Relative Equilibrium of Liquids

Relative Equilibrium of Liquids

$Learning Objectives$

Explain relative equilibrium in a non-inertial container frame and define effective gravity for translational acceleration.
Determine pressure gradients and free-surface slope under horizontal, vertical, and combined linear acceleration.
Distinguish no-spill, spill, and loss-of-contact regimes before applying rectangular-tank formulas.
Explain the free-fall limit without confusing zero pressure gradient with zero absolute pressure.
Derive the forced-vortex pressure gradients and parabolic free surface for rigid-body rotation.
Apply volume conservation, incipient-spill, angular-speed conversion, and rotating-pressure-field relations with explicit coordinate conventions.
Recognize when open-tank geometry, dry-core formation, confinement, cavitation, or other physical limits invalidate the elementary idealization.

$Relative Equilibrium$

A liquid is in relative equilibrium when, after transients decay, it has no motion relative to its translating or rotating container. In the container frame the liquid behaves as a rigid body, so hydrostatic-type pressure relations can be written using the appropriate effective body-force field.

$\vec g_{\text{eff}}$

For a container translating with acceleration $\vec a_0$ , effective gravity is the gravitational acceleration vector combined with the opposite inertial acceleration perceived in the accelerating container frame.

Effective Gravity for Translational Acceleration

Defines the effective body-force acceleration in a frame attached to a translating container.

\vec g_{\text{eff}}=\vec g-\vec a_0

Variables

Symbol	Description	Unit
$\vec g_{\text{eff}}$	Effective gravity vector in the container frame	$m/s^2$
$\vec g$	True gravitational acceleration vector	$m/s^2$
$\vec a_0$	Container translational acceleration vector	$m/s^2$

Pressure Gradient in Translational Relative Equilibrium

Relates static pressure gradient in the container frame to effective gravity.

\nabla p=\rho\vec g_{\text{eff}}

Variables

Symbol	Description	Unit
$p$	Fluid pressure	Pa
$\rho$	Fluid density	$kg/m^3$
$\vec g_{\text{eff}}$	Effective gravity vector	$m/s^2$

Effective gravity in an accelerating frame

Gravity and frame acceleration define the effective body-force direction.

$Free Surfaces Are Equipressure Surfaces$

For an open liquid free surface exposed to one atmosphere, pressure is constant along the surface. Its tangent direction is therefore perpendicular to $\nabla p$ and to $\vec g_{\text{eff}}$ whenever $\vec g_{\text{eff}}\ne0$ .

$Horizontal Acceleration$

Let $x$ be positive in the direction of a constant horizontal container acceleration $a_x$ , and let $z$ be positive upward. Effective gravity then has a component opposite the acceleration. Pressure decreases in the positive $x$ direction, so the free surface rises at the rear and falls at the front.

Free surface under horizontal acceleration

The free surface tilts normal to effective gravity.

Pressure Gradients under Horizontal Acceleration

Gives the horizontal and vertical pressure gradients for constant horizontal acceleration.

\frac{\partial p}{\partial x}=-\rho a_x, \qquad \frac{\partial p}{\partial z}=-\rho g

Variables

Symbol	Description	Unit
$a_x$	Horizontal acceleration magnitude in the positive x direction	$m/s^2$
$x$	Horizontal coordinate	m
$z$	Vertical coordinate positive upward	m

Free-Surface Slope under Horizontal Acceleration

Determines the plane free-surface slope while the liquid remains in contact with the tank and before spilling changes the volume.

\frac{dz}{dx}=-\frac{a_x}{g}

\tan\theta=\frac{|a_x|}{g}

Variables

Symbol	Description	Unit
$\theta$	Magnitude of free-surface angle from horizontal	degrees or rad
$a_x$	Horizontal container acceleration	$m/s^2$
$g$	Gravitational acceleration magnitude	$m/s^2$

End-to-End Surface Difference in a Rectangular Tank

Relates horizontal acceleration to free-surface elevation difference over tank length L.

\Delta h=\frac{|a_x|L}{g}

Variables

Symbol	Description	Unit
$\Delta h$	Difference in free-surface elevation between the tank ends	m
$L$	Inside tank length measured parallel to the horizontal acceleration	m

$No-Spill End Depths$

For a rectangular tank with initial uniform depth $h_0$ and no volume loss, the mean depth remains $h_0$ . If the plane free surface remains inside the tank and liquid contact is maintained, the raised and lowered end depths are

h_{\text{high}}=h_0+\frac{\Delta h}{2}, \qquad h_{\text{low}}=h_0-\frac{\Delta h}{2}

These formulas must not be continued after the raised end crosses the rim or the lowered end predicts a negative depth.

$Check Geometry before Using No-Spill Formulas$

First compare $h_{\text{high}}$ with tank height and $h_{\text{low}}$ with zero. If the high side reaches the rim, liquid spills and the remaining volume must be recomputed with the rim as a boundary. If the low-side depth reaches zero, the liquid loses contact with part of the tank bottom and a different geometric regime begins.

$Vertical Acceleration$

A purely vertical acceleration leaves the free surface horizontal while changing the effective downward acceleration. Upward container acceleration increases apparent gravity to $g+a_v$ . Downward acceleration less than $g$ reduces apparent gravity to $g-a_v$ .

Vertical acceleration and pressure gradient

Vertical frame acceleration changes effective gravity and pressure gradient.

Pressure under Upward Acceleration

Computes gage pressure at depth h below an open free surface during uniform upward acceleration.

p_g=\rho(g+a_v)h

Variables

Symbol	Description	Unit
$p_g$	Gage pressure relative to the open free surface	Pa
$a_v$	Upward container acceleration magnitude	$m/s^2$
$h$	Depth below the free surface	m

Pressure under Downward Acceleration

Computes gage pressure at depth h while downward container acceleration remains less than g and contact is maintained.

p_g=\rho(g-a_v)h

Variables

Symbol	Description	Unit
$p_g$	Gage pressure relative to the open free surface	Pa
$a_v$	Downward container acceleration magnitude	$m/s^2$
$h$	Depth below the free surface	m

$Free Fall Means Zero Pressure Gradient, Not Zero Absolute Pressure$

At downward acceleration $a_v=g$ , $\vec g_{\text{eff}}=0$ and therefore $\nabla p=0$ . Pressure becomes spatially uniform within a connected liquid under the idealization. Its value is still set by the applicable boundary or confinement condition. If a connected boundary fixes pressure at $95\ \text{kPa}$ absolute, the ideal free-falling liquid is uniformly at $95\ \text{kPa}$ absolute—not at zero absolute pressure. An open atmospheric boundary would similarly set the uniform liquid pressure to atmospheric pressure while the boundary remains physically connected.

$Downward Acceleration Greater than Gravity$

When a nominally open container accelerates downward faster than $g$ , effective gravity points upward in the container frame. The ordinary assumption that liquid remains on the bottom with a conventional upper free surface may fail. Use the simple formula only when confinement and contact conditions actually support the assumed geometry.

Combined Horizontal and Vertical Acceleration

Determines free-surface angle using the magnitude of the effective downward component.

\tan\theta=\frac{|a_x|}{g_{\text{eff}}}

Variables

Symbol	Description	Unit
$g_{\text{eff}}$	Positive effective downward acceleration, equal to g+a_v upward or g-a_v downward while contact is maintained	$m/s^2$
$a_x$	Horizontal acceleration magnitude	$m/s^2$
$\theta$	Free-surface angle magnitude	degrees or rad

$Interactive Relative-Equilibrium Exploration$

Use the simulation to vary translational acceleration and rotation. Treat the display as a visualization of the governing equations and separately verify whether the selected state would spill, uncover the bottom, or violate the pressure assumptions.

Relative Equilibrium Simulator

Learning objective: Observe how translational acceleration or rigid-body rotation changes effective gravity, free-surface geometry, and the pressure field.

Open tank: 2.0 m long, 1.8 m high, initially filled to 1.0 m.

Horizontal acceleration (

a_x

)0.0 m/s²

Vertical acceleration (

a_v

)0.00 m/s²

Positive is upward. At −9.81 m/s² the tank is in free fall.

Effective downward gravity9.81 m/s²

Surface angle0.00°

End-to-end elevation difference0.000 m

Left / right depth1.000 / 1.000 m

The free surface is perpendicular to effective gravity. It falls in the direction of horizontal acceleration, with $\left|dz/dx\right|=|a_x|/(g+a_v)$ .

$Forced Vortex$

A forced vortex is rigid-body rotation of a liquid at a common angular velocity after transients decay under an imposed rotating container or equivalent forcing.

Pressure Gradients in Rigid-Body Rotation

Gives radial and vertical pressure gradients for rotation about a vertical axis.

\frac{\partial p}{\partial r}=\rho\omega^2r, \qquad \frac{\partial p}{\partial z}=-\rho g

Variables

Symbol	Description	Unit
$r$	Radial distance from the rotation axis	m
$z$	Vertical coordinate positive upward	m
$\omega$	Angular velocity	rad/s

$Parabolic Free Surface$

In rigid-body rotation about a vertical axis, an open liquid free surface is a paraboloid of revolution because constant-pressure points satisfy a quadratic relation between elevation and radius.

Parabolic Free-Surface Equation

Gives free-surface elevation relative to the vertex on the rotation axis.

z-z_0=\frac{\omega^2r^2}{2g}

Variables

Symbol	Description	Unit
$z$	Free-surface elevation at radius r	m
$z_0$	Free-surface elevation at the axis	m
$r$	Radial coordinate	m
$\omega$	Angular velocity	rad/s

Forced-vortex free surface

Rigid-body rotation produces a parabolic free surface.

Center-to-Wall Elevation Difference

Evaluates the free-surface elevation difference from axis to wall in a cylindrical tank.

\Delta h=\frac{\omega^2R^2}{2g}

Variables

Symbol	Description	Unit
$\Delta h$	Wall elevation minus center elevation	m
$R$	Tank radius	m
$\omega$	Angular velocity	rad/s

$Interactive 3D Relative Equilibrium & Rotating Paraboloid$

Manipulate linear acceleration or rotating angular velocity in 3D. Observe the 3D paraboloid of revolution, verify volume conservation ( $h_{\text{wall}} = h_0 + \Delta h / 2$ and $h_{\text{center}} = h_0 - \Delta h / 2$ ), and test for incipient spillover.

Relative Equilibrium Fluid Surfaces 3D

Interactive 3D free surfaces, isobaric layers, linear translation tilt, and centrifugal paraboloid of revolution.

Loading 3D Relative Equilibrium Simulation…

Isobaric Surfaces (Equal Pressure)Acceleration / Velocity Vectors

Linear Acceleration Parameters

Horizontal Accel (

a_x

)2.5 m/s²

Vertical Accel (

a_z

)0.0 m/s²

Rest Water Depth (

d_0

)1.20 m

Tank Length (

L = 2R

)2.40 m

Surface Tilt

\theta

14.3°

Rear Wall Depth1.51 m

Front Wall Depth0.89 m

Effective Gravity9.81 m/s²

Spill StatusContained

Floor ExposureFully Submerged

Angular-Speed Conversion

Converts rotational speed N in revolutions per minute to angular velocity.

\omega=\frac{2\pi N}{60}

Variables

Symbol	Description	Unit
$N$	Rotational speed	rpm
$\omega$	Angular velocity	rad/s

$No-Spill Volume Conservation in a Cylindrical Tank$

While the paraboloidal surface remains entirely within the tank and the vertex stays above the bottom, volume conservation keeps the plan-area average liquid level equal to the original depth. Because the area-average of $r^2$ over a circle is $R^2/2$ , the center falls by $\Delta h/2$ and the wall rises by $\Delta h/2$ .

No-Spill Center and Wall Depths

Relates initial depth to the axis and wall depths for a fully wetted rotating cylindrical tank before spilling.

h_c=h_0-\frac{\Delta h}{2}, \qquad h_w=h_0+\frac{\Delta h}{2}

Variables

Symbol	Description	Unit
$h_c$	Liquid depth at the rotation axis	m
$h_w$	Liquid depth at the tank wall	m
$h_0$	Initial uniform liquid depth	m

$Spill and Dry-Core Regimes Must Be Treated Separately$

If $h_w$ reaches the rim, further rotation causes spillage and the wall elevation becomes constrained by tank height; remaining volume must be recomputed. If the calculated axis depth $h_c$ reaches zero, the paraboloid intersects the tank bottom and a dry central region can form. The simple symmetric rise/drop relation no longer applies beyond either transition.

Rotating-Liquid Pressure Field with Upward-Positive Elevation

Gives gage pressure relative to an atmospheric free-surface vertex at r=0, z=0.

p_g=\rho\left(\frac{\omega^2r^2}{2}-gz\right)

Variables

Symbol	Description	Unit
$p_g$	Gage pressure relative to the atmospheric vertex pressure	Pa
$r$	Radial coordinate	m
$z$	Elevation from the vertex, positive upward	m
$\omega$	Angular velocity	rad/s

$Keep the Vertical Coordinate Sign Consistent$

With upward-positive $z$ , points below the vertex have negative $z$ , so the term $-gz$ increases pressure. If a downward-positive depth $z_d$ is used instead, write $p_g=\rho(\omega^2r^2/2+gz_d)$ . Mixing the two conventions reverses the hydrostatic contribution.

$Closed Rotating Tank Pressure Field$

A completely filled closed rotating tank has no atmospheric free surface to set the integration constant. The same radial and vertical pressure gradients apply, but one known pressure at a specified location is required to determine absolute pressure everywhere else.

$Check Minimum Absolute Pressure in Closed Rotation$

After determining the pressure field in a closed rotating liquid, compare the minimum absolute pressure with the liquid vapor pressure at the relevant temperature. If the predicted pressure drops too low, the assumed single-phase liquid field can fail through cavitation or vapor-pocket formation.

$Engineering Applicability Checks$

Confirm the liquid has reached relative equilibrium; transient sloshing is outside this idealization.
Check freeboard before using constant-volume translational or rotational formulas.
Check for bottom uncovering or dry-core formation when predicted depths approach zero.
Use explicit pressure references in free fall and closed tanks.
Convert rpm to rad/s before using formulas containing $\omega$ .
Compare minimum absolute pressure with vapor pressure where low-pressure regions are possible.
State whether vertical coordinates are positive upward or downward before evaluating a rotating pressure field.

Key Takeaways

Relative equilibrium replaces ordinary gravity with an effective body-force field in the accelerating container frame.
A horizontally accelerating open liquid develops a plane free surface that rises opposite the acceleration direction.
Vertical acceleration changes apparent gravity but does not tilt the free surface.
Free fall eliminates the hydrostatic pressure gradient; it does not universally set absolute pressure to zero.
No-spill formulas are valid only while the predicted surface remains within the tank and liquid contact is maintained.
Forced-vortex rotation produces a paraboloidal free surface and an outward radial pressure increase proportional to $\rho\omega^2r$ .
Cylindrical no-spill volume conservation gives equal center drop and wall rise of $\Delta h/2$ only while the bottom remains fully wetted.
Spillage, dry-core formation, confinement, coordinate choice, and vapor-pressure limits must be checked before accepting an ideal solution.
Closed rotating tanks require one pressure reference point because there is no atmospheric free surface to fix the integration constant.

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Learning Objectives

Relative Equilibrium

Effective Gravity, g⃗eff\vec g_{\text{eff}}g​eff​

Effective Gravity for Translational Acceleration

Pressure Gradient in Translational Relative Equilibrium

Free Surfaces Are Equipressure Surfaces

Horizontal Acceleration

Pressure Gradients under Horizontal Acceleration

Free-Surface Slope under Horizontal Acceleration

End-to-End Surface Difference in a Rectangular Tank

No-Spill End Depths

Check Geometry before Using No-Spill Formulas

Vertical Acceleration

Pressure under Upward Acceleration

Pressure under Downward Acceleration

Free Fall Means Zero Pressure Gradient, Not Zero Absolute Pressure

Downward Acceleration Greater than Gravity

Combined Horizontal and Vertical Acceleration

Interactive Relative-Equilibrium Exploration

Relative Equilibrium Simulator

Forced Vortex

Pressure Gradients in Rigid-Body Rotation

Parabolic Free Surface

Parabolic Free-Surface Equation

Center-to-Wall Elevation Difference

Interactive 3D Relative Equilibrium & Rotating Paraboloid

Relative Equilibrium Fluid Surfaces 3D

Linear Acceleration Parameters

Angular-Speed Conversion

No-Spill Volume Conservation in a Cylindrical Tank

No-Spill Center and Wall Depths

Spill and Dry-Core Regimes Must Be Treated Separately

Rotating-Liquid Pressure Field with Upward-Positive Elevation

Keep the Vertical Coordinate Sign Consistent

Closed Rotating Tank Pressure Field

Check Minimum Absolute Pressure in Closed Rotation

Engineering Applicability Checks

$Learning Objectives$

$Relative Equilibrium$

$\vec g_{\text{eff}}$

$Free Surfaces Are Equipressure Surfaces$

$Horizontal Acceleration$

$No-Spill End Depths$

$Check Geometry before Using No-Spill Formulas$

$Vertical Acceleration$

$Free Fall Means Zero Pressure Gradient, Not Zero Absolute Pressure$

$Downward Acceleration Greater than Gravity$

$Interactive Relative-Equilibrium Exploration$

$Forced Vortex$

$Parabolic Free Surface$

$Interactive 3D Relative Equilibrium & Rotating Paraboloid$

$No-Spill Volume Conservation in a Cylindrical Tank$

$Spill and Dry-Core Regimes Must Be Treated Separately$

$Keep the Vertical Coordinate Sign Consistent$

$Closed Rotating Tank Pressure Field$

$Check Minimum Absolute Pressure in Closed Rotation$

$Engineering Applicability Checks$