Lateral hydrodynamics in supported membranes: the Evans-Sackmann model and its extensions.

We review the theoretical development and modern applications of the Evans-Sackmann hydrodynamic model for lateral transport in supported fluid membranes. We first cover the original formulation, emphasizing the linear momentum decay term that captures membrane-substrate coupling mediated by a thin lubricating fluid layer. This coupling term enables quantitative interpretation of tracer diffusion measurements in supported bilayers. We then survey theoretical extensions that relax standard boundary conditions at the inclusion perimeter. Here, inclusions refer to embedded objects such as proteins, lipid domains, or tracer particles within the membrane. We discuss the drag on a disk and on a liquid domain, as well as the dynamics of membrane phase separation. We also show that the supported-membrane mobility tensor provides a unified framework for correlated diffusion, polymer dynamics, phase separation kinetics, and many-body interactions. Finally, we discuss recent extensions to active and chiral membranes, where odd viscosity provides a transverse hydrodynamic response and offers a possible route for detecting chirality in two-dimensional fluids.