Another question on Bernoulli's principle

[PhysicsDaoist]

Homework Statement

I have read this thread https://www.physicsforums.com/showthread.php?p=3185555&posted=1"

I am still a bit confused. So I am putting this quantitatively with a slightly different scenario.
Assuming water coming out of water pipe with Area A₁ with a water flow rate of J where J=A₁V₁.
What happen if my thumb blocks 1/2 the area? That is, A₂=0.5A₁
What is my pressure inside the pipe in terms of J, A₁ and A₂?
How are the inside pressure different in the 2 cases (with thumb) and not with thumb restricting water flow?

Homework Equations

Assume no gravitational potential difference, I am ignoring the rho*g*h term.
P₁+1/2*rho*v₁² = P₂+1/2*rho*v₂²
J=A₁V₁=A₂V₂

The Attempt at a Solution

With thumb blocking 1/2 the area of the opening,
P₂ = 1 atm

P₁ = 1 + 1/2* rho * (v₂² - v₁²)

Since J=A₁V₁=A₂V₂
V₂ = J/(A₂)
V₁ = J/(A₁)

P₁ = 1 + 1/2* rho * J² *(1/A₂² - 1/A₁²)

For the case of no thumb, I got
P₁ = P₂ = 1 atm ??
It is because V₁=V₂

Now questions -
a) how do I show the jet pressure coming out from the 1/2 closed opening?
b) how do I compare the two?
c) I am aware that the kinetic energy is higher with the 1/2 closed case, but what about the total energy comparison?

 

[anathema]

it is important to understand the concepts and equations being used in the scenario presented. It appears that the main question being asked is how the pressure inside the water pipe changes when the area of the opening is reduced by half due to a thumb blocking the flow.

To answer this, we can use the Bernoulli's equation, which states that the total energy of a fluid in motion is constant along a streamline. In this case, we can ignore the gravitational potential difference and focus on the kinetic energy and pressure terms.

With the thumb blocking half the area, the velocity of the water will increase in order to maintain the same flow rate. This means that the kinetic energy term (1/2 * rho * v^2) will increase, resulting in a decrease in pressure. This can be seen in the equation P1 + 1/2 * rho * v1^2 = P2 + 1/2 * rho * v2^2.

To compare the two cases, we can plug in the values for v1 and v2 and see the difference in pressure. Without the thumb blocking the flow, v1 and v2 will be equal, resulting in the same pressure on both sides of the opening. However, with the thumb blocking half the area, v2 will be greater than v1, resulting in a decrease in pressure on the thumb side of the opening.

To show the jet pressure coming out from the half-closed opening, we can use the equation P2 = 1 atm and plug in the values for v2 and A2. This will give us the pressure on the thumb side of the opening.

In terms of total energy comparison, we can see that the kinetic energy will be higher in the half-closed case due to the increased velocity. However, the total energy will still be constant as per the Bernoulli's equation.

I hope this helps clarify the concepts and equations involved in this scenario. It is important as a scientist to fully understand the principles behind a situation before attempting to solve it quantitatively.