Control volume, Fluid mechanics

[Kqwert]

Homework Statement

I have a situation as illustrated by the image above, where the red box illustrates the CV. The pressure on the left side of CV is P1, while the pressure on the right side of CV is P2. Here I´ve directed the pressure forces acting inwards on CV. The problem arises with Fk. I am interested in the force acting from the water ON the wall. Sum of forces in x-direction:

Fsigma = P1*A1 + Fk - P2*A2.

The answer manual however want´s Fk to be negative. Where am I wrong?

 

[Chestermiller]

Homework Statement

View attachment 225725

I have a situation as illustrated by the image above, where the red box illustrates the CV. The pressure on the left side of CV is P1, while the pressure on the right side of CV is P2. Here I´ve directed the pressure forces acting inwards on CV. The problem arises with Fk. I am interested in the force acting from the water ON the wall, i.e. my force balance in the x-direction becomes:

P1*A1 + Fk - P2*A2.

The anser manual however want´s Fk to be negative. Where am I wrong?

I don't see an equal sign in your force balance.

 

[Kqwert]

I don't see an equal sign in your force balance.

I am sorry, bad formulation. I have edited my post now.

 

[Chestermiller]

According to your diagram and equation, ##F_k## is the force exerted by the CV on the fluid in the positive x direction. Thus a negative value indicates that the force exerted by the CV on the fluid is actually in the negative x direction. Thus, by Newton's 3rd law, the force exerted by the fluid on the CV is in the positive x direction.

 

[Kqwert]

According to your diagram and equation, ##F_k## is the force exerted by the CV on the fluid in the positive x direction. Thus a negative value indicates that the force exerted by the CV on the fluid is actually in the negative x direction. Thus, by Newton's 3rd law, the force exerted by the fluid on the CV is in the positive x direction.

So it´s the CV exerting a force on the fluid and not the fluid exerting a force on the walls inside the CV?

 

[Chestermiller]

So it´s the CV exerting a force on the fluid and not the fluid exerting a force on the walls inside the CV?

You're aware of Newton's 3rd law, correct?

In your diagram, with the vector arrows drawn the why you have shown them, what are you calling Fk?
(a). the force exerted by the control volume on the fluid or
(b). the force exerted by the fluid on the control volume

 

[Kqwert]

In your diagram, with the vector arrows drawn the why you have shown them, what are you calling Fk?
(a). the force exerted by the control volume on the fluid or
(b). the force exerted by the fluid on the control volume

Yes I am. The way I have drawn Fk I mean alternative (b)

 

[Chestermiller]

Yes I am. The way I have drawn Fk I mean alternative (b)

In that case, the force in the positive x direction exerted by the control volume on the fluid is -Fk. So the macroscopic balance on the fluid should read: $$P_1A_1-F_k-P_2A_2=\dot{m}(v_2-v_1)$$where ##\dot{m}## is the mass flow rate, given by:
$$\dot{m}=\rho v_1A_1=\rho v_2A_2$$The right hand side of this equation represents the rate of change of momentum of the fluid within the control volume.

 

[Kqwert]

In that case, the force in the positive x direction exerted by the control volume on the fluid is -Fk. So the macroscopic balance on the fluid should read: $$P_1A_1-F_k-P_2A_2=\dot{m}(v_2-v_1)$$where ##\dot{m}## is the mass flow rate, given by:
$$\dot{m}=\rho v_1A_1=\rho v_2A_2$$The right hand side of this equation represents the rate of change of momentum of the fluid within the control volume.

Thank you. Just to be sure: by "force exerted by control volume on fluid", would that be the same as the force from the wall (inside the CV) on the fluid?

 

[Chestermiller]

Thank you. Just to be sure: by "force exerted by control volume on fluid", would that be the same as the force from the wall (inside the CV) on the fluid?

Sure.

 

[Ray Vickson]

Homework Statement

View attachment 225725

I have a situation as illustrated by the image above, where the red box illustrates the CV. The pressure on the left side of CV is P1, while the pressure on the right side of CV is P2. Here I´ve directed the pressure forces acting inwards on CV. The problem arises with Fk. I am interested in the force acting from the water ON the wall. Sum of forces in x-direction:

Fsigma = P1*A1 + Fk - P2*A2.

The answer manual however want´s Fk to be negative. Where am I wrong?

What is a "CV"?

 

[Chestermiller]

What is a "CV"?

CV stands for”control volume.” It is usually a specified section of a device through which a fluid is flowing.

 

[Kqwert]

In that case, the force in the positive x direction exerted by the control volume on the fluid is -Fk. So the macroscopic balance on the fluid should read: $$P_1A_1-F_k-P_2A_2=\dot{m}(v_2-v_1)$$where ##\dot{m}## is the mass flow rate, given by:
$$\dot{m}=\rho v_1A_1=\rho v_2A_2$$The right hand side of this equation represents the rate of change of momentum of the fluid within the control volume.

Just to be completely sure.

In the example I have given, where Fk is pointing in the positive x direction. If we solve for Fk and it is positive, we have found the force from the fluid ON the wall. If this Fk turns out to be negative, we have found the force from the wall ON the fluid. By Newton´s 3rd law the force from the fluid would then be equally large in the positive direction and be positive. Is my reasoning correct?

 

[Chestermiller]

Just to be completely sure.

In the example I have given, where Fk is pointing in the positive x direction. If we solve for Fk and it is positive, we have found the force from the fluid ON the wall. If this Fk turns out to be negative, we have found the force from the wall ON the fluid. By Newton´s 3rd law the force from the fluid would then be equally large in the positive direction and be positive. Is my reasoning correct?

The first thing we need to do is declare at the outset whether the force vector we have drawn in the figure is the force exerted by the wall on the fluid or the force exerted by the fluid on the wall. These constitute an action-reaction pair. If you are doing a macroscopic momentum balance on the fluid, it would be better to show only those forces that are acting on the fluid (not on the wall). Then, if you next want to do a force balance on the wall, it would be better to have a second diagram showing only those forces that are acting on the wall (not on the fluid). This will help you avoid any ambiguity. After all, this is the procedure we used in freshman physics when we did force balances using free body diagrams.