CELE Hydraulics & Fluid Mechanics — Relative Equilibrium of LiquidsConcept Map

Concept

Horizontal Acceleration

Sub Concepts

• Free surface tilts at angle θ
• Formula: tan(θ) = a/g
• Surface slopes down toward direction of acceleration
• Pressure still follows hydrostatic distribution below tilted surface
• Applications: Tanker trucks, moving containers

Relationship To Central

First category of relative equilibrium where tank accelerates parallel to ground

Concept

Vertical Acceleration

Sub Concepts

• Free surface remains horizontal
• Effective gravity changes
• Upward acceleration: p = γh(1 + a/g)
• Downward acceleration: p = γh(1 - a/g)
• Free fall (a = g down): gauge pressure = 0
• Applications: Elevators, vibrating equipment

Relationship To Central

Second category where tank accelerates perpendicular to ground

Concept

Rotation (Rotating Vessel)

Sub Concepts

• Paraboloid free surface forms
• Height equation: z = ω²r²/(2g)
• Rise from center to wall: ω²R²/(2g)
• Paraboloid volume = half the cylinder volume
• Pressure at any point: γ × depth below surface
• Angular velocity in rad/s (convert from rpm)
• Applications: Centrifuges, rotating tanks, separation devices

Relationship To Central

Third category where tank rotates about vertical axis

Concept

Fundamental Principles

Sub Concepts

• No relative motion between fluid particles (rigid body motion)
• No shear stress in fluid
• Fluid experiences inertial (pseudo) forces
• Free surface always perpendicular to effective gravity
• Pressure distribution determined by vertical depth below free surface

Relationship To Central

Core physics governing all relative equilibrium cases

Concept

Mathematical Framework

Sub Concepts

• Horizontal: tan(θ) = a/g
• Vertical: p = γh(1 ± a/g)
• Rotation: z = ω²r²/(2g)
• Total pressure at depth: p = p_atm + ρgh(effective)
• Paraboloid volume: V = πR²h_cylinder / 2

Relationship To Central

Quantitative tools for solving relative equilibrium problems

Concept

Pressure Distribution

Sub Concepts

• Always measured perpendicular to free surface
• Increases linearly with depth below surface
• Gauge vs absolute pressure considerations
• Pressure contours parallel to free surface
• Isobars are perpendicular to effective gravity

Relationship To Central

How pressure changes in each acceleration scenario

Concept

Free Surface Behavior

Sub Concepts

• Horizontal: plane tilted at angle θ
• Vertical: plane remains horizontal but pressure changes
• Rotation: paraboloid of revolution
• Surface always perpendicular to effective gravity direction
• Volume conservation constraints

Relationship To Central

How the liquid-air interface deforms under acceleration

Concept

Problem-Solving Considerations

Sub Concepts

• Determining acceleration magnitude and direction
• Converting angular velocity units (rpm to rad/s)
• Handling spill-over in rotating vessels
• Volume conservation in open systems
• Pressure at arbitrary points in the fluid
• Closed vs open tank configurations

Relationship To Central

Practical issues in analyzing relative equilibrium systems

To

Free Surface Behavior

From

Horizontal Acceleration

Strength

strong

Relationship

Horizontal acceleration causes the free surface to tilt at an angle determined by the ratio of acceleration to gravity

To

Pressure Distribution

From

Vertical Acceleration

Strength

strong

Relationship

Vertical acceleration modifies the effective gravitational field, changing the rate at which pressure increases with depth

To

Paraboloid Formation

From

Rotation

Strength

strong

Relationship

Centrifugal acceleration in rotating systems creates a quadratic relationship between radius and surface height, forming a paraboloid shape

To

Horizontal Acceleration

From

Fundamental Principles

Strength

strong

Relationship

The principle of no relative motion between fluid particles underlies all three acceleration cases, including horizontal scenarios

To

Vertical Acceleration

From

Fundamental Principles

Strength

strong

Relationship

Rigid body motion and inertial effects determine the pressure changes seen in vertically accelerating systems

To

Rotation

From

Fundamental Principles

Strength

strong

Relationship

No shear stress in rotating liquids means the surface must orient perpendicular to the combined effect of gravity and centrifugal acceleration

To

Horizontal Acceleration

From

Mathematical Framework

Strength

strong

Relationship

The tangent formula tan(θ) = a/g directly quantifies the surface tilt angle for horizontal acceleration

To

Vertical Acceleration

From

Mathematical Framework

Strength

strong

Relationship

The pressure formula p = γh(1 ± a/g) captures the effect of vertical acceleration on hydrostatic pressure distribution

To

Rotation

From

Mathematical Framework

Strength

strong

Relationship

The paraboloid equation z = ω²r²/(2g) describes the complete shape of the free surface in rotating vessels

To

Pressure Distribution

From

Free Surface Behavior

Strength

strong

Relationship

The orientation of the free surface determines the reference level from which pressure is measured; pressure always increases with depth below this surface

To

Horizontal Acceleration

From

Problem-Solving Considerations

Strength

moderate

Relationship

Properly identifying acceleration magnitude and direction is critical for calculating the tilt angle and pressure distribution

To

Rotation

From

Problem-Solving Considerations

Strength

moderate

Relationship

Unit conversion (rpm to rad/s) and volume conservation are essential for solving rotating vessel problems correctly

To

Pressure Distribution

From

Horizontal Acceleration

Strength

moderate

Relationship

Although the surface tilts, pressure still follows p = γh below the tilted surface; the hydrostatic law remains valid

To

Free Surface Behavior

From

Vertical Acceleration

Strength

moderate

Relationship

While vertical acceleration keeps the free surface horizontal, it modifies the effective gravity that the surface must be perpendicular to

To

Pressure Distribution

From

Rotation

Strength

moderate

Relationship

In rotating vessels, pressure at any point depends on the vertical distance below the paraboloid free surface