1.0. RELATIVE EQUILIBRIUM OF LIQUIDS

The last discussion of hydrostatics is the preliminary topic for hydraulics. Although the fluid system experiences motion, this is still considered a part of hydrostatics with the reason that the whole system is moving as a unit and the liquid particles are still in equilibrium or at rest.

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There are two kinds of relative equilibrium:

1. Rectilinear acceleration where the vessel moves either horizontally, vertically or along an incline.

2. Curvilinear acceleration where the vessel rotates.

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1.1. TRANSLATION (RECTILINEAR ACCELERATION)

A. HORIZONTAL ACCELERATION

Considering a water tank mounted on a vehicle, if the vehicle moves horizontally, vertically or diagonally, the particles of the fluid are not in motion but still are affected by the motion of the whole vessel.

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If we take one fluid particle, the forces affecting it are:

• Its weight which is always applied vertically downwards
• The normal force applied perpendicular to the fluid surface
• The reverse effected force (REF) which is influenced by the motion of the vehicle.

From these forces, we are able to form a force triangle where we can derive our equation.

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The angle of fluid inclination can be solved as:

                           





Example 1.1.1. Relative  Equilibrium on horizontal acceleration
Example 1.1.2. Relative Equilibrium on horizontal acceleration
Example 1.1.3. Relative Equilibrium on horizontal acceleration


B. VERTICAL ACCELERATION

When fluid vessels are transported vertically, the forces acting on a fluid particle will only be vertical.

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Equating forces in equilibrium:






Converting m to W to create similar terms:





Simplifying the volume into area and height:

    

      

Where + is considered for upward motion and - for downward motion OR (+) for acceleration and (-) for deceleration.



Example 1.1.4. Relative  Equilibrium on vertical acceleration
Example 1.1.5. Relative  Equilibrium on vertical acceleration
Example 1.1.6. Relative  Equilibrium on vertical acceleration



C. INCLINED MOTION

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In inclined motion, the angle of surface inclination, , is different from the fluid inclination, .

From the force triangle, we can deduce the formula for the surface inclination and fluid inclination:

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Using the right triangle:

From the surface inclination triangle, mass will stay constant and the only parameter having components is the acceleration.





Thus, the formula can be simplified as:





Considering (+) for upward motion and (-) for downward motion.


Example 1.1.7. Relative  Equilibrium on inclined acceleration
Example 1.1.8. Relative  Equilibrium on inclined acceleration
Example 1.1.9. Relative  Equilibrium on inclined acceleration

1.2. ROTATION

When a vessel rotates, fluid particles also move with the motion.