Flow of fluids through an infinitely thin hole

In summary, the conversation discusses the possibility of finding a formula for the flow of fluid through a small hole or opening. It is mentioned that there may be a formula that takes into account factors such as surface tension and viscosity, but it is difficult to find an analytical solution for this problem. Instead, numerical simulations are often used. The conversation also references resources for understanding the calculations involved in this problem, such as the concept of "vena contracta" and specific websites. Ultimately, it is stated that this remains an unsolved problem with no current analytical solution.
  • #1
danielarbib
1
0
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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  • #2
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.
 
  • #4
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?
 
  • #6
I would add searching on the term "vena contracta" as well.
 
  • #7
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."
 

1. What is the definition of "flow of fluids through an infinitely thin hole"?

The flow of fluids through an infinitely thin hole refers to the movement of a fluid (liquid or gas) through a small opening or hole that has negligible thickness. This phenomenon is also known as laminar flow, where the fluid particles move in a smooth, parallel manner without any turbulence.

2. How does the flow rate through an infinitely thin hole affect the pressure of the fluid?

The flow rate through an infinitely thin hole is directly proportional to the pressure of the fluid. This means that as the flow rate increases, the pressure of the fluid decreases, and vice versa. This relationship is described by the Bernoulli's principle, which states that an increase in the velocity of a fluid results in a decrease in its pressure.

3. What is the equation for calculating the flow rate through an infinitely thin hole?

The equation for calculating the flow rate through an infinitely thin hole is Q = A * V, where Q represents the flow rate, A is the cross-sectional area of the hole, and V is the velocity of the fluid. This equation is known as the continuity equation and is based on the principle of conservation of mass.

4. How does the viscosity of a fluid affect its flow through an infinitely thin hole?

The viscosity of a fluid, which is its resistance to flow, plays a significant role in determining the flow rate through an infinitely thin hole. Higher viscosity fluids, such as honey, have a lower flow rate through a tiny hole compared to lower viscosity fluids, such as water. This is because higher viscosity fluids have a greater resistance to flow, while lower viscosity fluids flow more easily through small openings.

5. What factors can affect the flow of fluids through an infinitely thin hole?

Several factors can affect the flow of fluids through an infinitely thin hole, including the fluid's viscosity, density, and velocity, as well as the size and shape of the hole. Other external factors such as temperature, pressure, and surface tension can also impact the flow rate of a fluid through a small opening. The presence of any obstacles or turbulence can also affect the flow of fluids through an infinitely thin hole.

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