A small air bubble in glycerine ( against air) is momentarily stretched by a stirring…

PHYSICS · Multi-part · 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

A small air bubble in glycerine (γ=0.063 N m−1\gamma = 0.063\,\mathrm{N\,m^{-1}} against air) is momentarily stretched by a stirring flow into a prolate ellipsoidal shape. At the pointed tip the surface is spherical in cross-section with both principal radii equal, R1=R2=0.60 mmR_1=R_2=0.60\,\mathrm{mm}. At a point on the rounded equator the principal radii are R1=0.60 mmR_1=0.60\,\mathrm{mm} (in the meridional plane) and R2=1.8 mmR_2=1.8\,\mathrm{mm} (round the girth). While the flow acts, viscous normal stresses in the glycerine add to the pressure jump, so only its capillary part is asked for. ((a)) Find the capillary contribution γ ∇⋅n^\gamma\,\nabla\cdot\hat{\mathbf n} to the normal-stress jump at the tip. ((b)) Find the same contribution at the equatorial point.

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