The equations governing lipid membrane dynamics in planar, spherical, and
cylindrical geometries are presented here. Unperturbed and first-order
perturbed equations are determined and non-dimensionalized. In membrane systems
with a nonzero base flow, perturbed in-plane and out-of-plane quantities are
found to vary over different length scales. A new dimensionless number, named
the Scriven--Love number, and the well-known F\"oppl--von K\'arm\'an number
result from a scaling analysis. The Scriven--Love number compares out-of-plane
forces arising from the in-plane, intramembrane viscous stresses to the
familiar elastic bending forces, while the F\"oppl--von K\'arm\'an number
compares tension to bending forces. Both numbers are calculated in past
experimental works, and span a wide range of values in various biological
processes across different geometries. In situations with large Scriven--Love
and F\"oppl--von K\'arm\'an numbers, the dynamical response of a perturbed
membrane is dominated by out-of-plane viscous and surface tension forces---with
bending forces playing a negligible role. Calculations of non-negligible
Scriven--Love numbers in various biological processes and in vitro experiments
show in-plane intramembrane viscous flows cannot generally be ignored when
analyzing lipid membrane behavior.Comment: 16 pages, 7 figures, 5 table
