Question

hey srry guys about last link , I didn't realise my attachment wasn't uploaded so : it's a theoretical question and would love to know ur opinion :)

Answer 1

you have left out background material.
Is the liquid going in at a fixed rate?
if so, you are putting in (or emptying) a fixed volume per unit time
Volume = Area of base * height
and
h= Volume/Area
if volume changes linearly with time and the base area is fixed, then h changes linearly with time.

Answer 2

in that case, a cylinder such as a "rain meter" is an example.

Answer 3

do you have to empty it? what about filling the container?

Answer 4

this sounds more like physics than math. Off-hand I don't know how fast fluid will drain from a hole. Does the rate depend on the water pressure (hence the height) ?
It does sound like if you make a really tall container and measure the change in height only over the "top part", the level will be linear or very close to linear.

Answer 5

next for a

excusing the crap art, what we have is a vessel that has an as yet undefined contour but which is symmetrical in that it revolves around the h axis, so this is just obe class of solution.

fluid leaves at the bottom of the vessel at h = 0 through a pipe of x-sectional area a, moving at velocity v.

thus for a given height of water we can say \( \dot V = -av \) and as \(v = \sqrt{2gh} \) we have \( \dot V = -a\sqrt{2gh} \)\)

if we say that the radius of the vessel ir related to h by \(r = f(h)\), then an elemental volume of this shape is given by \(dV = \pi f^2(h) dh \implies \dot V = \pi f^2(h) \dot h\)

you seem to want \(\dot h = const. \) if i understand correctly

so we can say that \(\pi f^2(h) \dot h = \pi f^2(h)k = a\sqrt{2gh}\)

meaning \[f(h) = \sqrt[4]{\frac {2ga^2}{\pi^2 k^2}}\times h^{\frac{1}{4}} \\= const. \times h^{\frac{1}{4}}\]

that's a bit of a stream of consciousness that may have some merit but needs verification, devil will be in detail

specifically an easy way to check is to do it in reverse. imagine a vessel with contour given by
\(f(h) = \sqrt[4]{\frac {2ga^2}{\pi^2 k^2}}\times h^{\frac{1}{4}}\)
or if you like
\( f(h) = const. \times h^{\frac{1}{4}}\)]

with a leak cross section area a at it base. what is \(\dot h\) for that vessel given that flow velocity through the hole is governed by \(v = \sqrt{2gh}\)?