Relative Equilibrium

When a fluid-contained vessel moves with a constant acceleration it will be transmitted to the fluid. The fluid particles will move to a new position and remain in such position in equilibrium, relative to the vessel. Such equilibrium is known as the Relative Equilibrium of a fluid.

Relative Equilibrium under linear acceleration

No flow of the fluid (relative to the fluid particles). No shear forces, and all forces are normal to the surface they act on. Hence, fluid statics equations can be used in relative equilibrium.

Variation of pressure

Let .

Consider the fluid element containing point which is under an acceleration of in the directions.

By applying Newton’s second law of motion in all 3 directions:

Substituting all the terms:

Integrating both sides:

Shape of free surface

On the free surface because gauge pressure is considered.

Free surface is a plane surface in 3D.

Inclination with horizontal plane

Let a vessel be in acceleration in in directions. .

If are the angles in directions.

Differentiating the equation of the free surface with respect to .

Relative Equilibrium under Horizontal Acceleration

Equation of the free surface

Is a straight line in axes. The straight line is at an inclination of :

Vertical Pressure Distribution

Varies only in direction. Increases with height. Isobars are horizontal.

Relative Equilibrium under Vertical Acceleration

Equation of the free surface

Horizontal straight line.

Vertical Pressure Distribution

Here:

• - hydrostatic pressure
• - due to

Varies only in direction. Increases with height. Isobars are horizontal.