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Young–Laplace Equation Calculator

The Young–Laplace equation quantifies the pressure difference across curved fluid interfaces—essential for predicting capillary rise, wetting behaviour, and fluid transport in porous media. Engineers, geochemists, and materials scientists rely on this relationship to understand how surface tension and meniscus curvature drive liquid movement in narrow spaces, often without external mechanical assistance. This calculator solves the equation for capillary pressure, meniscus radius, and equilibrium column height.

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Creators Davide Borchia, PhD
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Surface Tension and Contact Angle

Capillary phenomena emerge from two interfacial properties: surface tension and contact angle.

Surface tension (γ) arises because fluid molecules at the liquid–air boundary experience asymmetric intermolecular forces. Molecules within the bulk liquid are surrounded uniformly by cohesive attractions, but those at the surface have neighbours only on one side. This imbalance creates an effective

The Young–Laplace equation links capillary pressure (Δp) to surface tension and meniscus radius. Below are the key forms:

Δp = 2γ / R

Δp = poutside − pinside

R = a / cos(θ)

Δp = ρ × g × h

  • Δp — Capillary (Laplace) pressure, the pressure jump across the meniscus in pascals (Pa).
  • γ — Surface tension of the liquid in joules per square metre (J/m²), often given in mN/m (millinewtons per metre).
  • R — Radius of curvature of the meniscus in metres. For a cylindrical tube, R depends on the tube radius and contact angle.
  • a — Inner radius of the tube in metres.
  • θ — Contact angle between the liquid and tube wall in degrees. Ranges from 0° (perfectly wetting) to 180° (perfectly non-wetting).
  • p<sub>outside</sub> — Pressure of the external phase (usually air) in pascals.
  • p<sub>inside</sub> — Pressure of the internal phase (the rising liquid) in pascals.
  • ρ — Density of the liquid in kilograms per cubic metre (kg/m³).
  • g — Gravitational acceleration, typically 9.81 m/s² at Earth's surface.
  • h — Height of the liquid column at equilibrium (Jurin height) in metres.

Capillary Rise at Equilibrium

When a vertical tube is immersed in a liquid bath, capillary action draws liquid upward until two opposing forces balance: the capillary pressure pulling upward and the hydrostatic pressure of the rising column pushing downward.

At equilibrium, the Laplace pressure equals the weight of the liquid column:

ρ × g × h = 2γ × cos(θ) / a

Rearranging for height:

h = 2γ × cos(θ) / (ρ × g × a)

This relationship shows why fine tubes (small a) produce dramatic rises: a 0.5 mm capillary can lift water 30 cm or more, whereas a 5 mm tube raises it only 3 cm. Real-world systems rarely reach perfect equilibrium because air may be trapped, thermal gradients develop, and dynamic viscosity slows ascent.

Applications in Industry and Science

The Young–Laplace equation governs phenomena across disciplines:

  • Petroleum engineering: Capillary pressure profiles in rock cores predict oil–water contact zones and inform enhanced recovery strategies. Permeability and fluid saturation depend critically on meniscus curvature.
  • Microfluidics: Lab-on-a-chip devices exploit precise pressure differences to route nanolitre droplets without pumps, relying entirely on surface tension and channel geometry.
  • Soil science: Capillary rise in clay and silt affects plant root water availability, salt accumulation, and foundation stability.
  • Coating and printing: Ink wetting on substrates, paint levelling, and adhesive spreading all hinge on contact angle and interfacial tension.
  • Emulsion stability: Smaller emulsion droplets have higher interfacial pressure, making them thermodynamically less stable and prone to coalescence—a key consideration in pharmaceuticals and cosmetics.

Frequently Asked Questions

What is the capillary pressure for water in a 2 mm diameter tube?

For water with surface tension γ = 0.0729 J/m² and a contact angle of approximately 20°, a tube with 2 mm diameter (1 mm radius) produces a capillary pressure of roughly 137 Pa. This is calculated using Δp = 2γ cos(θ) / a. At this modest pressure, the water column rises to about 1.4 cm. The exact value depends on temperature, tube cleanliness, and whether the glass is fresh or has absorbed organic contaminants—surface roughness and contamination can shift the contact angle by 10–20°.

How do I find capillary pressure if I don't know the meniscus radius?

Use the simplified form Δp = 2γ cos(θ) / a, where a is the tube radius and θ is the contact angle. The meniscus radius R is derived from these parameters (R = a / cos θ), so you avoid measuring curvature directly. This is the most practical approach in laboratory settings, since contact angle is readily determined by a goniometer or estimated from literature for common liquid–solid pairs.

Why does capillary rise matter in petrochemical exploration?

Capillary pressure measurements on core samples reveal the wetting state of reservoir rock and the capillary entry pressure—the threshold at which oil invades water-saturated pores. These data constrain fluid saturation profiles, identify cap-rock sealing capacity, and predict relative permeability curves. Engineers use this information to model hydrocarbon migration, estimate recoverable reserves, and design extraction schemes. Without capillary analysis, pressure and saturation predictions in subsurface models are unreliable.

What physically causes capillary pressure?

Capillary pressure arises from imbalanced molecular forces at the liquid–solid–fluid interface. At the meniscus, the liquid–air (or liquid–liquid) interfacial tension acts at an angle set by the contact angle, generating a net inward force that pulls the meniscus into a curve. This curvature creates a pressure jump: the concave side (inside the meniscus) is at lower pressure than the convex side. Wetting liquids curve inward (concave), amplifying the pressure drop and driving upward flow. Non-wetting liquids curve outward (convex), raising the internal pressure and suppressing rise.

How does temperature affect capillary rise?

Surface tension decreases with temperature for virtually all liquids—roughly 0.1–0.2 mN/m per °C for water and oils. Since capillary height is proportional to γ cos(θ), a 50 °C rise reduces γ by ~5 %, lowering h by a similar margin. Contact angle also shifts slightly with temperature, though the effect is smaller. In precision laboratory work, especially for small tubes where capillary forces dominate, temperature control to ±2 °C is prudent.

Can capillary pressure be negative?

Yes. When a non-wetting fluid (contact angle > 90°) is at an interface, cos(θ) becomes negative, flipping the sign of Δp. Mercury in a glass tube experiences negative capillary pressure—the meniscus bulges downward, and the pressure inside the mercury is higher than outside. This is why mercury columns are depressed rather than elevated. In soil mechanics, organic non-aqueous phase liquids (NAPLs) create negative capillary pressure, trapping water above them and complicating contaminant remediation.

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