CELE Hydraulics & Fluid Mechanics — Relative Equilibrium of LiquidsRevision Notes

Formulas

Exam Tips

• Draw a clear sketch of the tank, mark the new free surface, and label all relevant dimensions before applying any formula.
• Always confirm whether the problem asks for gauge pressure or absolute pressure; board problems typically ask for gauge pressure.
• When the problem gives γ in kN/m³ (e.g., 9.81 kN/m³), the resulting pressure will be in kPa directly — a common computational shortcut.

Key Points

• Relative equilibrium is a state where a fluid body accelerates as a unit — no particle moves relative to another, so viscous shear is absent.
• Because there is no shear, the equations of static fluid pressure (p = γh) still apply, but measured from the NEW free surface (tilted or curved).
• The free surface is always perpendicular to the resultant of all body forces acting on the fluid (gravity + inertial body force).
• Pressure at any point equals γ × vertical depth below the free surface, regardless of the direction of acceleration.
• Three standard cases appear in PRC board exams: horizontal rectilinear acceleration, vertical rectilinear acceleration, and uniform rotation about a vertical axis.

Definitions

Section Title

Fundamentals of Relative Equilibrium

Common Mistakes

• Measuring depth h along the slant of the free surface instead of vertically — always use vertical depth.
• Forgetting that the free surface is the zero-pressure datum; pressure at the bottom of a tilted tank must use the vertical distance from the (tilted) surface to the bottom.
• Assuming the pressure distribution is the same as static when the tank is accelerating — it is NOT; only the formula form p = γh is preserved, but h changes with the new surface geometry.

Formulas

Exam Tips

• Memorize: 'Surface tilts down toward the acceleration.' This is your quick sanity check.
• For a rectangular tank of width W perpendicular to acceleration: pressure at the bottom is uniform across W (no variation in that direction).
• Spill-check: compute Δh; if Δh > (tank height − initial depth at front), liquid spills from the front. If spill occurs, recompute with the condition that the surface just touches the front top edge.
• In board problems, a = g tan θ is frequently needed in reverse: given θ, find a.

Key Points

• The free surface tilts — it slopes downward in the direction of acceleration and upward on the trailing side.
• The angle θ of the free surface with the horizontal is given by tan θ = a/g.
• Pressure still follows p = γh where h is vertical depth below the (now tilted) free surface.
• For a closed, fully-filled tank, no surface tilt is visible but pressure increases at the back and decreases at the front; the imaginary free surface concept (piezometric surface) is used.
• Horizontal acceleration does NOT change the total volume of liquid; it merely redistributes it.

Definitions

Section Title

Horizontal Acceleration

Common Mistakes

• Using tan θ = g/a instead of a/g — the acceleration is in the numerator because it is the inertial 'horizontal component' of effective gravity.
• Not checking whether the tilted surface goes below the tank bottom on the front side — if it does, the front portion has zero pressure (air pocket), and a separate analysis is needed.
• Applying Δh = L tan θ / 2 when the tank is NOT initially full; only valid if the original surface is at mid-level or you account for the actual initial depth.
• Forgetting the direction: surface goes DOWN at the FRONT (direction of a), UP at the REAR.

Formulas

Exam Tips

• Memory aid: 'Going up → pressure goes up; going down → pressure goes down.'
• When a problem says 'decelerating at X m/s² while moving upward,' the net acceleration is DOWNWARD at X m/s² — use the minus formula.
• When a problem says 'decelerating at X m/s² while moving downward,' the net acceleration is UPWARD at X m/s² — use the plus formula.
• Board exam favorite: find the acceleration required so that the pressure at the bottom equals zero — solve (1 − a/g) = 0, giving a = g = 9.81 m/s² downward (free fall).

Key Points

• The free surface remains horizontal — only the magnitude of effective pressure changes.
• Upward acceleration increases pressure (equivalent to higher g); downward acceleration decreases pressure.
• Free fall (a = g downward) produces zero gauge pressure throughout — weightlessness condition for the liquid.
• This case directly models elevators, rockets at launch, and crane-lifted tanks of water.
• The specific weight effectively becomes γ_eff = γ(1 ± a/g).

Definitions

Section Title

Vertical Acceleration

Common Mistakes

• Using + sign for downward acceleration (the most common sign-error in board exams for this topic).
• Computing a = g for free fall and getting a negative pressure — gauge pressure cannot be negative for normal liquids (cavitation would occur first). The answer should be zero, not negative.
• Confusing 'decelerating upward' with 'accelerating downward' — both produce effective downward acceleration and reduce pressure. Always resolve the net acceleration direction first.
• Neglecting to add atmospheric pressure when absolute pressure is asked.

Formulas

Exam Tips

• Always convert N (rpm) to ω (rad/s) as the very first step — write it explicitly to avoid errors.
• No-spill check: Is h_rise/2 ≤ (tank height − initial liquid depth)? If yes, no spill (vertex rises, volume redistributes within tank). If no, spill occurs.
• More precisely: The vertex drops by h_rise/2 and the rim rises by h_rise/2 from the undisturbed level. So check if h_rise/2 ≤ freeboard.
• For closed, fully-filled rotating tanks, the paraboloid concept applies to the pressure distribution — no actual free surface, but piezometric surface is a paraboloid.
• Memorize: Volume of paraboloid = ½ × base area × height. This is identical to: V_para = ½ × V_enclosing_cylinder.

Key Points

• A liquid in an open cylinder rotating at constant angular velocity ω forms a paraboloid free surface.
• The shape is a paraboloid of revolution: z = ω²r²/(2g), where z is measured upward from the vertex of the paraboloid.
• The rise from the center (vertex) to the rim (radius R) is h_rise = ω²R²/(2g).
• Volume of the paraboloid = (1/2) × volume of its bounding cylinder — a key property for spill-check problems.
• ω must be in rad/s; convert from rpm: ω = 2πN/60.
• Pressure at any interior point (r, z_point) is p = γ × (vertical depth from the paraboloid surface directly above that point to the point itself).

Definitions

Section Title

Rotation About a Vertical Axis (Rotating Vessel)

Common Mistakes

• Using N (rpm) directly in the formula instead of converting to ω (rad/s).
• Confusing h_rise (vertex-to-rim) with the absolute elevation of the free surface.
• In spill problems, forgetting that the volume of liquid is conserved — the paraboloid volume above the original level equals the volume removed from below it.
• Assuming the vertex stays at the bottom of the tank — the vertex can drop below the floor if rotation is fast enough (indicating dry zone at center bottom).
• Using r = diameter instead of r = radius in the formula.

Formulas

Exam Tips

• In board exams, 'spill problems' are usually flagged by phrases like 'the tank is initially full' or 'find the rotational speed that just causes the liquid to spill from the rim.'
• The critical (just-about-to-spill) angular velocity for a full tank: h_rise = H (full tank height): ω_critical = √(2gH/R²).
• For partial-fill rotation (liquid depth d, tank height H): no-spill condition is h_rise/2 ≤ H − d. The vertex drops by h_rise/2.

Key Points

• Spill analysis is required whenever computed Δh (horizontal case) or h_rise/2 (rotation case) exceeds the available freeboard.
• For horizontal acceleration with spill: Set the tilted surface to pass through the top-rear corner; recompute effective length and volume constraints.
• For rotation with spill: The vertex of the paraboloid descends below the tank bottom, creating a dry annular zone at the bottom. The problem becomes: find the dry radius.
• Conservation of liquid volume is always enforced — the volume of paraboloid above original level equals volume 'vacated' below.
• Closed-tank rotation: use the imaginary free surface (piezometric paraboloid) with pressure = γ × vertical distance to the imaginary surface.

Definitions

Section Title

Spill Analysis and Special Cases

Common Mistakes

• Skipping the spill check and using the no-spill formula when the tank has actually spilled — leads to gross errors in computed pressures.
• In rotation-with-spill problems, forgetting that the vertex may descend below the bottom, creating a dry ring zone — treating the bottom as fully wetted when it is not.