Flow of fluids through an infinitely thin hole

[danielarbib]

Every student knows the flow through a long tube ignoring end effects but my "A" level son has asked if there is a formula for the flow of a fluid through a this hole (or out of a bucket with a hole at the bottom). I think such a formula would include surface tension, viscosity and other parameters and could be discontinious as individual drops or even chaotical. I have never seen a solution - even an approximation so can anyone help, please?

 

[nasu]

You can find the general equation of motion for fluid with viscosity. I don't know if the surface tension effect is considered. Anyway it does not matter because even without surface tension, analytical solutions are known for very few and very simple cases.
So if by "formula" you mean an analytical solution of the equation of motion for a viscous fluid going through a hole and then flowing freely under gravity... I don't think you have a chance.
What people do (if interested) is to solve the equations numerically, using computer programs for simulations.

 

[Topher925]

When holes get small enough, then usually the effects of capillary action take over.

http://en.wikipedia.org/wiki/Capillary_action

 

[FredGarvin]

When you say "thin" I think of a plate with a hole in it, i.e. a typical orifice calculation. Do you mean with a very small diameter?

 

[mgb_phys]

This site has a good explanation of the calculations for common setups.
http://www.mcnallyinstitute.com/13-html/13-12.htm

 

[vector2]

I would add searching on the term "vena contracta" as well.

 

[mordechai9]

Actually, it is surprising but this is an unsolved problem. At least, no analytical solution exists to the flow of a liquid through a hole in a bucket. For example, see White, Viscous Fluid Flow, pg. 23, "It is an intriguing fact that the flow of a viscous liquid out of the bottom ofa cup is a difficult problem for which no analytic solution exists at present."