For each flow below

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity · Viscous flow and dynamical similarity

Fluid Dynamics, Ch The stress tensor completed: constitutive law and fluid properties

Problem

For each flow below, estimate the Knudsen number

Kn = \ell_{\rm mfp}/L

and the Deborah number

De = t_{\rm relax}/t_{\rm flow}

, and decide whether it may be treated as a Newtonian continuum with no slip. For a liquid, use the molecular size in place of the mean free path. ((a)) Air flowing past an aircraft wing of chord

L = 2.0\,\mathrm{m}

10\,\mathrm{km}

altitude, where the mean free path is

\ell_{\rm mfp} = 1.0\times10^{-6}\,\mathrm{m}

, speed

U = 250\,\mathrm{m\,s^{-1}}

(take

t_{\rm flow}=L/U

), and molecular relaxation time

t_{\rm relax} \approx 1\times10^{-10}\,\mathrm{s}

. ((b)) Air in a MEMS sensor channel of half-width

h = 500\,\mathrm{nm}

at sea level, where

\ell_{\rm mfp} = 68\,\mathrm{nm}

, with a gas residence time

t_{\rm flow} = 1\times10^{-4}\,\mathrm{s}

and the same molecular relaxation time as in (a). ((c)) A polymer melt in injection moulding, with molecular relaxation time

t_{\rm relax} = 5.0\,\mathrm{s}

, mould-filling time

t_{\rm flow} = 0.20\,\mathrm{s}

, molecular size

\approx1\,\mathrm{nm}

and mould channel size

L \approx 1\,\mathrm{mm}

. ((d)) Water in a clean, smooth, well-wetted catheter of diameter

L = 2.0\,\mathrm{mm}

flowing at mean speed

U = 0.05\,\mathrm{m\,s^{-1}}

over a length of

20\,\mathrm{mm}

(take

t_{\rm flow}

from this length and speed), water molecular size

\approx0.3\,\mathrm{nm}

, and

t_{\rm relax} \approx 1\times10^{-12}\,\mathrm{s}

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