Fluctuation-supported membrane--particle gap ($\delta$)

Thermal membrane undulations generate Helfrich repulsion from a nearby particle. For a sphere of radius $R$ tightly enclosed by a high-tension membrane,

For a sphere of radius $R$ moving through a membrane tube with a uniform narrow gap $\delta \ll R$, lubrication theory givesIt exceeds the unbounded-fluid Stokes drag law coefficient by $2R^2/{\delta }^2$.