A stripe of dye thick is to be thinned to

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Kinematics, conservation laws and ideal flow

Fluid Dynamics, Ch Kinematics in three dimensions and Cauchy's equation of motion

Problem

A stripe of dye 1 cm1\,\mathrm{cm} thick is to be thinned to 10 μm10\,\mu\mathrm{m}, a factor of 10001000, by a steady flow of velocity-gradient magnitude S=1 s−1S=1\,\mathrm{s^{-1}}. Estimate the time this takes if the flow is ((a)) a pure strain, with the stripe laid along the stretching principal axis, so that its thickness lies along the compressing axis; ((b)) a plane shear of the same gradient magnitude, with the stripe laid across the flow direction. Use ln⁡1000=6.9\ln 1000=6.9.

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