A thin layer of viscous fluid between two plates undergoes simple shear

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Manifolds, charts and tangent spaces

Problem

A thin layer of viscous fluid between two plates undergoes simple shear, with velocity field X=k y ∂xX=k\,y\,\partial_x on the plane R2\mathbb{R}^2, where k=2 s−1k=2\,\mathrm{s^{-1}} is a constant shear rate. A dye particle starts at (x0,y0)=(1,1) m(x_0,y_0)=(1,1)\,\mathrm{m} at time t=0t=0. Find the flow φt(x0,y0)\varphi_t(x_0,y_0) of XX, and give the particle's position at t=3 st=3\,\mathrm{s}.

Browse the problem library