Dynamics of filaments and membranes in a viscous fluid

Motivated by the motion of biopolymers and membranes in solution, this article presents a formulation of the equations of motion for curves and surfaces in a viscous fluid. The focus is on geometrical aspects and simple variational methods for calculating internal stresses and forces, and the full nonlinear equations of motion are derived. In the case of membranes, particular attention is paid to the formulation of the equations of hydrodynamics on a curved, deforming surface. The formalism is illustrated by two simple case studies: (1) the twirling instability of straight elastic rod rotating in a viscous fluid and (2) the pearling and buckling instabilities of a tubular liposome or polymersome.

Figure 1

A two-dimensional being (after 10) considers the relation between the circumference and radius of a circle in his curved world; $K$ is a number he can determine by making careful measurements of length.

Figure 2

(Color online) The directors for a rod are chosen along the principal axes of the cross section of the rod.

Figure 3

The basis vectors of the material and Serret-Frenet frames in the plane of the rod cross section.

Figure 4

Two rods with the same curvature $\kappa$ and the same twist ${\Omega }_3$. For each rod, the black line traces the path of the tip of ${\widehat{\mathbf{d}}}_1$. Note that the fact that one rod is closed and one is open is immaterial for determining the twist.

Figure 5

The variation of a curve stretches the curve.

Figure 6

Rotation of directors under a variation with $\delta {\chi }_1\ne 0$, $\delta {\chi }_2=\delta {\chi }_3=0$.

Figure 7

An elastic filament immersed in a liquid with viscosity $\eta$, and twirled about the $z$ axis with angular speed ${\omega }_0$. The straight state is unstable above a critical angular speed.

Figure 8

(Color online) The height near the origin given by the second fundamental form: $h\approx K_{\alpha \beta }\Delta {\xi }^{\alpha }\Delta {\xi }^{\beta }$, where $\Delta {\xi }^1=\Delta x$ and $\Delta {\xi }^2=\Delta y$.

Figure 9

Equilibrium shapes of a vesicle subject deformed with an optical tweezer. The magnitude of the force increases from left to right. From 43.

Figure 10

(Color online) A section of a tubule, showing the forces per unit length ${\mathbf{F}}^z$ and $R{\mathbf{F}}^{\phi }$.

Figure 11

Dimensionless growth rate $\lambda \mu a^2∕\kappa$ vs dimensionless wave number $qa$ for the pearling instability of a tubular polymersome, in which membrane surface viscosity dominates the dissipation. The solid lines have ${\mathcal{l}}_{\mathrm{SD}}∕a=1000$ and the dashed line has ${\mathcal{l}}_{\mathrm{SD}}∕a=\infty$.

Figure 12

Dimensionless growth rate $\lambda \mu a^2∕\kappa$ vs dimensionless wave number $qa$ for the buckling instability of a tubular polymersome, in which membrane surface viscosity dominates the dissipation. The solid lines have ${\mathcal{l}}_{\mathrm{SD}}∕a=1000$ and the dashed line has ${\mathcal{l}}_{\mathrm{SD}}∕a=\infty$.