Pulsating Microstructure
Homogenization & multiscale media
Overview
Fast oscillations of a microstructure can modify macroscopic transport. In particular, an oscillating geometry may induce an effective advective contribution in a diffusion equation.
Pulsating microstructure at three scales
A travelling bending wave deforms the obstacle in each periodic cell. The fast temporal period is exactly the spatial period: \(T_\varepsilon=\varepsilon\).
Microscopic domains \(\Omega_\varepsilon(t)\)
Effective medium
Reference cell \(Y_x(t,\tau)\)
All fast phases at fixed macroscopic time
Click any microscopic domain or the effective domain to select the zoom location. The three magnified cells beneath the microscopic domains show the same location at each ε-scale, making their different oscillation speeds easy to compare. The limit reference cell shows the region swept out over all fast phases at the selected macroscopic position and time. Freeze macroscopic time to animate the local microscopic bending instead. The effective medium is schematic; no homogenized equation or fluid velocity is computed.
The fast bending cycle uses \(\tau=t/\varepsilon\), while the two optional checkboxes modify a slow rotation \(\theta(x,t)\) of the same obstacle. The drawing period follows \(T_\varepsilon=\varepsilon\) for all three scales.