A small toy boat of length floats on clean water ( , )

PHYSICS · Subjective · Fermi University Physics · Fluids & Continuum Physics · Fluid statics and the continuum picture · Viscous flow and dynamical similarity

Fluid Dynamics, Ch Interfaces: capillary statics and the stress conditions at a surface

Problem

A small toy boat of length L=4.0 cmL = 4.0\,\mathrm{cm} floats on clean water (γwater=0.0728 N m−1\gamma_{\rm water} = 0.0728\,\mathrm{N\,m^{-1}}, ρ=998 kg m−3\rho = 998\,\mathrm{kg\,m^{-3}}). A drop of soap solution released at the stern lowers the surface tension there to γsoap=0.030 N m−1\gamma_{\rm soap} = 0.030\,\mathrm{N\,m^{-1}}, so the boat experiences an unbalanced pull, per unit width, of magnitude Δγ=γwater−γsoap\Delta\gamma = \gamma_{\rm water}-\gamma_{\rm soap} towards the bow, where the surface tension is still that of clean water. Estimate the boat's steady forward speed, assuming its Marangoni thrust per unit width is eventually balanced by an inertial drag per unit width of order ρU2L\rho U^2 L, where UU is the boat's speed.

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