Lilli suggests that they explore the simulation starting with varying only a single parameter in order to understand the role of each variable independently. Joseph agrees that this is a good strategy. After running the simulation several times and viewing the results, they decide to set the height difference between the tubes, Δy, to zero so that they can view the changes that occur with the cross-sectional area of the second tube. Joseph then suggests that they set A2 to twice A1 and predict the resulting pressure in tube 2, P2.

Joseph asks Lilli, "Should we expect that the pressure in the second tube should increase or decrease if A2 is increased?"

Which is the correct response to this question?

A.)

"If we increase A2 then the speed of the fluid should also increase so that the tube remains full of fluid. Bernoulli's equation then shows that if the velocity of the fluid increases (at constant height conditions) then the pressure of the fluid should decrease."

B.)

"If we increase A2 then by the continuity equation the speed of the fluid should decrease. Bernoulli's equation then shows that if the velocity of the fluid decreases (at constant height conditions) then the pressure of the fluid should increase."

C.)

"If we increase A2 then the speed of the fluid should also increase so that the tube remains full of fluid. Since the speed increases the pressure should be larger in order to keep the fluid moving at this higher speed."

D.)

"If we increase A2 then by the continuity equation the speed of the fluid should decrease. Since the speed of the fluid is smaller the pressure will be smaller since the force required on any part of fluid to keep it in motion at a lower speed should be smaller."

One of the ways to address this issue is through the options given by the statement. The concepts related to the continuity equation and the Bernoulli equation.

Through these two equations it is possible to observe the behavior of the fluid, specifically the velocity at a constant height.

By definition the equation of continuity is,

[tex]A_1V_1=A_2V_2[/tex]

In the problem [tex]A_2[/tex] is [tex]2A_1[/tex], then

[tex]A_1V_1=2A_1V_2[/tex]

[tex]V_2 = \frac{V_1}{2}[/tex]

Here we can conclude that by means of the continuity when increasing the Area, a decrease will be obtained - in the diminished times in the area - in the speed.

For the particular case of Bernoulli we have to

[tex]P_1 + \frac{1}{2}\rho V_1^2 = P_2 +\frac{1}{2}\rho V_2^2[/tex]

[tex]P_2-P_1 = \frac{1}{2} \rho (V_1^2-V_2^2)[/tex]

For the previous definition we can now replace,

[tex]P_2-P_1 = \frac{1}{2} \rho (V_1^2-(\frac{V_1}{2})^2)[/tex]

[tex]\Delta P = \frac{3}{8} \rho V_1^2[/tex]

Expressed from Bernoulli's equation we can identify that the greater the change that exists in pressure, fluid velocity will tend to decrease

The correct answer is B: "If we increase A2 then by the continuity equation the speed of the fluid should decrease. Bernoulli's equation then shows that if the velocity of the fluid decreases (at constant height conditions) then the pressure of the fluid should increase"