bashforx

Pendant-drop tensiometry à la Bashforth & Adams (1883). The axisymmetric Young–Laplace equation is solved with Diffrax and fitted to a drop's edge points by Levenberg–Marquardt with Optimistix. The result is a² = γ / (Δρ g), the surface tension relative to specific gravity.

The model

Bashforth & Adams wrote the profile in terms of arc length. Here s runs from the apex, z points up towards the needle, and φ is the angle of the tangent:

dx/ds = cos φ
dz/ds = sin φ
dφ/ds = 2/b − c·z − sin φ / x        c = Δρg/γ = 1/a²,  β = c·b²

b is the radius of curvature at the apex, and β is Bashforth & Adams' shape factor. At the apex, sin φ / x tends to 1/b. The right-hand side uses a double jnp.where there, so neither the value nor its forward-mode derivative becomes NaN.

Fitting

• The unknowns are (b, c, x0, z0) plus one signed arc length per observed point. Points left of the axis take negative values, using the symmetry x(−s) = −x(s) and z(−s) = z(s). The residual is model(sᵢ) − pointᵢ. That is smooth, so Levenberg–Marquardt applies directly. The alternative, distance to the nearest point on the curve, is not smooth.
• Each residual evaluation does one ODE solve with dense output, and then sol.evaluate(|sᵢ|) is vmapped over the points.
• adjoint=diffrax.ForwardMode() is required. Optimistix's least-squares solvers build Jacobians in forward mode, and Diffrax's default RecursiveCheckpointAdjoint supports only reverse mode.
• The starting point mimics a lookup in Bashforth & Adams' tables. A quadratic fitted to the bottom of the drop gives the apex and b. A grid search over β then picks the closest shape.
• Uncertainties come from the linearised covariance σ²(JᵀJ)⁻¹.

Install

With uv:

git clone https://github.com/gdmcbain/bashforx
cd bashforx
uv sync --all-extras
uv run pytest
uv run examples/demo.py

To use it from another uv project: uv add git+https://github.com/gdmcbain/bashforx.

Usage

import bashforx as bfx

points, _ = bfx.synthetic_drop(b=1.6, capillary_length_sq=7.35, noise=0.005, seed=0)
fit = bfx.fit_drop(points)  # points: (N, 2) array of (x, z), z up towards the needle

fit.capillary_length_sq, fit.capillary_length_sq_std  # a² and its 1σ, in data units²
fit.params.b, fit.b_std                               # apex radius
fit.surface_tension(998.0)                            # γ in N/m, for Δρ in kg/m³ and data in mm

Edge points taken from an image usually have y pointing down, so flip it first. fit_drop raises RuntimeError if Levenberg–Marquardt does not converge, or if it converges to an unphysical shape (b ≤ 0 or c ≤ 0).

Importing bashforx turns on JAX's 64-bit mode.

Example

examples/demo.py fits a synthetic water drop (150 points, 5 µm noise):

	Fitted	True
b (apex radius)	1.5985 ± 0.0008 mm	1.6
a² = γ/(Δρg)	7.309 ± 0.017 mm²	7.35
γ (Δρ = 998 kg/m³)	71.5 mN/m	72.0

The quoted uncertainties hold up over many noise seeds. Across 40 seeds, the error divided by its reported σ averaged 0.03 with a spread of 1.10.

Status

So far it has been tested only on synthetic drops, not on edge points from real images.

Reference

F. Bashforth & J. C. Adams (1883). An Attempt to Test the Theories of Capillary Action by Comparing the Theoretical and Measured Forms of Drops of Fluid. Cambridge University Press. This is also where the Adams–Bashforth multistep method first appeared, as the way Adams integrated this same equation.

License

Apache-2.0, the same as Diffrax and Optimistix.