FLD-25

Young–Laplace pressure

Δp = σ (1/R₁ + 1/R₂). Pressure jump across a curved interface.

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FluidsYoung–Laplace

Governing equation

\Delta p=\sigma\left(\dfrac1{R_1}+\dfrac1{R_2}\right)

$R_1 \in \left[0.05, 20\right]$

where

\sigma: Surface tension (N/m)
R_1: Radius 1 (mm)
R_2: Radius 2 (mm)
\Delta p: Pressure jump (Pa)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\Delta p=\sigma\left(\dfrac1{R_1}+\dfrac1{R_2}\right)

$R_1 \in \left[0.05, 20\right]$

$R_1 = 0.05$ $\Delta p = 1512$

Along the curve

②

Domain of the variable — blue

R_1_{\min} = 0.05,\quad R_1_{\max} = 20

$R_1 \in \left[0.05, 20\right]$

$R_1 = 0.05$ $R_1 = 0.05$

Along the curve

③

Response curve — green

\Delta p=\sigma\left(\dfrac1{R_1}+\dfrac1{R_2}\right)

$R_1 \in \left[0.05, 20\right]$

$R_1 = 0.05$ $\Delta p = 1512$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\Delta p}{\mathrm{d}R_1}

$R_1 \in \left[0.05, 20\right]$

$R_1 = 0.175$ $\Delta p^{\prime} = -4785$

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-25 — Young–Laplace pressure) is the form associated with Young–Laplace. Working symbols: $\sigma$ , $R_1$ , $R_2$ $\rightarrow$ $\Delta p$ . A sphere has R₁ = R₂ so Δp = 2σ/R. Soap bubbles have two surfaces: 4σ/R.

Purpose

Purpose: compute

\Delta p

from

\sigma

R_1

R_2

in Fluid physics via

\Delta p=\sigma\left(\dfrac1{R_1}+\dfrac1{R_2}\right)

Δp = σ (1/R₁ + 1/R₂). Pressure jump across a curved interface. 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}

R_1 = 1.000\,\mathrm{mm}

R_2 = 1.000\,\mathrm{mm}

, the governing relation

\Delta p=\sigma\left(\dfrac1{R_1}+\dfrac1{R_2}\right)

yields

\Delta p = 144.0\,\mathrm{Pa}

. A meniscus, two radii, a Δp. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Surface tension \sigma0.072 N/mRadius 1 R_11 mmRadius 2 R_21 mm

Outputs

Pressure jump \Delta p144.0 Pa

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Narration of this film

A meniscus, two radii, a Δp.

A sphere has R₁ = R₂ so Δp = 2σ/R. Soap bubbles have two surfaces: 4σ/R.

Reading speed

Watch on YouTube