  We discuss computational analysis of phononic thermal conduction in the
suspended membrane geometry, in the experimentally commonly appearing case
where heat can flow out radially in two dimensions from a central source. As we
are mostly interested in the low-temperature behavior where bulk scattering of
phonons becomes irrelevant, we study the limit where all phonon scattering
takes place at the membrane surfaces. Moreover, we limit the discussion here to
the case where this surface scattering is fully diffusive, the so called
Casimir limit. Our analysis shows that in the two-dimensional case, no analytic
results are available, in contrast to the well known 1D Casimir limit.
Numerical solutions are presented for the temperature profiles in the membrane
radial direction, for several different membrane thicknesses. Our results can
be applied, for example, in the design of membrane-supported bolometric
radiation detectors.
