Hydrodynamics in curved membranes: The effect of geometry on particulate mobility

We determine the particulate transport properties of fluid membranes with nontrivial geometries that are surrounded by viscous Newtonian solvents. Previously, this problem in membrane hydrodynamics was discussed for the case of flat membranes by Saffman and Delbrück [P. G. Saffman and M. Delbrück, Proc. Natl. Acad. Sci. U.S.A. 72, 3111 (1975)]. We review and develop the formalism necessary to consider the hydrodynamics of membranes with arbitrary curvature and show that the effect of local geometry is twofold. First, local Gaussian curvature introduces in-plane viscous stresses even for situations in which the velocity field is coordinate-independent. Secondly, even in the absence of Gaussian curvature, the geometry of the membrane modifies the momentum transport between the bulk fluids and the membrane. We illustrate these effects by examining in detail the mobilities of particles bound to spherical and cylindrical membranes. These two examples provide experimentally testable predictions for particulate mobilities and membrane velocity fields on giant unilamellar vesicles and membrane tethers. Finally, we use the examples of spherical and cylindrical membranes to demonstrate how the global geometry and topology of the membrane influences the membrane velocities and the particle mobilities.

Figure 1

(Color online) Membrane fluid velocity for a spherical membrane immersed in a symmetric embedding fluid environment $\left({\eta }_{+}={\eta }_{-}\right)$ due to a force ${\vec{F}}_0=F_{\text{np}}\widehat{y}$ at the north pole, for decreasing values of $R/{\ell }_0$. The magnitudes of the velocity vectors in each plot are scaled by following scale factors: (a) 15; (b) 2; and (c) $\frac{1}{2}$.

Figure 2

(Color online) Dimensionless mobilities of a pointlike particle embedded in a sphere as a function of the ratio $R/{\ell }_{+}$ for ${\ell }_{+}/a={10}^3$ and ${\ell }_{-}=0.1{\ell }_{+}$ [green (gray) curves], ${\ell }_{-}={\ell }_{+}$ (black curves), and ${\ell }_{-}=10{\ell }_{+}$ (dotted curves). The dot-dashed curves are the appropriate analytic expressions for the mobilities in the limit $R/{\ell }_{+}\to 0$ (see text); the dashed curves are the flat-space limits, Eq. (49). (a) “Free” mobility ${\mu }_{\text{free}}$; (b) mobility in a pinned membrane ${\mu }_{\text{pin}}$; and (c) mobility in the co-rotating frame of reference, ${\mu }_{\text{corot}}$.

Figure 3

(Color online) Membrane fluid velocity for a cylindrical membrane immersed in a symmetric external fluid environment $\left({\eta }_{+}={\eta }_{-}\right)$ due to a force (a) ${\vec{F}}_0=F_0{\widehat{\theta }}_0$ and (b) ${\vec{F}}_0=F_0\widehat{z}$ at the origin, for $R/{\ell }_0=0.01$ and 1. The magnitudes of the velocity vectors in the plot for ${\vec{F}}_0=F_0\widehat{\theta }$, $R/{\ell }_0=0.01$ are scaled by a factor of 0.1 relative to the other plots. In addition, the region of the cylinder pictured for each cases is variable; the length of the cylinder is indicated next to each figure. (c) Detail of the velocity magnitudes along the line $\theta =0$, $\zeta >0$ for the velocities for $R/{\ell }_0=0.01$. The points are the exact numerical solutions for ${\vec{F}}_0=F_0{\widehat{\theta }}_0$ (squares) and ${\vec{F}}_0=F_0\widehat{z}$ (circles); the solid curves are the analytic expressions, Eq. (85) [green (gray) curve] and Eq. (86) (black curve), respectively.

Figure 4

(Color online) Dimensionless mobilities of a pointlike particle embedded in a cylindrical membrane as a function of the ratio $R/{\ell }_{-}$ for ${\ell }_{-}/a=4\ast {10}^3$ and ${\ell }_{+}=0.1{\ell }_{-}$ [green (gray) curves], ${\ell }_{+}={\ell }_{-}$ (black curves), and ${\ell }_{+}=10{\ell }_{-}$ (dotted curves). The dot-dashed curves are the analytic expressions for the mobilities in the limit $R/{\ell }_{+}\to 0$ (see text); the dashed curves are the flat-space limits, Eq. (49). (a) Mobility ${\mu }_z$ for a particle subject to a force in the $\widehat{z}$ direction. (b), (c), and (d) Mobilities for a particle subject to a force in the $\widehat{\theta }$ direction: (b) in a “free” membrane; (c) in a pinned membrane; (d) in a locally co-rotating frame of reference.