Fig. 3. Heat transfer under 2D and 3D density modulation.
a Density ρ and thermal conductivity κ profiles (color maps) of 2D traveling waves such that the profiles move at speed v0 in x direction (marked by dashed lines). The density is modulated by mass movement in y direction at speed v0y (green arrows). The upper and lower boundaries are periodic. b A three-dimensional (3D) model in a (r,θ,x) cylindrical coordinate system. Similar ρ and κ modulations are achieved with fixed (white) and moving (yellow) fan-shaped plates. c–f Backward (c, e) and forward (d, f) temperature distributions on the top line r = R2, θ = π/2 extracted from simulated results for the 2D (lines) and 3D (scatters) models at t = N2π/Ω (N is a large enough integer to achieve time-harmonic steady state). The insets show the entire temperature distributions. The results in e and f use a hypothetical 3D model where the moving plates have time-varying masses, violating the law of mass conservation.