Stationary Sphere Meniscus Calculator

Stationary sphere meniscus calculator

Estimate the gravity-capillary meniscus around a sphere touching a deep bath of the same liquid.

Geometry and liquid

Meniscus geometry

How to use

Use this single stationary sphere-at-bath mode only when a sphere touches a deep horizontal bath of the same liquid.
Enter sphere radius b in mm and the equilibrium contact angle θ on the sphere in degrees, measured through the liquid.
Enter liquid-gas surface tension γ in mN/m, liquid density ρ in kg/m³, and positive gravity g in m/s².
Set the profile cutoff q between 0 and 1; it changes only how far the graph extends toward the bath.
Select Calculate to update the rise, pressure, curvature, metrics, and graph. Select Reset to restore the worked example.
Interpret y₀ with b/a: the analytical sphere-boundary approximation is strongest when b/a is much greater than one.
Review warnings for small b/a or limiting contact angles; contact-angle hysteresis can dominate real measurements.
Do not use this mode for two-solid liquid bridges, confined gaps, dynamic wetting, evaporation, rough contact lines, force or volume prediction, or final component design.

Equations used

The capillary length, sphere contact radius, and boundary equation are:

a = √[γ/(ρg)]

r₀ = √[y₀(2b − y₀)]

1 − y₀²/(2a²) = sin{θ − atan[(b − y₀)/r₀]}

The physical root satisfies 0 < y₀ < min(2b, √2a). The plotted large-sphere profile is:

X(y) = r₀ + G(y₀) − G(y)

G(y) = −a acosh(2a/y) + √2a √[2 − y²/(2a²)]

Δp = p_liquid − p_gas = −ρgy₀ = γκ

Symbols and units: b, a, r₀, y, y₀, and X are metres internally; θ is radians internally; γ is N/m; ρ is kg/m³; g is m/s²; Δp is Pa; κ is m⁻¹. Sign convention: height is positive upward from the undisturbed bath, radial X is outward from the sphere axis, and a raised concave meniscus has negative liquid-minus-gas pressure and curvature.

Assumptions and limits: static axisymmetric equilibrium, deep horizontal bath, same liquid on the sphere and in the bath, zero bath contact angle, constant γ and ρ, and negligible inertia. The analytical profile assumes b ≫ a. It does not compute bridge volume or capillary force.

Reference: S. Champmartin, A. Ambari, and J. Y. Le Pommelec, “New procedure to measure simultaneously the surface tension and contact angle,” Review of Scientific Instruments 87 (2016), doi:10.1063/1.4948736.