Relative Equilibrium of Liquids

Analysis of fluids subjected to uniform acceleration where fluid particles remain at rest relative to the container.

Concept of Relative Equilibrium

When a vessel containing a liquid is at rest or moving with constant velocity, the liquid is in static equilibrium. The surface is horizontal, and the pressure depends only on the depth.
However, if the vessel is subjected to a uniform acceleration (linear or rotational), the liquid surface will adjust to a new position. Once the liquid surface stabilizes and there is no longer any relative motion between fluid particles or between the fluid and the vessel, the liquid is said to be in relative equilibrium.

Uniform Linear Acceleration

Fluids in a container accelerating in a straight line.

Horizontal Acceleration

When a container moves with a constant horizontal acceleration aa, the liquid surface inclines downwards in the direction of acceleration. The angle of inclination θ\theta with the horizontal is determined by the ratio of the horizontal acceleration to the acceleration due to gravity gg.

Horizontal Acceleration

Fluids in a container accelerating in a straight line.

tan⁡θ=ag\tan \theta = \frac{a}{g}

Vertical Acceleration

When a container moves with a constant vertical acceleration ava_v (upwards or downwards), the liquid surface remains horizontal, but the pressure within the fluid changes.
For upward acceleration (+av+a_v):

Vertical Acceleration

P=γh(1+avg)P = \gamma h \left(1 + \frac{a_v}{g}\right)
For downward acceleration (−av-a_v):
P=γh(1−avg)P = \gamma h \left(1 - \frac{a_v}{g}\right)
If the container is falling freely (av=ga_v = g downwards), the pressure at any depth is zero (weightlessness).

Note

In problems involving both horizontal and vertical acceleration, you can analyze them by combining their effects vectorially. The slope of the surface becomes tan⁡θ=ahg±av\tan \theta = \frac{a_h}{g \pm a_v}.

Relative Equilibrium Simulator

0.0 m/s²
0.0 m/s²

-9.81 m/s² represents free-fall (weightlessness).

Rotation (Forced Vortex)

Fluids in a container rotating at a constant angular velocity.

Paraboloid of Revolution

When a cylindrical container filled with liquid is rotated at a constant angular velocity ω\omega (rad/s) about its vertical axis, the liquid surface forms a curved shape called a paraboloid of revolution. The volume of the paraboloid is half the volume of its circumscribing cylinder.

Height of the Paraboloid (y)

The elevation yy of the liquid surface at any radial distance xx from the axis of rotation is given by:

Height of the Paraboloid (y)

Fluids in a container rotating at a constant angular velocity.

y=ω2x22gy = \frac{\omega^2 x^2}{2g}
At the edge of the container of radius rr, the maximum height hh is:
h=ω2r22gh = \frac{\omega^2 r^2}{2g}

Acceleration on an Inclined Plane

Fluids in a container moving along a sloped path.

Inclined Acceleration

When a container accelerates up or down an inclined plane making an angle α\alpha with the horizontal at acceleration aa, we resolve aa into horizontal (axa_x) and vertical (aya_y) components:
  • ax=acos⁡αa_x = a \cos\alpha
  • ay=asin⁡αa_y = a \sin\alpha
The angle θ\theta that the liquid surface makes with the horizontal is then found by combining the effects of both components.

Inclined Acceleration

Fluids in a container moving along a sloped path.

tan⁡θ=axg±ay\tan \theta = \frac{a_x}{g \pm a_y}
Use +ay+a_y if the container is accelerating upwards, and −ay-a_y if accelerating downwards.

Pressure Variation in Rotating Fluids

Pressure at any point

For a fluid rotating at constant angular velocity ω\omega, the pressure at any radial distance rr from the axis of rotation and depth zz below the free surface is given by the combination of hydrostatic pressure and the centrifugal effect.

Pressure at any point

P=γω2r22g−γzP = \frac{\gamma \omega^2 r^2}{2g} - \gamma z
Where zz is measured positively downwards from the vertex of the paraboloid (lowest point of the free surface).

Closed Tanks in Rotation

If a closed tank completely filled with liquid is rotated, a paraboloid of revolution still governs the pressure distribution, but the "free surface" becomes an imaginary surface located above the tank.
  • The pressure at the top center of the tank is determined by the fluid expansion (if any) or initial pressurization.
  • If there is a small air space, the real free surface forms a paraboloid within that space, and the imaginary paraboloid extends outwards from there.
Key Takeaways
  • In relative equilibrium, there is no relative motion between fluid particles; the fluid acts as a rigid body.
  • For linear horizontal acceleration aa, the fluid surface slopes at an angle tan⁡θ=a/g\tan \theta = a/g.
  • For vertical acceleration, the pressure distribution changes: P=γh(1±av/g)P = \gamma h (1 \pm a_v/g).
  • For rotation at angular velocity ω\omega, the fluid surface forms a paraboloid of revolution with height y=(ω2x2)/(2g)y = (\omega^2 x^2) / (2g).
  • The volume of a paraboloid of revolution is half the volume of its circumscribing cylinder.
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