Biological lipid membranes make up the boundary of the cell and many of its internal organelles, including the nucleus, endoplasmic reticulum, and Golgi complex. Such membranes are not simply static, semi-permeable barriers protecting their internal contents, but rather play a dynamic role in many cellular processes. Importantly, lipid membranes are unique materials where lipids flow in-plane as a two-dimensional fluid, yet the membrane bends out-of-plane as an elastic shell. Many past works neglected in-plane fluidity when describing lipid membrane shapes and their stability, thus ignoring important hydrodynamic couplings between in-plane and out-of-plane behavior. We develop an irreversible thermodynamic framework for arbitrarily curved and deforming lipid membranes and determine their dynamical equations of motion – including the aforementioned coupling1. We find in-plane viscous stresses arising from lipid flows lead to an out-of-plane force, despite the membrane bending elastically out-of-plane. We non-dimensionalize the dynamical equations and find a new dimensionless number, named the Scriven-Love number, comparing out-of-plane viscous forces to well-known bending forces2. Works ignoring membrane fluidity implicitly set this number to zero, however we calculate the Scriven-Love number in past experimental works and find many instances where it is large. In such cases, membrane dynamics are governed by out-of-plane viscous forces, bending plays a negligible role, and we can find novel membrane shape changes and instabilities. For example, in tubes the Scriven-Love number mediates a hydrodynamic instability involving in-plane lipid flows. At low Scriven-Love numbers, tubes undergo a pearling instability. At high Scriven-Love numbers, however, time-oscillating solutions are admitted and moreover the tube is convectively unstable: local perturbations are carried with a base flow of lipids to yield nontrivial membrane shapes3. Our results may be relevant in understanding various biological situations, such as lipid flows along an axon body and tubes shooting from the endoplasmic reticulum.
1A. Sahu, ..., K.K. Mandadapu. "Irreversible thermodynamics of curved lipid membranes". Phys. Rev. E 96 (2017), arXiv:1701.06495
2A. Sahu, A. Glisman, J. Tchoufag, K.K. Mandadapu. "Geometry and dynamics of lipid membranes: The Scriven-Love number". Phys. Rev. E 101 (2020), arXiv:1910.10693
3J. Tchoufag, A. Sahu, K.K. Mandadapu. "Absolute/convective instabilities and front propagation in lipid membrane tubes". (In preparation)
