Plate floating on oil - linear momentum equation

[Pietair]

Homework Statement

1: Determine the wall shear stress that acts at the lower side of the plate.
2: Determine the force Fx that is needed to give the plate a speed of u = 1m/s.
3: Determine the speed V, that is leaves an air jet that blows against the plate and which creates the same force Fx.

Question 1 and 2 were easy to solve, but I have got a problem with the third question.

Homework Equations

Linear momentum equation: $$\sum \vec{V}_{out} \rho_{out} A_{out} V_{out}-\sum \vec{V}_{in} \rho_{in} A_{in} V_{in}=\sum F$$

The Attempt at a Solution

The left part of the linear momentum equation is easy to find. However, I don't understand which forces I have to include when applying the linear momentum equation in the x-direction.
I know that I have to include a force -Fx, which is the force exerted by the plate on the control volume (which is the reaction force of the force exerted by the fluid on the plate).
But in the answer sheet they do not include the pressure force exerted on the left side of the control volume, which I think should be P_{atm} * the left area of the control volume (because the pressure in the air jet is simply P_{atm}. Why isn't this pressure force on the control volume included?

Thank you in advance.

 

[anathema]

Thank you for your post. It seems like you are on the right track with your solution for the third question. However, there are a few things to consider when applying the linear momentum equation in this scenario.

Firstly, you are correct in including the force -Fx in the equation as it is the reaction force of the fluid on the plate. However, you also need to consider the force exerted by the fluid on the control volume, which is equal to the force exerted by the plate on the control volume (-Fx). This is because of Newton's third law of motion.

Secondly, you are also correct in considering the pressure force exerted on the left side of the control volume. However, this force is already accounted for in the linear momentum equation as it is included in the term \sum \vec{V}_{in} \rho_{in} A_{in} V_{in}. This term represents the momentum of the fluid entering the control volume, which includes the pressure force.

In summary, when applying the linear momentum equation, you need to consider the force exerted by the fluid on the control volume (-Fx) and the momentum of the fluid entering the control volume, which includes the pressure force. I hope this helps clarify your doubts. Good luck with your calculations!