The discussion revolves around understanding the equations P=pgh and Bernoulli's equation, particularly in the context of calculating the height of a water fountain. Participants explore the derivations, applications, and implications of these equations in a specific problem scenario.
Participants generally do not reach a consensus on the interpretation of the equations and the relationships between the pressures at different points. Multiple competing views and uncertainties remain regarding the correct application of the equations and the definitions of variables involved.
Participants note limitations in their understanding of the derivations and the specific problem context. There are unresolved questions about the assumptions made in applying Bernoulli's equation and the definitions of pressure used in calculations.
This discussion may be useful for students and individuals interested in fluid dynamics, particularly those seeking to understand the application of Bernoulli's equation and hydrostatic pressure in practical scenarios.
$$p_B=p_A+\rho g h_2\tag{1}$$Physicist1011 said:What? How is PA on both sides of the 2nd equation? PB is on the other side.
Both are atmospheric pressure. The pressure at the outlet where the fountain is emerging is atmospheric also.Physicist1011 said:But I thought for the 2nd equation PA was the pressure of the water coming out of the fountain and in the first equation it was the pressure of the water in the open container A which would be atmospheric pressure.
The atmosphere imposes its own pressure on the fluid at the exit of a tube. What did you think?Physicist1011 said:What? how is that?
Not so. Imagine a jet of fluid coming out of the tube. What is the radial force per unit area exerted by the surrounding atmosphere on the free surface of the jet?Physicist1011 said:I thought the pressure would be different since it will have a different pressure in the tube where it is being pushed by air towards the exit of the tube.
Yes. The radial force per unit area on the free surface of the jet will be PA. And, this must match the fluid pressure within the jet. And, since, (according to Pascal's law) pressure acts equally in all directions, the pressure throughout the fluid jet at the exit of the tube will be PA.Physicist1011 said:The force will be PA so atmospheric pressure times the area of the tube.
Hi again I have another question to do with the one we just solved that I cannot figure out. What is the diameter of the tubes is changed, how would I calculate the change in height of the herons fountain using equations then?Chestermiller said:Yes. The radial force per unit area on the free surface of the jet will be PA. And, this must match the fluid pressure within the jet. And, since, (according to Pascal's law) pressure acts equally in all directions, the pressure throughout the fluid jet at the exit of the tube will be PA.
What are your thoughts on this? Your thinking should be along the lines of "what are the limitations of the derivation of the Bernoulli equation, and of how the Bernoulli equation was applied to this problem?Physicist1011 said:Hi again I have another question to do with the one we just solved that I cannot figure out. What is the diameter of the tubes is changed, how would I calculate the change in height of the herons fountain using equations then?
So the limitations are that it cannot detect a change in velocity due to a different diameter.Chestermiller said:What are your thoughts on this? Your thinking should be along the lines of "what are the limitations of the derivation of the Bernoulli equation, and of how the Bernoulli equation was applied to this problem?
You are aware that density is not the same thing as viscosity, right?Physicist1011 said:The viscosity of the fluid involved is used - p - density in the equation.
Also how would I calculate the height of the fountain then if I was looking at the effects of different diameters?