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. We here consider wrinkling of fluid deformable surfaces. In contrast with thin elastic sheets, with in-plane and out-of-plane elasticity, these surfaces are characterized by in-plane viscous flow and out-of-plane elasticity 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 by the continuous increase of the surface area. Both lead to almost identical results for wrinkle formation and the coarsening process, for which a 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.

Keywords: solid–fluid duality; surface viscosity; wrinkling.

Figure 1.

Wrinkling obtained by volume reduction with P_V = 0.02 and 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 Re (for P_V = 0.02) and (b) for different volume reduction rates P_V (for Re = 0.016).

Figure 3.

Number of wrinkles N_w over time for continuous area increase. (a) Considered for different Reynolds number Re (for P_A corresponding to P_V = 0.02) and (b) for different increasing area rates P_A (for 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 Re and the volume reduction rate P_V (a) and the area increase rate P_A (b). Instead of P_A, we indicate (3/2)|Ω|P_A, which are the corresponding values for P_V, for better comparison. The black dots highlight values of Re and P_V shown in figures 2 or 3, respectively.

Figure 5.

Coarsening of wrinkles analysed by the wavenumber $\tilde{\nu }$ as a function of time. (a) Considered for different Reynolds numbers 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 Re = 0.016. The data correspond to figures 2 and 3, but only those values which lead to wrinkling are considered. Both indicate a scaling law of t^−1/2 indicated by the black lines.

Figure 6.

Wrinkling instability and subsequent coarsening on the unit sphere for Re = 0.016 and P_A = 1.0. Time instances are shown for t = 0.0, 0.1, 0.2, 0.4, 0.6, from left to right. The colour coding corresponds to the mean curvature $\mathcal{H}$. A corresponding video of the evolution is provided in the electronic supplementary material.

Figure 7.

Evolution of the Helfrich energy E_H and the kinetic energy E_kin for different Reynolds numbers Re (a,c) for decreasing volume with P_V = 0.002 and (b,d) for increasing surface area with P_A = (3/2)|Ω|P_V. The corresponding plots are almost identical. As a reference, we plot the minimal Helfrich energy for the corresponding reduced volume (dashed green line).

Figure 8.

Visualization of the tangential fluid flow by the surface LIC filter. The colour demonstrates the magnitude of the tangential velocity. The plots correspond to the time instants of the maximal number of wrinkles highlighted in figure 2a for P_V = 0.02 and different Reynolds number (a) Re = 0.016, (b) Re = 0.025, (c) Re = 0.05, (d) Re = 0.1, (e) Re = 1.0.

Figure 9.

Convergence study for continuous volume reduction. The considered parameters are Re = 0.05 and P_V = 0.02. h denotes the mesh size. (a) Inextensibility error e indicates third-order convergence in space and first-order convergence in time. (b) Number of wrinkles over time. The time wrinkling and the coarsening process are almost indistinguishable for fine mesh sizes.