Surface tension from droplet deformation · RMPH 2021 #A.1

Physics · fluid-mechanics

Problem

All situations take place in an enclosure thermostated at 70°C with air and saturated water vapor. The pressure is constant at p0=1.0×105 N m−2p_0 = 1.0 \times 10^5 \, \text{N m}^{-2}. Physical constants: g=9.8 m s−2g = 9.8 \, \text{m s}^{-2}, μ0=4×10−7 H m−1\mu_0 = 4 \times 10^{-7} \, \text{H m}^{-1}, ε0=8.9×10−12 F m−1\varepsilon_0 = 8.9 \times 10^{-12} \, \text{F m}^{-1} Water properties: Density ρ=1.0×103 kg⋅m−3\rho = 1.0 \times 10^3 \, \text{kg} \cdot \text{m}^{-3}, Saturated vapor pressure at 70°C ps=3.1×104 N⋅m−2p_s = 3.1 \times 10^4 \, \text{N} \cdot \text{m}^{-2}, Surface tension coefficient at 70°C σ=6.5×10−2 N⋅m−1\sigma = 6.5 \times 10^{-2} \, \text{N} \cdot \text{m}^{-1}, Magnetic susceptibility χ=−9.0×10−6\chi = -9.0 \times 10^{-6} You will study the falling with constant speed of water droplets. An ultra-fast, enlarged photograph shows that the falling water droplet remains approximately spherical, retains its symmetry of rotation around the vertical axis but undergoes a slight elongation along this axis, reaching a length 2h=2.0×10−3 m2h = 2.0 \times 10^{-3} \, \text{m} (measured along the axis of vertical symmetry). At the ends of the longitudinal axis of symmetry of the droplet, the radii of curvature of its surface are respectively R1=h−εR_1 = h - \varepsilon and R2=h+εR_2 = h + \varepsilon with ε=7.5×10−5 m\varepsilon = 7.5 \times 10^{-5} \, \text{m}. Using the data obtained from the ultrafast photo and physical quantities indicated in the statement, determine the value of σ\sigma - the surface tension coefficient of water.

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