Free-molecule mobility of polyhedra and other convex hard-bodies

The free-molecule drag of hard convex bodies can be written in dyadic notation in terms of two purely geometrical integrals over the closed body surface, A=∫dA and N=A−1∫nndA, where n is the outward normal to the surface element dA at a given point. For sufficiently symmetric bodies, N is isotropic, and the drag is exactly proportional to the total surface area A, with a proportionality coefficient β independent of the objectʹs geometry and equal to the well-known value βs for a sphere. This result yields effortlessly the drag for all regular and semi-regular polyhedra, previously known only for cubes. β differs generally from βs for bodies with anisotropic drag tensors; but it is a local maximum with respect to (small) arbitrary shape changes away from that of any body with an isotropic drag tensor. Hence, moderate departures from drag isotropy shift β very slightly below βs. This maximum property provides a rationale for the common assumption of an approximate relation between area and drag. However, the relation involves the total surface area rather than projected areas, and leads to accurate predictions only for bodies with moderately anisotropic drag tensors. The tensorial method introduced leads also to simple results for less symmetric bodies, such as pyramids, double pyramids, parallelepipeds and axisymmetric figures. When collisions are predominantly inelastic, the ratio β/βs departs far less from unity than for elastic collisions. Similar properties are obtained for the thermophoretic force.