1. Folding of Viscous Sheets and Filaments
Maksim Skorobogatiy
L. Mahadevan
MIT, Department of Mechanical Engineering

2. Examples of Folding of Viscous Filaments
The buckling of solids is a well established subject whose origins date back to the work of L. Euler and Joh. Bernoulli. Take an example of a simple pillar. If one puts a big enough weight on it the pillar will buckle and eventually break. This instability which arises as a result of the competition between axial compression and bending in slender objects is not restricted to solids; it can also occur in the flows of fluids. As in the case of solids, the buckling, folding, and coiling of thin sheets and filaments of fluids occurs on length scales spanning several orders of magnitude, from geophysics and materials processing to soft-matter physics. On the right figure part a) a flow of the melted quartz on the surface of the rock is photographed. Because of the high viscosity of quartz the axial compression generated during the downward flow under the force of gravity is high enough to lead to the buckling instability and instead of flowing straight down the jet of quartz starts meandering. Everyday examples of the buckling instability phenomena abound. In the kitchen, when a sheet of honey, maple syrup or cake batter is poured onto a surface from a sufficient height, near the surface the thin sheet b) is laid out in a series of 2D folds, while in the bathroom analogous phenomena can be observed with shampoo. Analogous phenomena also occur when the sheet is replaced by a slender jet or filament which coils in 3D instead of folding. It is also possible to confine such a filament to lie in a vertical plane, as in a soap film c) and instead of coiling a slender filament will again exhibit 2D folding.
While buckling in solids is a very developed subject analogous phenomena in liquids received a much smaller attention. Partly it is the case because buckling phenomena in solids is a time independent problem, while the same phenomena in liquids is inherently time dependent. The aim of our work is to provide a theoretical formulation and computational algorithm for the general problem of 2D folding of viscous filaments and sheets. To our knowledge this is the first time such formulation is presented. We confirm our formulation by experimental and computational justification of the scaling laws that can be derived for the 2D folding from the independent general arguments.
Examples of Folding of Viscous Filaments

3. Hydrodynamic Description
For example, we consider a 2D folding of a viscous jet confined to a 2D vertical soup interface. When jet first touches the substrate it starts buckling, thus leading to the curving of the jet. Because of the curving and the high viscosity of the jet that leads to the effective viscose torque counteracting the gravitational force. Competition of the viscose and gravitational forces lead to the periodic folding of a jet.
At each moment of time t we characterize the shape of the jet by its inclination angle to the horizontal axis f(s,t) along the length of the curve s. Inertia of a jet is ignored. We formulate the time dependent problem of the jet dynamics via a set of time dependent partial differential equations with time dependent boundary conditions. Other parameters involved in the equations are the components of the stress in the horizontal and vertical directions n1(s,t) and n2(s,t).
Hydrodynamic Description

4. We were able to develop a stable algorithm for solving the above mentioned PDE’s with time dependent boundary conditions that allowed us to simulate periodic time dependent profiles of the folding jet. On the left figure one can see a simulation of half a complete fold. Going from profiles 1 to 5 the point where the jut touches a substrate migrate with time, and half a fold is complete when the jet touches itself as in the profile 5.
To test our theoretical and computational formulations we verified computationally the validity of the scaling law predicting that the length of the periodic fold is linearly proportional the 1/4 power of the jet’s velocity. Experimental and computational methods both confirmed this prediction with a good accuracy.
Thus, we are confident that our theoretical and computational formulation for the 2D folding of slender filaments indeed valid.
Scaling Laws