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The Bernoullis Equation for an Incompressible Inviscid flow (say water) is as follows:
p + (1/2).ρV^2 + ρgh = constant
where,
In words,
static pressure + dynamic pressure + pressure energy due to height = constant
It gives the Energy Balance or Conservation Of Energy in terms of pressure, naming the Kinetic Energy of the flowing fluid as the Dynamic Pressure to unify it with the other purely pressure terms.
I have found the equation always obscure in relating to the two terms Static Pressure and Dynamic Pressure especially in an Incompressible Flow.
I am attatching a diagram for refernce and discussion please refer to it.
It is a simple pipe of varying area of cross section and water flows within it in an enclosed volume.
The volume of water is enclosed within a piston on one side and a movable solid disc made up of an elastic material such that it can slide along the inner diameter of the pipe adjusting its own periphery (under elastic force  imagine it to be under some sort of spring type force) to meet up with the pipe diameter inorder to enclose the volume of water.
The piston is pushed forward with a force F and the pressure created at the piston is F/A where A is the area of the piston and section aa.
Question : does Pascal Law be applicable to the flowing enclosed volume of water ?
Also note that if we consider Pascals Law or very basis of Hydraulics then pressure at section aa must be EQUALLY transmitted at section bb but Bernoulli suggests that pressure at section bb must be higher than aa. Explain this.
Further on:
The understanding and (lack of understanding) that I have regarding the concept behind this seemingly simple equation is as follows and please check if each of the points is correct in terms of concept and please explain the part I havent much understood.
We will go in a sequence.
1. The Bernoulli Equation is based on two very basic principles of nature:
a. The Law Of Conservation Of Mass  to start with
b. The Law Of Conservation Of Energy  most importantly
2. The Law Of Conservation Of Mass suggests that the Flow Rate Of Mass across any cross sectional area remains the same for a flow at any instant.
This leads to the important conclusion that as the area of cross section increases the velocity of the fluid must decrease in order to maintain the equal mass flow rate across the section
And the velocity of the fluid must Increase at the decreasing cross section area again to conserve the mass flow rate.
3. The deduction in the second step above leads to the actual derivation of the Bernoullis Equation in an attempt to find the reason behind the observed phenomena of mass conservation.
And this could be explained only by considering the fact that Energy is also always conserved.
4. The Total Energy of the flow was ofcourse the Kinetic Energy and the Potential Energy of the fluid.
And as the velocity of the fluid was required to Increase and Decrease as per the reduction and increment in the cross sectional area of the flow it meant that the Kinetic Energy associated with the flow (actually streamline) must Increase and decrease accordingly.
But as an Universal fact that the Energy is always conserved then the Increase in the Kinetic Energy must lead to corresponding decrease in the Potential Energy of the fluid as these are the only two types of energies associated with the flow
(pressure due to height is considered negligible as elevation is negligible)
5. Thus the Static Pressure of the fluid must decrease when the velocity of the fluid increases at the smaller cros section and the Static Pressure must increase as the velociy decreases as the cross sectional area increases.
Fine  as per the Law of Conservation Of Mass and the leading reason for it found in the Law Of Conservation Of Energy is considered the theory goes absolutely flawless in explaination and matches the practically observed effects of velocity anbd pressure in a flow.
But the real reason behind this transformation of energies within themselves has always remained very obscure to me.
My question is:
a. What does Static Pressure clearly mean in an Incompressible fluid like water? and is it the same as the pressure in a gas due to thermal state?
b. How actually does the pressure Increase or reduce itself almost intutively at the increasing or decreasing cross sectional area so as to let the velocity decrease or increase
OR
What is the ACTUAL INTERMOLECULAR CAUSE or WHAT PROMPTS the molecules in the streamline to reduce their pressure so as the velocity can increase or decrease to maintain the mass flow rate? After all the molecules dont have a mind of their own to think about this conservation laws?
So can anyone please explain the actual Reason or Cause that brings about the observed change in pressure when velocity changes.
This concept needs to be Actually understood at depth by me for knowing the reason that actually brings about such a change in pressure and veloity to compiment each other rather than simply muttering the fact that energy and mass are always conserved and being satisfied with the observed effects.
After all we are engineers and we MUST answer all questions in terms of HOW, WHY and WHAT at the very base of anything rather than being happy with just laws and seeing them be applicable.
So please answer me elaborated.
p + (1/2).ρV^2 + ρgh = constant
where,
In words,
static pressure + dynamic pressure + pressure energy due to height = constant
It gives the Energy Balance or Conservation Of Energy in terms of pressure, naming the Kinetic Energy of the flowing fluid as the Dynamic Pressure to unify it with the other purely pressure terms.
I have found the equation always obscure in relating to the two terms Static Pressure and Dynamic Pressure especially in an Incompressible Flow.
I am attatching a diagram for refernce and discussion please refer to it.
It is a simple pipe of varying area of cross section and water flows within it in an enclosed volume.
The volume of water is enclosed within a piston on one side and a movable solid disc made up of an elastic material such that it can slide along the inner diameter of the pipe adjusting its own periphery (under elastic force  imagine it to be under some sort of spring type force) to meet up with the pipe diameter inorder to enclose the volume of water.
The piston is pushed forward with a force F and the pressure created at the piston is F/A where A is the area of the piston and section aa.
Question : does Pascal Law be applicable to the flowing enclosed volume of water ?
Also note that if we consider Pascals Law or very basis of Hydraulics then pressure at section aa must be EQUALLY transmitted at section bb but Bernoulli suggests that pressure at section bb must be higher than aa. Explain this.
Further on:
The understanding and (lack of understanding) that I have regarding the concept behind this seemingly simple equation is as follows and please check if each of the points is correct in terms of concept and please explain the part I havent much understood.
We will go in a sequence.
1. The Bernoulli Equation is based on two very basic principles of nature:
a. The Law Of Conservation Of Mass  to start with
b. The Law Of Conservation Of Energy  most importantly
2. The Law Of Conservation Of Mass suggests that the Flow Rate Of Mass across any cross sectional area remains the same for a flow at any instant.
This leads to the important conclusion that as the area of cross section increases the velocity of the fluid must decrease in order to maintain the equal mass flow rate across the section
And the velocity of the fluid must Increase at the decreasing cross section area again to conserve the mass flow rate.
3. The deduction in the second step above leads to the actual derivation of the Bernoullis Equation in an attempt to find the reason behind the observed phenomena of mass conservation.
And this could be explained only by considering the fact that Energy is also always conserved.
4. The Total Energy of the flow was ofcourse the Kinetic Energy and the Potential Energy of the fluid.
And as the velocity of the fluid was required to Increase and Decrease as per the reduction and increment in the cross sectional area of the flow it meant that the Kinetic Energy associated with the flow (actually streamline) must Increase and decrease accordingly.
But as an Universal fact that the Energy is always conserved then the Increase in the Kinetic Energy must lead to corresponding decrease in the Potential Energy of the fluid as these are the only two types of energies associated with the flow
(pressure due to height is considered negligible as elevation is negligible)
5. Thus the Static Pressure of the fluid must decrease when the velocity of the fluid increases at the smaller cros section and the Static Pressure must increase as the velociy decreases as the cross sectional area increases.
Fine  as per the Law of Conservation Of Mass and the leading reason for it found in the Law Of Conservation Of Energy is considered the theory goes absolutely flawless in explaination and matches the practically observed effects of velocity anbd pressure in a flow.
But the real reason behind this transformation of energies within themselves has always remained very obscure to me.
My question is:
a. What does Static Pressure clearly mean in an Incompressible fluid like water? and is it the same as the pressure in a gas due to thermal state?
b. How actually does the pressure Increase or reduce itself almost intutively at the increasing or decreasing cross sectional area so as to let the velocity decrease or increase
OR
What is the ACTUAL INTERMOLECULAR CAUSE or WHAT PROMPTS the molecules in the streamline to reduce their pressure so as the velocity can increase or decrease to maintain the mass flow rate? After all the molecules dont have a mind of their own to think about this conservation laws?
So can anyone please explain the actual Reason or Cause that brings about the observed change in pressure when velocity changes.
This concept needs to be Actually understood at depth by me for knowing the reason that actually brings about such a change in pressure and veloity to compiment each other rather than simply muttering the fact that energy and mass are always conserved and being satisfied with the observed effects.
After all we are engineers and we MUST answer all questions in terms of HOW, WHY and WHAT at the very base of anything rather than being happy with just laws and seeing them be applicable.
So please answer me elaborated.
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