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In a previous publication, we discussed the formalism and some computationalresults for phononic thermal conduction in the suspended membrane geometry forradial heat flow from a central source, which is a common geometry for somelow-temperature detectors, for example. We studied the case where onlydiffusive surface scattering is present, the so called Casimir limit, which canbe experimentally relevant at temperatures below $\sim$ 10 K in typicalmaterials, and even higher for ultrathin samples. Here, we extend our studiesto much thinner membranes, obtaining numerical results for geometries which aremore typical in experiments. In addition, we interpret the results in terms ofa small signal and differential thermal conductance, so that guidelines fordesigning devices, such as low-temperature bolometric detectors, are moreeasily obtained. Scaling with membrane dimensions is shown to differsignificantly from the bulk scattering, and, in particular, thinning themembrane is shown to lead to a much stronger reduction in thermal conductancethan what one would envision from the simplest bulk formulas.

Radial phononic thermal conductance in thin membranes in the Casimir limit: Design guidelines for devices