Validity of hydrostatic pressure equation

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Discussion Overview

The discussion centers on the validity of the hydrostatic pressure equation, P=hρg, particularly in relation to the directionality of pressure in fluids. Participants explore the implications of weight acting downward while pressure is exerted in all directions, examining both theoretical and conceptual aspects of fluid behavior.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Technical explanation

Main Points Raised

  • Some participants assert that the hydrostatic pressure equation is derived from the relationship P=F/A, questioning how pressure can act in all directions if weight only acts downward.
  • One participant suggests that at equilibrium, fluids cannot sustain shear stress, implying that pressure is uniform in all directions.
  • Another participant uses the analogy of a water balloon to illustrate how pressure applied at the top results in expansion and pressure exerted outward, indicating that pressure is not limited to a downward direction.
  • A participant presents a scenario involving a vertical pipe connected to a horizontal pipe, questioning whether longitudinal pressure remains consistent across both segments when pressure is applied at the top.
  • One participant emphasizes that the forces exerted by adjacent water columns cancel each other out, leading to no net force in any direction, thus maintaining equilibrium.
  • Another participant challenges the notion that weight only acts downward within liquids, suggesting that this perspective may not apply in fluid dynamics.

Areas of Agreement / Disagreement

Participants express differing views on the implications of weight acting downward versus pressure acting in all directions. While some agree on the concept of pressure equilibrium, others question the foundational assumptions regarding the behavior of fluids, indicating that the discussion remains unresolved.

Contextual Notes

There are references to concepts such as Pascal's law and Torricelli's law, but the implications of these laws in relation to the hydrostatic pressure equation are not fully explored or agreed upon by participants.

pgirl1729
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The hydrostatic pressure equation, P=hρg is derived from P=F/A where F=mg and A=cross sectional area of the water column. As weight only acts downwards and it's force is only exerted in downward direction. Then why does pressure acts in all directions? How is the P=hρg still valid?
 
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pgirl1729 said:
The hydrostatic pressure equation, P=hρg is derived from P=F/A where F=mg and A=cross sectional area of the water column. As weight only acts downwards and it's force is only exerted in downward direction. Then why does pressure acts in all directions? How is the P=hρg still valid?
A short and not very revealing answer is that fluids are fluid. At equilibrium, they cannot sustain a shear stress. [Stress is a generalization of pressure that one may eventually learn about]

Perhaps more appealing is to think about a water balloon. If you press down on its top, it expands out the sides. Your top pressure has become a side pressure.

Or consider a vertical pipe connected by an elbow to a horizontal pipe. Seal the horizontal pipe at its free end. Fill the pipes with water and apply pressure at the top. Would you agree that the longitudinal pressure in both pipe segments should be the same? Remove the pipes and replace them with a bath of water. Maintain the pressure on the top of the bath. Would you expect the pressure anywhere to change?
 
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pgirl1729 said:
As weight only acts downwards and it's force is only exerted in downward direction. Then why does pressure acts in all directions?

The weight of the balls in the below image is also only exerted in downward direction, and yet they will push sideways against the walls. You don't need complex fluid dynamics to understand the basic principle qualitatively.

hex_packing.png
 
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pgirl1729 said:
Then why does pressure acts in all directions? How is the P=hρg still valid?
The pressure on which the water column fron the left is "pressed" is equal to the pressure with which it is fron the right and they cancel each other out.
 
Pisica said:
The pressure on which the water column fron the left is "pressed" is equal to the pressure with which it is fron the right and they cancel each other out.
Yes. This is a correct observation. If we imagine a small parcel of water (perhaps a cube), the force from the left will match the force from the right. So there is no net force in that direction. The parcel remains at rest, not accelerating.

This must be the case if the fluid is in an equilibrium state.

This is the sense in which the left and right pressures "cancel out". It is not that there is no pressure. It is that there is no acceleration of the fluid parcel.
 

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