Relative Equilibrium of Liquids - Theory & Concepts

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 are still analyzable as static relative to the accelerating container.

Horizontal Acceleration

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

Horizontal Acceleration Formula

Calculates the angle of inclination of the liquid surface under horizontal acceleration.

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

Variables

Symbol	Description	Unit
	Angle of inclination of the liquid surface with the horizontal	rad or degrees
$a$	Constant horizontal acceleration of the container	$m / s^{2}$
$g$	Acceleration due to gravity	$m / s^{2}$

Vertical Acceleration

When a container moves with a constant vertical acceleration $a_v$ (upwards or downwards), the liquid surface remains horizontal, but the pressure within the fluid changes.

If the container is falling freely ( $a_v = g$ downwards), the pressure at any depth is zero (weightlessness).

Pressure under Upward Vertical Acceleration

Calculates pressure in a fluid subjected to constant upward vertical acceleration.

P = \gamma h \left(1 + \frac{a_v}{g}\right)

Variables

Symbol	Description	Unit
$P$	Pressure at depth h	Pa
	Specific weight of the liquid	$N / m^{3}$
$h$	Depth below the free surface	m
$a_{v}$	Constant upward vertical acceleration	$m / s^{2}$
$g$	Acceleration due to gravity	$m / s^{2}$

Pressure under Downward Vertical Acceleration

Calculates pressure in a fluid subjected to constant downward vertical acceleration.

P = \gamma h \left(1 - \frac{a_v}{g}\right)

Variables

Symbol	Description	Unit
$P$	Pressure at depth h	Pa
	Specific weight of the liquid	$N / m^{3}$
$h$	Depth below the free surface	m
$a_{v}$	Constant downward vertical acceleration	$m / s^{2}$
$g$	Acceleration due to gravity	$m / s^{2}$

Note

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

Rotation (Forced Vortex)

Fluids in a container rotating at a constant angular velocity behave like a rigid body after relative motion dies out.

Paraboloid of Revolution

When a cylindrical container filled with liquid is rotated at a constant angular velocity $\omega$ (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 $y$ of the liquid surface at any radial distance $x$ from the axis of rotation depends on the angular velocity. At the edge of the container of radius $r$ , the liquid rises to a maximum height $h$ .

Height of the Paraboloid

Calculates the elevation of the liquid surface at any radial distance from the axis of rotation.

y = \frac{\omega^2 x^2}{2g}

Variables

Symbol	Description	Unit
$y$	Elevation of the liquid surface at distance x	m
	Constant angular velocity of rotation	rad/s
$x$	Radial distance from the axis of rotation	m
$g$	Acceleration due to gravity	$m / s^{2}$

Maximum Height of the Paraboloid

Calculates the maximum height of the paraboloid at the container's outer radius.

h = \frac{\omega^2 r^2}{2g}

Variables

Symbol	Description	Unit
$h$	Maximum height of the paraboloid at the outer edge	m
	Constant angular velocity of rotation	rad/s
$r$	Radius of the cylindrical container	m
$g$	Acceleration due to gravity	$m / s^{2}$

Acceleration on an Inclined Plane

Fluids in a container moving along a sloped path respond to both horizontal and vertical acceleration components.

Inclined Acceleration

When a container accelerates up or down an inclined plane making an angle $\alpha$ with the horizontal at acceleration $a$ , we resolve $a$ into horizontal ( $a_x$ ) and vertical ( $a_y$ ) components:

$a_x = a \cos\alpha$
$a_y = a \sin\alpha$

The angle $\theta$ that the liquid surface makes with the horizontal is then found by combining the effects of both components. Use $+a_y$ if the container is accelerating upwards, and $-a_y$ if accelerating downwards.

Inclined Acceleration Formula

Calculates the angle of the liquid surface when the container accelerates along a sloped path.

\tan \theta = \frac{a_x}{g \pm a_y}

Variables

Symbol	Description	Unit
	Angle the liquid surface makes with the horizontal	rad or degrees
$a_{x}$	Horizontal component of acceleration	$m / s^{2}$
$a_{y}$	Vertical component of acceleration	$m / s^{2}$
$g$	Acceleration due to gravity	$m / s^{2}$

Pressure at any point

For a fluid rotating at constant angular velocity $\omega$ , the pressure at any radial distance $r$ from the axis of rotation and depth $z$ below the free surface is given by the combination of hydrostatic pressure and the centrifugal effect, where $z$ is measured positively downwards from the vertex of the paraboloid (lowest point of the free surface).

Pressure at any Point in Rotating Fluid

Calculates fluid pressure at a given radial distance and depth from the vertex in a rotating container.

P = \frac{\gamma \omega^2 r^2}{2g} - \gamma z

Variables

Symbol	Description	Unit
$P$	Pressure at the specified point	Pa
	Specific weight of the liquid	$N / m^{3}$
	Constant angular velocity of rotation	rad/s
$r$	Radial distance from the axis of rotation	m
$z$	Vertical distance measured from the vertex of the paraboloid	m
$g$	Acceleration due to gravity	$m / s^{2}$

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 $a$ , the fluid surface slopes at an angle $\tan \theta = a/g$ .
For vertical acceleration, the pressure distribution changes: $P = \gamma h (1 \pm a_v/g)$ .
For rotation at angular velocity $\omega$ , the fluid surface forms a paraboloid of revolution with height $y = (\omega^2 x^2) / (2g)$ .
The volume of a paraboloid of revolution is half the volume of its circumscribing cylinder.

PreviousHydrostatics: Buoyancy & Stability - Examples & Applications

Quiz Me

NextRelative Equilibrium of Liquids - Examples & Applications