FLD-02

Continuity equation

A₁ v₁ = A₂ v₂. Volume flux is conserved for steady incompressible flow.

Reading speed

FluidsContinuity

Governing equation

A_1 v_1=A_2 v_2

$A_2 \in \left[0.001, 2\right]$

where

A_1: Area 1 (m²)
v_1: Speed 1 (m/s)
A_2: Area 2 (m²)
Q: Discharge (m³/s)
v_2: Speed 2 (m/s)

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.

00:0 / 00:08

①

Governing law — red

A_1 v_1=A_2 v_2

$A_2 \in \left[0.001, 2\right]$

$A_2 = 0.001$ $Q = 100$

Along the curve

②

Domain of the variable — blue

A_2_{\min} = 0.001,\quad A_2_{\max} = 2

$A_2 \in \left[0.001, 2\right]$

$A_2 = 0.001$ $A_2 = 0.001$

Along the curve

③

Response curve — green

A_1 v_1=A_2 v_2

$A_2 \in \left[0.001, 2\right]$

$A_2 = 0.001$ $Q = 100$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}Q}{\mathrm{d}A_2}

$A_2 \in \left[0.001, 2\right]$

$A_2 = 0.014$ $Q^{\prime} = -3824.9$

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-02 — Continuity equation) is the form associated with Continuity. Working symbols: $A_1$ , $v_1$ , $A_2$ $\rightarrow$ $Q$ , $v_2$ . Mass conservation in a stream tube. Compressible flow uses ρ A v instead.

Purpose

Purpose: compute

Q

v_2

from

A_1

v_1

A_2

in Fluid physics via

A_1 v_1=A_2 v_2

A₁ v₁ = A₂ v₂. Volume flux is conserved for steady incompressible flow. 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

A_1 = 0.050\,\mathrm{m^{2}}

v_1 = 2.000\,\mathrm{m/s}

A_2 = 0.020\,\mathrm{m^{2}}

, the governing relation

A_1 v_1=A_2 v_2

yields

Q = 0.1000\,\mathrm{m^{3}/s}

v_2 = 5.000\,\mathrm{m/s}

. A narrowing pipe, two area disks, a speeding core. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Area 1 A_10.05 m²Speed 1 v_12 m/sArea 2 A_20.02 m²

Outputs

Discharge Q0.1000 m³/s
Speed 2 v_25.000 m/s

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

A narrowing pipe, two area disks, a speeding core.

Mass conservation in a stream tube. Compressible flow uses ρ A v instead.

Reading speed

Watch on YouTube