Water in a shallow closed layer of depth has a surface temperature gradient of

PHYSICS · Subjective · Fermi University Physics · Fluids & Continuum Physics · Fluid statics and the continuum picture · Fluid statics and the continuum picture

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

Problem

Water in a shallow closed layer of depth h=2.0 mmh = 2.0\,\mathrm{mm} has a surface temperature gradient of 1 K cm−11\,\mathrm{K\,cm^{-1}}, with xx measured from the cold end towards the hot end, and dγ/dT≈−1.5×10−4 N m−1 K−1\mathrm{d}\gamma/\mathrm{d}T \approx -1.5\times10^{-4}\,\mathrm{N\,m^{-1}\,K^{-1}}, so that dγ/dx=−0.015 Pa\mathrm{d}\gamma/\mathrm{d}x = -0.015\,\mathrm{Pa}. Someone argues: “Writing the tangential stress balance at the free surface (zz upward, xx along the surface) as μ ∂u/∂z=− dγ/dx\mu\,\partial u/\partial z = -\,\mathrm{d}\gamma/\mathrm{d}x, with μ=1.0×10−3 Pa s\mu = 1.0\times10^{-3}\,\mathrm{Pa\,s} for water, gives ∂u/∂z=+15 s−1\partial u/\partial z = +15\,\mathrm{s^{-1}}, so the surface layer is dragged towards the hot end, from high tension to low tension, just as fluid flows from high pressure to low.” Find the error in this reasoning, state the correct direction of the surface flow, and use the closed-layer result us=h ∣dγ/dx∣/(4μ)u_s = h\,|\mathrm{d}\gamma/\mathrm{d}x|/(4\mu) to compute the correct surface speed.

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