Boundary conditions select the viscous operator on Riemannian hypersurfaces: formal analysis and rigorous thin-shell limits

Abstract

A viscous fluid confined to a thin layer around a curved surface is governed, as the layer thickness vanishes, by an effective viscous operator on the surface. We show that the wall conditions select this operator. The vorticity-free (free) and stress-free (Navier) conditions, which coincide on flat walls but differ on curved ones by the shape operator, yield respectively the Hodge Laplacian and the deformation Laplacian, and these differ universally, on any hypersurface, by twice the Ricci curvature; a one-parameter family of wall conditions joins them, with an effective operator that couples to the extrinsic geometry only in between. We prove this in two forms: formally, by matched asymptotics, on an arbitrary hypersurface, and rigorously, as Mosco convergence of the viscous energy forms, hence with resolvent, semigroup and spectral convergence, on surfaces of revolution. The stress-free limit on general surfaces is due to Miura and is recovered here; the rigorous vorticity-free limit beyond the sphere, via a uniform Gaffney inequality, together with the interpolating family and the spectral packaging, is new, and the analysis makes precise a conflation of the two conditions in the classical spherical treatment. The extension-dependence of the operator found on the ellipsoid is explained as a dependence on the wall condition.