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Hydrodynamics

Problem 1.337

When a sphere of radius \(r_{1}=1.2 \mathrm{~mm}\) moves in glycerin, the laminar flow is observed if the velocity of the sphere does not exceed \(v_{1}=23 \mathrm{~cm} / \mathrm{s}\). At what minimum velocity \(v_{2}\) of a sphere of radius \(r_{2}=5.5 \mathrm{~cm}\) will the flow in water become turbulent? The viscosities of glycerin and water are equal to \(\eta_{1}=13.9 \mathrm{P}\) and \(\eta_{2}\) \(=0.011 \mathrm{P}\) respectively.

Reveal Answer
v2=v1r1ρ1η2r2ρ2η1=5μm/sv_{2}=v_{1} \frac{r_{1} \rho_{1} \eta_{2}}{r_{2} \rho_{2} \eta_{1}}=5 \quad \mu \mathrm{m} / \mathrm{s}
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