Draining a Tank of Fluid | First Order Differential Equation example | Fluid Mechanics

#differentialequations #fluidmechanics
First Order Differential Equation example in Fluid Mechanics.

I absolutely love this example for understanding how differential equations work! In the video we derive the differential equation describing the draining of a bottle/vessel out of a small hole on its' side.

I have tried to talk through the problem in a way that I hope will seem intuitive. I wanted this to be a worked example that taught and refreshed the subject of differential equations for those that are studying for an exam, as well as introducing, those of you that have never heard of a differential equation, to the brilliance of them.

Please do comment below your thoughts on whether you'd prefer more maths, less maths, less talking through the maths, the maths done 'live' so you can follow along better... any feedback is much appreciated!

Further information:
1) At 6:01 in the note, it should be "...falling freely..".

2) Averaged timings: I did the experiment 4 more times after the one in the video. And calculated an average as follows: (55.4 + 53.73 + 55.32 + 53.35 + 52.67)/5 = 54.094... 54.094!! --- That's only 0.306s off our estimate!!

3) My uncertainty calculation: I am going to do another video on this, as a follow up to this video, since I did not practice it well whilst I was at university. For those of you that would like to check my calculation of the final uncertainty, I used a standard ruler with a minimum division of 1mm, but too say that I could measure the 6mm diameter of the jet hole to an uncertainty of +-1mm was a huge difference in comparison to the hole size. So I instead said that I could measure it to +-0.2mm and the diameter of the bottle to +-0.5mm and the height of the water too +-1mm. Using the relative errors I calculated ((Diameter of Bottle)^2)*sqrt(height)/((Diameter of the jet hole)^2) and used the upper and lower bounds of that in my calculation of the final time. I did not put any uncertainty into the gravitational acceleration 9.81m/s^2.

4) The fluid mechanics of the velocity coming out of the jet hole can be calculated using Bernoulli's equation and the continuity equation. With the assumption that the area of the jet hole divided by the area of the bottle is close to zero (negligible), you will get the same result as Torricelli did prior to modern understandings from Daniel Bernoulli. I chose to explain Torricelli's method since it was shorter and didn't detract too much from the differential equation at hand. Example 3.13 in Frank White's fluid mechanics explains this phenomenon.

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Видео Draining a Tank of Fluid | First Order Differential Equation example | Fluid Mechanics канала Ciaran McEvoy