Lake water carries gas nuclei which

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids · Solid continuum mechanics and complex fluids

Bending, Ringing, Breaking and Flowing, Ch Bubbles, drops and cavitation

Problem

Lake water carries gas nuclei which, at the surface pressure of

101.3\,\mathrm{kPa}

, sit at an equilibrium radius

R_0 = 1.0\,\mathrm{\mu m}

(water:

\rho = 1000\,\mathrm{kg\,m^{-3}}

\gamma = 0.072\,\mathrm{N\,m^{-1}}

p_v = 2.3\,\mathrm{kPa}

;

g = 9.8\,\mathrm{m\,s^{-2}}

). A hydrofoil with minimum pressure coefficient

C_{p,\min} = -1.2

runs at a depth of

3\,\mathrm m

. Treat each nucleus as isothermal, holding a fixed amount of gas, and suppose it follows the local pressure quasi-statically as it passes the low-pressure region of the foil. ((a)) Show that the Blake threshold of one of these nuclei depends only on the amount of gas it holds, so that it is the same at

3\,\mathrm m

as at the surface, and find it. ((b)) Find the foil speed at which the lowest pressure on the foil first reaches the vapour pressure, and the speed at which it first reaches the nuclei's Blake threshold. ((c)) The foil runs at

16\,\mathrm{m\,s^{-1}}

. Does the vapour-pressure rule predict cavitation? Do these nuclei cavitate? Which of the two answers would change if the water were full of

10\,\mathrm{\mu m}

nuclei instead, and why?

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