FLD-07

Centripetal Euler acceleration

n = v² / r. Normal acceleration on a curved streamline.

Reading speed

FluidsEuler

Governing equation

n=\dfrac{v^2}{r}

$r \in \left[0.05, 40\right]$

where

v: Speed (m/s)
r: Radius of curvature (m)
n: Normal acceleration (m/s²)

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

n=\dfrac{v^2}{r}

$r \in \left[0.05, 40\right]$

$r = 0.05$ $n = 1280$

Along the curve

②

Domain of the variable — blue

r_{\min} = 0.05,\quad r_{\max} = 40

$r \in \left[0.05, 40\right]$

$r = 0.05$ $r = 0.05$

Along the curve

③

Response curve — green

n=\dfrac{v^2}{r}

$r \in \left[0.05, 40\right]$

$r = 0.05$ $n = 1280$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}n}{\mathrm{d}r}

$r \in \left[0.05, 40\right]$

$r = 0.301$ $n^{\prime} = -2316.7$

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-07 — Centripetal Euler acceleration) is the form associated with Euler. Working symbols: $v$ , $r$ $\rightarrow$ $n$ . Euler's inviscid equation projected on the principal normal. Cyclones and pipe bends.

Purpose

Purpose: compute

n

from

v

r

in Fluid physics via

n=\dfrac{v^2}{r}

n = v² / r. Normal acceleration on a curved streamline. 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

v = 8.000\,\mathrm{m/s}

r = 2.000\,\mathrm{m}

, the governing relation

n=\dfrac{v^2}{r}

yields

n = 32.00\,\mathrm{m/s^{2}}

. A curved pipe, a particle, an inward arrow. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Speed v8 m/sRadius of curvature r2 m

Outputs

Normal acceleration n32.00 m/s²

Reading speed

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Narration of this film

A curved pipe, a particle, an inward arrow.

Euler's inviscid equation projected on the principal normal. Cyclones and pipe bends.

Reading speed

Watch on YouTube