Understanding Fluid Dynamics: Free Surface of Liquid and Uniform Acceleration

[siddharth5129]

A container filled up to about 2/3rd's with water moves with a unifrom acceleration of 'a' due to a force 'F'. Considering the fact that the defining property of a fluid requires that it's free surface be normal to the net force acting on it, how can the liquid's free surface make an angle given by tan(theta) = g/a with the horizontal, when it should be completely perpendicular to the horizontal (as this is the direction in which the force is acting).

 

[boneh3ad]

That isn't the direction the force is acting. Remember there is also gravity acting straight down.

 

[siddharth5129]

That isn't the direction the force is acting. Remember there is also gravity acting straight down.

The NET force is in the horizontal direction. The weight of the fluid is balanced out by the normal force. Further, since there is no acceleration in the vertical direction, I think it is safe to say that there is no net force in that direction.

 

[boneh3ad]

There is no net force horizontally either then. The acceleration is balanced by the glass.

Trust me, there is still a gravity force and shill a force due to acceleration.

 

[nasu]

The NET force is in the horizontal direction. The weight of the fluid is balanced out by the normal force. Further, since there is no acceleration in the vertical direction, I think it is safe to say that there is no net force in that direction.

Then why is the surface horizontal when the container is not moving? According to you own post, the free surface is perpendicular to the net force. Maybe this statement needs a little refinement or revision? The net force on the fluid is zero for any fluid in equilibrium, isn't it?

 

[olivermsun]

There is no net force once the free surface is "tilted" to its "stable" position.

 

[siddharth5129]

There is no net force horizontally either then. The acceleration is balanced by the glass.

Trust me, there is still a gravity force and shill a force due to acceleration.

The liquid as a whole is accelerating to the right. Newton's second law says that the net force on the liquid can be along no direction other than towards the right. And I don't understand what you meant by 'the acceleration is balanced by the glass'. If you meant that the horizontal force on the liquid is exactly equal and opposite to the normal force on the liquid exerted by the glass container, then how does the liquid accelerate towards the right?

Then why is the surface horizontal when the container is not moving? According to you own post, the free surface is perpendicular to the net force. Maybe this statement needs a little refinement or revision? The net force on the fluid is zero for any fluid in equilibrium, isn't it?

Hmm... fair point. As far as the statement is concerned, it comes from my introductory fluid mechanics text which states that the defining property of a fluid is that it cannot sustain a static response to a shear stress. I am assuming that this amount's to the same thing as 'a fluid's free surface must be perpendicular to the net force acting on it.' I suppose the free surface is horizontal because any other orientation and there would be a shear stress on the free surface of the fluid, and it would flow until it became horizontal. Then my question becomes , how do you apply that principle to the scenario I have mentioned?

There is no net force once the free surface is "tilted" to its "stable" position.

I'm sorry, I don't understand this. The free surface is still accelerating to the right, so, tilted or not, there has to be a net force on it.

 

[Michael C]

The NET force is in the horizontal direction.

No, the net force is diagonal, since it is the result of the force of gravity and the force of the horizontal acceleration.

When the fluid is at rest in the container, the forces acting on it are:

1. Force of gravity.
2. Forces from the bottom and sides of the container that sum up to a force equal and opposite to that of gravity.

When you accelerate the container in a horizontal direction, this adds a horizontal force. If you add the vectors of the horizontal force due to acceleration and the vertical force due to gravity together, they are equivalent to a force pointing diagonally downwards. If the acceleration is constant, the liquid will come to a stable position where the surface is perpendicular to this diagonal force. At this point, the bottom and sides of the container are producing a net force that is equal and opposite to the diagonal force.

 

[boneh3ad]

The liquid as a whole is accelerating to the right. Newton's second law says that the net force on the liquid can be along no direction other than towards the right. And I don't understand what you meant by 'the acceleration is balanced by the glass'. If you meant that the horizontal force on the liquid is exactly equal and opposite to the normal force on the liquid exerted by the glass container, then how does the liquid accelerate towards the right?

Because you are completely ignoring gravity. The fluid is accelerating "right", but it is also being pulled down by the fluid. If it was only experiencing a force to the right, then the glass container would not hold it anymore because the surface would tend to be vertical. Clearly that doesn't happen. Just draw a free body diagram of the fluid. You'll see.

 

[olivermsun]

The free surface is still accelerating to the right, so, tilted or not, there has to be a net force on it.

In the frame of the container (which is the only frame which is in some sort of "steady" state), there is no net force. That's why it appears to be in a "steady" state, with a fixed tilt.

You may also think it in a fixed frame in which case the total force on the thing is m*(g + horizontal acceleration) as everyone seems to be telling you in the thread.