What is Relative Equilibrium of Fluids?

Relative equilibrium occurs when a fluid mass moves as a rigid body – meaning there’s no relative motion between fluid particles, even though the entire fluid mass may be accelerating or rotating. Think of it as the fluid “going along for the ride” with its container without sloshing around internally.

What you’ll learn

Types of Relative Equilibrium of Fluids

Linear Acceleration
– Horizontal Motion
– Vertical Motion
Rotational Motion

Horizontal Motion

Solve for ACCELERATION:

$tan \theta =\frac{a}{g}$

where:
$\theta$ = angle of free surface
a = horizontal acceleration $(m/s^2)$
g = gravitational acceleration $(9.81 m/s^2)$

Solve for FORCE:

$F = ma$

where:
a = acceleration
m = mass of the tank + mass of the fluid

Inclined Motion

Solve for ACCELERATION:

$tan \theta = \frac{a \: cos\alpha}{g \pm a \: sin \alpha}$

Vertical Motion

In a closed tank:

Rotational Motion

Slope of the Fluid Surface:

$tan \theta = \frac{\omega^2x}{g}$

$\frac{dy}{dx} = \frac{\omega^2 x}{g}$

$y_p = \frac{\omega^2x^2}{2g}$

$y = \frac{\omega^2r^2}{2g}$

Paraboloid Equation:

$y = \frac{\omega^2 r^2}{2g}$

where:
y = Height above the lowest point of the parabola
$\omega$ = Angular velocity (rad/s)
r = Radial distance from rotation axis
g = Gravitational Acceleration (9.81 m/s^2)