Flow of fluids through an infinitely thin hole

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Discussion Overview

The discussion centers on the flow of fluids through an infinitely thin hole, particularly in the context of fluid dynamics and the challenges associated with deriving a formula for such flow. Participants explore various factors that may influence the flow, including viscosity, surface tension, and capillary action, while considering both analytical and numerical approaches to the problem.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that while students learn about flow through long tubes, there seems to be no established formula for flow through a hole, suggesting it may involve complex factors like surface tension and viscosity.
  • Another participant mentions that analytical solutions for viscous fluid flow through a hole are rare and that numerical simulations are often used instead.
  • Capillary action is highlighted as a significant factor when the hole size is small enough, indicating a shift in the dynamics of fluid flow.
  • A clarification is sought regarding the definition of "thin," with one participant suggesting it may refer to a typical orifice calculation with a small diameter.
  • Participants are directed to resources that provide explanations of calculations for fluid flow setups.
  • The existence of the term "vena contracta" is suggested as a relevant search term for further exploration of the topic.
  • One participant asserts that the problem of fluid flow through a hole in a bucket remains unsolved, citing a reference that states no analytical solution currently exists.

Areas of Agreement / Disagreement

Participants express differing views on the existence of analytical solutions for fluid flow through a hole, with some asserting that no such solutions exist while others suggest numerical methods may provide insights. The discussion remains unresolved regarding the overall complexity and solvability of the problem.

Contextual Notes

Participants acknowledge the limitations of existing models and the dependence on specific conditions, such as hole size and fluid properties, which may affect the applicability of any proposed solutions.

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

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