now its 12:30) with a pencil and apper searching online for info and taking derivatives and integrals and using pressure and potential energy and reading up on CV factors, etc. Please, if anyone can help I would be so relieved.

Alos, someone has told me that a diiferential equation must be used in this case, which makes sense since water flow depends on volume which decreases over time due to water flow.

Thanks,
Mike.

siddharth
Homework Helper
Gold Member
The simplest way to do it, is to use bernoulli's equation and the conservation of mass (assuming density is constant. ie, V1A1=V2A2)

If you're stuck somewhere, you can post what you've tried till then and it'll be easy to help.

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What I would try is:

1. Come up with a time dependant expression of the volume of water in the bottle, based on the starting volume of water minus the volumetric flow rate of water leaving the bottle.

2. Then, find an expression for the hydrostatic pressure the water is exerting on the slot for any given volume of water (this will be based on the constant cross-sectional area of the bottle times the height of water in the bottle above the slot.)

3. Come up with an expression for the velocity of the water out of the slot based on the column height and the pressure it is exerting. This is where bernoulli is useful.

4. Combine all that into one expression, which given an intital volume in the bottle, density of water, and gravity, should allow you to solve for mass flow rate based on time.

Well I have used bernoulli's equation to derive a general equation for water level height versus time, which is H = (8E-05)*t^2 - 0.0062*t + 0.13. Now I can munipulate that formula by multiplying it by the cross section area of the bottle to get the volume as a function of time, which is what Peregrine has suggested to do first.

However, from here I am slightly confused on how to determine the pressure of the water exerted on the slot. And also, m data shows that the total time required to empty the bottle is exactly inversely proportional to the cross section area of the drainage slot (Total time elasped is proportional to 1/A or, T=k/A where K is found to be 0.0169), but there must be a simple way to determine k depending on initial height of the bottle and, if the fluid chages, the density and gravity.

How could I do this?

Thanks

Please anyone, I give up, I have been trying to solve the relation between total time requiore to empty a bottle versus the slot size and the relation between volume in the bottle versus time elasped after draining started. I cannot do it. I simply can't. If anyone can please provide the answer I would be so garteful. I have tried for the past two days and I give up. I'm done.

Thanks anyway,
Mike.

If anyone wants to see where we left off in the other physics forum (sory i double threaded) heres the link https://www.physicsforums.com/showthread.php?t=144490"

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Just brainstorming:
I have not done physics for a long time, however, given the statement of Bernoulli's equation: "If the speed of a fluid particle increases as it travels along a streamline, the pressure of the fluid must decrease, and conversely", the flow rate at the surface of the slot must be a function of the pressure at that point. Since pressure is a function of the height of the fluid, as the height of the fluid decreases, pressure must decrease. I would be looking at formulating pressure as a function of height, that is dP/dh and once this is done, compare the pressure at the surface of the fluid (atmospheric pressure) with that of the pressure at the slot. The comparison might produce the needed differential equation.

Well I have used bernoulli's equation to derive a general equation for water level height versus time, which is H = (8E-05)*t^2 - 0.0062*t + 0.13. Now I can munipulate that formula by multiplying it by the cross section area of the bottle to get the volume as a function of time, which is what Peregrine has suggested to do first.

However, from here I am slightly confused on how to determine the pressure of the water exerted on the slot. And also, m data shows that the total time required to empty the bottle is exactly inversely proportional to the cross section area of the drainage slot (Total time elasped is proportional to 1/A or, T=k/A where K is found to be 0.0169), but there must be a simple way to determine k depending on initial height of the bottle and, if the fluid chages, the density and gravity.

How could I do this?

Thanks