Calculate pressure difference across curved interfaces using the Young-Laplace equation. Supports bubbles, droplets, capillary rise, meniscus curvature, and contact angle calculations.
The Young-Laplace equation describes the pressure difference (Laplace pressure) across a curved interface between two fluids due to surface tension. For a general surface with two principal radii of curvature R₁ and R₂: ΔP = γ(1/R₁ + 1/R₂), where γ is the surface tension.
This fundamental equation of capillarity governs soap bubbles (ΔP = 4γ/r for two interfaces), liquid droplets (ΔP = 2γ/r), capillary rise in narrow tubes, meniscus shape in containers, and the stability of foams and emulsions. It connects macroscopic surface tension to the microscopic curvature of interfaces.
This calculator computes Laplace pressure for spheres, cylinders, general ellipsoids, and capillary tubes. It includes surface tension data for common liquids, predicts capillary rise heights, and provides context for applications in microfluidics, lung surfactant physics, inkjet printing, and nanoparticle stability.
For best results, combine calculator output with direct observation and periodic check-ins with a veterinarian or qualified advisor. Small adjustments made early usually improve comfort, safety, and long-term outcomes more than large corrective changes made later.
Understanding Laplace pressure is vital in microfluidics design, pharmaceutical emulsion stability, pulmonary medicine (alveolar mechanics), inkjet and 3D printing, enhanced oil recovery, and nanotechnology — wherever curved fluid interfaces determine system behavior. It also improves communication between theory and experiments by letting teams translate radius and surface-tension measurements into pressure impacts that can be validated in practical setups.
General: ΔP = γ(1/R₁ + 1/R₂). Sphere: ΔP = 2γ/r. Soap bubble: ΔP = 4γ/r. Capillary rise: h = 2γ cos θ / (ρgr), where θ is contact angle, ρ is liquid density, g = 9.81 m/s², and r is tube radius.
Result: ΔP = 145.6 Pa (1.09 mmHg)
A water droplet of radius 1 mm has ΔP = 2 × 0.0728 / 0.001 = 145.6 Pa. This is small relative to atmospheric pressure, but becomes significant for micrometer-sized droplets.
For a **sphere** (droplet or gas bubble), R₁ = R₂ = r, giving ΔP = 2γ/r. For a **soap bubble**, the factor doubles to 4γ/r because of two air-liquid interfaces. For a **cylinder** (e.g., a liquid jet), one principal radius is r and the other is infinite, giving ΔP = γ/r. For a **saddle-shaped surface**, the two radii have opposite signs, potentially giving ΔP = 0.
The lungs contain approximately 300 million alveoli with radii around 0.1-0.15 mm. Without pulmonary surfactant, the Laplace pressure in the smallest alveoli would be ~3 kPa, causing them to collapse. Surfactant reduces surface tension from ~70 mN/m to ~25 mN/m at end-expiration, preventing atelectasis. In premature infants, surfactant deficiency causes Respiratory Distress Syndrome (RDS).
The capillary length κ⁻¹ = √(γ/(ρg)) defines the scale below which surface tension dominates gravity. For water, κ⁻¹ ≈ 2.7 mm. Droplets smaller than this are approximately spherical; larger ones flatten. This is why raindrops flatten but fog droplets are spherical.
Surface tension acts to minimize surface area, compressing the gas inside. The smaller the bubble, the higher the internal pressure — this is why small bubbles dissolve faster than large ones (Ostwald ripening).
A soap bubble has two surfaces (inner and outer). Each applies 2γ/r, so the total is 4γ/r. A liquid droplet or gas bubble in liquid has only one interface.
In a narrow tube, the meniscus curves due to adhesion between the liquid and tube wall. The pressure difference from this curvature pulls the liquid up (for wetting liquids) or pushes it down (for non-wetting liquids like mercury).
The contact angle θ is the angle between the liquid surface and the solid wall at their intersection. θ < 90° means wetting (e.g., water on glass), θ > 90° means non-wetting (e.g., mercury on glass).
It assumes a sharp interface, neglects gravity (except in capillary rise), and assumes constant surface tension. It fails at molecular scales (<10 nm) and near the critical point where the interface vanishes.
Water: 72.8 mN/m, ethanol: 22.1 mN/m, mercury: 485 mN/m, blood: ~55 mN/m, lung surfactant: 25-70 mN/m depending on compression.