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Let us consider a real life situation.

The bucket in figure below (drawn using MS PAINT) has an irregular crack of essentially small area. And I decide to open a tap above it and start filling it with water.

CRACKED BUCKET

Say, the crack starts at a distance $x\,\mathrm{cm.}$ from bottom surface and ends about $y\,\mathrm{cm.}$ from below. The area covered by crack is $q\,\mathrm{sq.cm.}$

My questions are:

  1. Is there any constant speed, to which if the flow of water from tap be set, the water level in bucket will reach the brim irrespective of water continuing to flow out through crack?
  2. If such speed exists, does it depend on shape of crack?
  3. If such speed does not exist, is there any other way to control and vary only the speed of water flow and fill water to brim?
  4. Finally, what happens if the speed of flow of water tends to c(speed of light)?

P.S. If possible, consider flow of water to be turbulent. That is what I meant by real life situation. Again IF POSSIBLE ONLY.

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  • $\begingroup$ Have you actually tried filling a bucket with a hole in it? $\endgroup$
    – Kyle Kanos
    Commented Sep 24, 2015 at 17:00
  • $\begingroup$ Incidentally, this is more of a question for Engineering.SE than Physics.SE. IMO. I'm comfortable with moderation in either direction. If you regularly have questions like this, you may want to try there, too. I think the (subset) question that would better suited here would regard how one models the flow characteristics through a crack / irregular geometry. $\endgroup$
    – New Alexandria
    Commented Sep 24, 2015 at 17:10
  • $\begingroup$ @KyleKanos Yes I tried. I succeeded when the hole was small and located near the open end of the bucket but failed miserably when the hole was rectangular in shape . $\endgroup$
    – SchrodingersCat
    Commented Sep 24, 2015 at 19:41
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    $\begingroup$ Given that you've succeeded in some configurations, I think the question could benefit fro my of the conditions you found in experimentation. How close to the exact brim of the bucket do you need to be? Why does speed-of-light ever come into the question? You've already answered that the general-case is possible, but the details of the specific case(s) are not clear. $\endgroup$
    – New Alexandria
    Commented Sep 24, 2015 at 20:27

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You certainly can accomplish what you want. Whether you can calculate it with sufficient accuracy is largely a function of need, and access to CFD codes and resources.

The shape of the crack certainly has a reasonable influence on what you want to engineer. You cannot reasonably consider the fluid inside the bucket to be a purely turbulent regime. The shape of the crack, the head pressure above it, and the characteristics of the in-flow of fluid will all impact whether any laminar flow characteristics can form in the outlet through the crack.

If a 'purely' laminar flow were to form, then you could presume a constancy in outflow. The same expectation would occur in a 'purely' turbulence regime. You will be in the middle, which will lead to some nonlinearity in the outflow. Easiest way to determine this is through experimentation. From experience, oscillation from the inflow (splashing, etc) will create the biggest nonlinearity... Unless the bucket is also vibrating.

If you must model the outflow, seek to break-down the crack into sections that have more-well-defined characteristics as a 'nozzle'. The 'edges' on which they 'join' will be more laminar and change the rest of the transfer.

I'm going to leave out this bit about approach C. The mechanics of your gedanken do not translate to relativistic system (in any way that is apparent in your question)

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  • $\begingroup$ Good answer. Probably worth adding a bit about the fact that the circumference and the area of the hole both influence the flow rate. This is captured by your "shape" comment, but I think it's possible to be a little more specific about it. $\endgroup$
    – Floris
    Commented Sep 24, 2015 at 17:15
  • $\begingroup$ @New Alexandria Thanks for your answer. Can you explain a bit more about how you feel that I can accomplish what I want? You haven't backed that statement of yours with adequate evidence. And why do you not consider that the fluid inside the bucket to be a purely turbulent regime? You see we have water falling from the tap, splashing, outflow through an irregular crack located anywhere on the side and the artificial convection currents due to water falling. I don't understand why the fluid molecules don't have turbulent motion. $\endgroup$
    – SchrodingersCat
    Commented Sep 24, 2015 at 19:56
  • $\begingroup$ I didn't say the system lacks turbulence, I said it's not a purely turbulent regime. Your convection currents will have some layered or laminar properties, regionally. As well, the channel (longitudinally for the crack) will experience hysteresis of flow. $\endgroup$
    – New Alexandria
    Commented Sep 24, 2015 at 20:22

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