Wrinkling of fluid deformable surfaces

Abstract

Wrinkling instabilities of thin elastic sheets can be used to generate periodic structures over a wide range of length scales. Viscosity of the thin elastic sheet or its surrounding medium has been shown to be responsible for dynamic processes. While this has been explored for solid as well as liquid thin elastic sheets we here consider wrinkling of fluid deformable surfaces, which show a solid-fluid duality and have been established as model systems for biomembranes and cellular sheets. We use this hydrodynamic theory and numerically explore the formation of wrinkles and their coarsening, either by a continuous reduction of the enclosed volume or the continuous increase of the surface area. Both lead to almost identical results for wrinkle formation and the coarsening process, for which a universal scaling law for the wavenumber is obtained for a broad range of surface viscosity and rate of change of volume or area. However, for large Reynolds numbers and small changes in volume or area wrinkling can be suppressed and surface hydrodynamics allows for global shape changes following the minimal energy configurations of the Helfrich energy for corresponding reduced volumes.

Figures (9)

• Figure 1: Wrinkling obtained by volume reduction with $P_V = 0.02$ and $\mathrm{Re} = 0.025$. (a) Snapshots for $t=0.0,0.4,0.6,1.2$ colour coded by mean curvature $\mathcal{H}$. The wrinkles are analysed along the equator denoted by the white line. (b) Minima and maxima of wrinkle profile along the rotational angle of the equator over time. (c) Mean curvature $\mathcal{H}$ of wrinkle profile along the rotational angle of the equator over time.
• Figure 2: Number of wrinkles $N_w$ over time for continuous volume reduction. (a) Considered for different Reynolds number $\mathrm{Re}$ (for $P_V=0.02$) and (b) for different volume reduction rates $P_V$ (for $\mathrm{Re}=0.016$).
• Figure 3: Number of wrinkles $N_w$ over time for continuous area increase. (a) Considered for different Reynolds number $\mathrm{Re}$ (for $P_A$ corresponding to $P_V=0.02$) and (b) for different increasing area rates $P_A$ (for $\mathrm{Re}=0.016$). Instead of $P_A$, we indicate the corresponding values for $P_V$ for better comparison.
• Figure 4: Phase diagram of the maximal number of wrinkles $N_{max}$ over the Reynolds number $\mathrm{Re}$ and the volume reduction rate $P_V$ (a) and the area increase rate $P_A$ (b). Instead of $P_A$ we indicate $\frac{3}{2}\vert\Omega\vert P_A$, which are the corresponding values for $P_V$, for better comparison. The black dots highlight values of $\mathrm{Re}$ and $P_V$ shown in Figures \ref{['fig:viscosity']} or \ref{['fig:viscosity2']}, respectively.
• Figure 5: Coarsening of wrinkles analysed by the wavenumber $\tilde{\nu}$ as a function of time. (a) Considered for different Reynolds numbers $\mathrm{Re}$ and $P_V=0.02$ (solid lines) and corresponding $P_A$ (dashed line). (b) Considered for different volume reduction rates $P_V$ (solid lines) and corresponding area increase rates $P_A$ (dashed lines) and $\mathrm{Re}=0.016$. The data correspond to Figures \ref{['fig:viscosity']} and \ref{['fig:viscosity2']}, but only those values which lead to wrinkling are considered. Both indicate a scaling law of $t^{-1/2}$ indicated by the black lines.