# Shape and orientation of liquid surface in accelerating rocket tank

I have a hard time explaining myself what would be the shape and the orientation of the liquid surface inside a rocket tank during acceleration (while the rocket motor are in operation). an example would be the surface of the Liquid oxygen inside the LOX tank of the Falcon 9 rocket first stage: is the surface perpendicular to the axis of the rocket (perpendicular to the thrust) or perpendicular to the relative acceleration (meaning including the gravity acceleration)?

The intuition from the ground would say that it is perpendicular to the relative acceleration. for example a car fuel tank when the car is linearly accelerating on an horizontal plane: the fluid surface tilts toward the direction of the acceleration and settle at an angle $\theta$ whose tangent is equal to the ratio of the horizontal acceleration over the gravity: $\tan(\theta) = \frac{a}{g}$

However, the rocket itself is submitted to the gravity, without any support to counteract it: if we stop the motor, the rocket is in free-fall (if we discard the air friction). So if we start our reasoning from the free-fall state, the fluid inside the tank doesn't see any forces acting on it. it is in a state of weightlessness in the tank and without any perturbation will stay where it is (with respect to the tank). Now if we apply the thrust, this is the only force that the liquid will see and will thus settle perpendicularly to this force, without any impact from the gravity. if this reasoning is correct, it means that the previous intuitive example of the car doesn't apply here. Most likely because the gravity is taken into account somewhere else.

I strongly think the right answer is the 2nd case, meaning the fluid will settle perpendicular to the axis of the rocket. I just cannot get around the equations to make up my mind correctly.

Does anybody could help me settle this problem? maybe with other examples or even the equations of the movement (that would be the best! I'm a bit rusted with non inertial frame...)?

Bastien

• This rocket you're imagining - in what direction is it accelerating? – M. Enns Aug 17 '16 at 14:12
• In whatever direction, this should not make any change: either the gravity must be taken into account, in which case the fluid will be perpendicular to the relative acceleration, the vector addition of the two vectors thrust and gravity, or the fluid remains perpendicular to the thrust, in whichever direction the thrust is directed, with the gravity taken into account somewhere else. – BHaemmer Aug 18 '16 at 6:42

You are right to consider the rocket-fuel system as in free-fall.
The moment the rocket leaves the ground, the only force that dictates the positioning of the fuel relative to the rocket is the engine's thrust. The only reason (as you explained) that the fuel sits flat on the ground is because it is being pulled down towards the earth and is forced to lie flat because it cannot move the rocket out of the way as the earth is holding it up.

The thing to consider here is the relative forces acting between the fuel and rocket, not the acceleration.

From the reference frame of the rocket, the moment it leaves the ground suddenly the acceleration that was pushing it up has increased. The thrust of the rocket replaces the electrostatic (contact) force and then some extra to enable a net acceleration. So long as the rocket was vertical to begin with, the direction of the net force has not changed.

The electrostatic force is what stops the rocket from falling through the ground, and as such is equal to the force of gravity. To overcome gravity, the rocket must first reach a thrust that can cancel gravity, this is the replacing part, and then to accelerate the rocket it must be come greater than gravity.

Thus even as the rocket turns over the fuel must remain perpendicular to the direction of thrust.

Edit: Tried to work out the mathematics for it. The easiest way is to show that the only way for the liquid to be disturbed within the rocket is for the fuel tank to actually push against it. As both the fuel and rocket have the same acceleration due to gravity, there is no way for gravity to press the tank and fuel together. Only the thrust which accelerates the rocket, and through contact forces, accelerates the fuel can have an effect on it's shape/orientation. Therefore when considering it's shape the only force that should be considered is the one that moves the fuel by moving the rocket against the fuel.

When it's on the ground you can think of the ground as pushing the rocket upwards towards the fuel.

Incidentally the equation for this contact force is: F(rocket on fuel)=Ma
Where, a is the acceleration of the rocket and fuel combined due to the thrust, and M is the mass of the fuel.