Geometry-memory interaction in time-fractional convection within Julia-set cavities

Temporal memory and escape-time-resolved boundary roughness are examined as distinct sources of nonlocal behavior in natural convection. Two bounded prefractal domains are constructed from the quadratic Julia map, representing the San Marco and Siegel-disk geometries. On each domain, a Boussinesq model with Caputo time derivatives is discretized by the L1 formula and a semi-implicit finite-element procedure. The reported calculations span 1 ≤ n ≤ 60 , 0.6 ≤ α ≤ 0.99 , 0.01 ≤ Pr ≤ 10 , and 1 0 2 ≤ Ra ≤ 1 0 6 . The fractional order changes the relaxation path and the persistence of the transient response, whereas any limiting equilibrium satisfies the classical steady balance after the Caputo history has decayed. Increasing the escape-time cutoff decreases the resolved cavity area and increases a perimeter-to-area roughness measure. At t = 1 , the reported kinetic-energy fits scale as n − 1.44 for the San Marco sequence and n − 1.65 for the Siegel-disk sequence. A pooled set of 1580 realizations exhibits an empirical energy-enstrophy collapse. The analysis distinguishes conventional wall Nusselt numbers from the volume-averaged heat-flux indicators plotted in the original figures. All fitted relations are interpreted as descriptive over finite parameter and geometric-resolution ranges, not as universal fractal laws.