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Index/Fluid physics/FLD-04

FLD-04

Stokes drag

F = 6π μ r v. Linear drag on a slow sphere, Re ≪ 1.

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FluidsStokes

Governing equation

Fd=6πμrvF_d=6\pi\mu r v

v∈[1×10−4,1]v \in \left[1\times 10^{-4}, 1\right]

where

\mu
Viscosity (Pa·s)
r
Radius (mm)
v
Speed (m/s)
F_d
Drag force (µN)

The figure, part by part

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①

Governing law — red

Fd=6πμrvF_d=6\pi\mu r v

v∈[1×10−4,1]v \in \left[1\times 10^{-4}, 1\right]

v=1×10−4v = 1\times 10^{-4}Fd=9.42×10−4F_d = 9.42\times 10^{-4}

②

Domain of the variable — blue

vmin⁡=1×10−4,vmax⁡=1v_{\min} = 1\times 10^{-4},\quad v_{\max} = 1

v∈[1×10−4,1]v \in \left[1\times 10^{-4}, 1\right]

v=1×10−4v = 1\times 10^{-4}v=1×10−4v = 1\times 10^{-4}

③

Response curve — green

Fd=6πμrvF_d=6\pi\mu r v

v∈[1×10−4,1]v \in \left[1\times 10^{-4}, 1\right]

v=1×10−4v = 1\times 10^{-4}Fd=9.42×10−4F_d = 9.42\times 10^{-4}

④

Rate of change — amber

dFddv\frac{\mathrm{d}F_d}{\mathrm{d}v}

v∈[1×10−4,1]v \in \left[1\times 10^{-4}, 1\right]

v=0.006v = 0.006Fd′=9.425F_d^{\prime} = 9.425

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-04 — Stokes drag) is the form associated with Stokes. Working symbols: μ\mu, rr, vv →\rightarrow FdF_d. Creeping-flow solution of Navier–Stokes around a sphere. Millikan used it.

Purpose

Purpose: compute FdF_d from μ\mu, rr, vv in Fluid physics via Fd=6πμrvF_d=6\pi\mu r v F = 6π μ r v. Linear drag on a slow sphere, Re ≪ 1. 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 μ=0.001 Pa⋅s\mu = 0.001\,\mathrm{Pa·s}, r=0.500 mmr = 0.500\,\mathrm{mm}, v=0.020 m/sv = 0.020\,\mathrm{m/s}, the governing relation Fd=6πμrvF_d=6\pi\mu r v yields Fd=0.188 μNF_d = 0.188\,\mathrm{\mu N}. A sphere, a viscous halo, a linear force arrow. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Outputs

  • Drag force F_d0.188 µN
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Narration of this film

A sphere, a viscous halo, a linear force arrow.

Creeping-flow solution of Navier–Stokes around a sphere. Millikan used it.

Reading speed

Watch on YouTube