FLD-19

Capillary rise

h = 2 σ cosθ / (ρ g r). Jurin's law in a circular tube.

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FluidsJurin

Governing equation

h=\dfrac{2\sigma\cos\theta}{\rho g r}

$r \in \left[0.02, 5\right]$

where

\sigma: Surface tension (N/m)
\theta: Contact angle (°)
\rho: Density (kg/m³)
r: Radius (mm)
h: Rise (mm)

The figure, part by part

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①

Governing law — red

h=\dfrac{2\sigma\cos\theta}{\rho g r}

$r \in \left[0.02, 5\right]$

$r = 0.02$ $h = 689.7$

Along the curve

②

Domain of the variable — blue

r_{\min} = 0.02,\quad r_{\max} = 5

$r \in \left[0.02, 5\right]$

$r = 0.02$ $r = 0.02$

Along the curve

③

Response curve — green

h=\dfrac{2\sigma\cos\theta}{\rho g r}

$r \in \left[0.02, 5\right]$

$r = 0.02$ $h = 689.7$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}h}{\mathrm{d}r}

$r \in \left[0.02, 5\right]$

$r = 0.051$ $h^{\prime} = -8345.5$

Along the curve

Lecture brief

Historical brief

Bernoulli, Navier–Stokes, Reynolds (1883) and Stokes drag made continuum flow a dimensionless craft. The sheets compute head, drag, Re and a pedagogical NS snapshot. This sheet (FLD-19 — Capillary rise) is the form associated with Jurin. Working symbols: $\sigma$ , $\theta$ , $\rho$ , $r$ $\rightarrow$ $h$ . Young–Laplace pressure jump balanced by the hydrostatic column. Depression if θ > 90°.

Purpose

Purpose: compute

h

from

\sigma

\theta

\rho

r

in Fluid physics via

h=\dfrac{2\sigma\cos\theta}{\rho g r}

h = 2 σ cosθ / (ρ g r). Jurin's law in a circular tube. Use it when a real fluid physics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\sigma = 0.072\,\mathrm{N/m}

\theta = 20.000\,\mathrm{^{\circ}}

\rho = 1000.000\,\mathrm{kg/m^{3}}

r = 0.300\,\mathrm{mm}

, the governing relation

h=\dfrac{2\sigma\cos\theta}{\rho g r}

yields

h = 45.98\,\mathrm{mm}

. A meniscus, a thin tube, a climb. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Surface tension \sigma0.072 N/mContact angle \theta20 °Density \rho1000 kg/m³Radius r0.3 mm

Outputs

Rise h45.98 mm

Reading speed

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Narration of this film

A meniscus, a thin tube, a climb.

Young–Laplace pressure jump balanced by the hydrostatic column. Depression if θ > 90°.

Reading speed

Watch on YouTube