# Bending and stretching dissipation potentials of a Newtonian sheet

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## Thin-sheet theory: essential concepts

Because our subduction model involves a thin viscous sheet, we summarize here the essential concepts of thin-sheet theory (Ribe, 2001) in Fig. 2.1. At each point on the sheet, let d1 be the tangent vector to the sheet’s midsurface, d2 the unit normal to the midsurface, and d3 the unit vector pointing out of the page. The Cartesian coordinates of the midsurface are x0(s), and those of an arbitrary point on the sheet are x(s; z) x0(s) + zd2; (􀀀H=2 z H=2), where H is the sheets thickness. Let the inclination of the midsurface to the horizontal be (s), and the curvature of the midsurface be K(s) @=@s. The balance of forces and moments on an element of the sheet requires N0 + gh + F = 0; M0 + d1 N = 0.

### Boundary-integral equation for an immersed uid sheet

Let V1 and V2 be the volumes occupied by uids 1 and 2, respectively, and let C be the interface between them. The general integral represen where J and K are the velocity and stress Green functions for Stokes ow satisng the relevant boundary conditions which will be discussed in more detail in the next chapter. Let 1(x) = 1, 1=2 or 0 if x is in V1, right on the contour, or in V2, respectively, and dene 2(x) similarly with the subscripts 1 and 2 interchanged. The unit normal vector n is directed out of uid 2 and into uid 1.
On the contour C, the velocity is continuous while the modied normal stress undergoes a jump proportional to the dierence of the densities of the two uids. Symbolically, u1 = u2; (2.41) f2(y) = f1(y) + (gy)n.

#### The discrete viscous thin sheet model

In this section, discrete forms of the smooth equations discussed above are introduced, using concepts from discrete dierential geometry developed by Audoly et al. (2013).
The rst step is to represent the sheet’s midsurface by a collection of discrete vertices and connecting edges, as shown in Fig. 2.2. The positions of the (n + 2) vertices are X0(t);X1(t); : : : ;Xn+1(t). The material tangent T becomes the segments between the vertices T 0(t); T 1(t); : : : ,T n(t), T i(t) = Xi+1(t) 􀀀Xi(t): (2.51).
Here and henceforth, subscripts and superscripts denote variables that are dened on vertices and edges, respectively.

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The Green’s function for sheet subduction

For the geometry of Fig. 3.1, the free-slip surface (mirror symmetry) condition at x2 = 0 can be automatically satised by adding to the innite- uid Green’s function the corresponding Green’s function for the image point located above the free surface. The resulting symmetrized form of J no longer has a logarithmic singularity at r ! 1, indicating that the presence of the boundary has resolved Stokes’s paradox. The Green’s function Jij = J(Xj 􀀀Xi) that satises the free-slip boundary
conditions is J(Xj 􀀀Xi) = J(Xj 􀀀Xi).

Acknowledgements
Abstract
Resume
1 Introduction
1.1 Free subduction
1.2 Buckling instabilities in microchannels
2 Thin Newtonian sheets: The discrete approach
2.1 Thin-sheet theory: essential concepts
2.2 Lagrangian description of a thin sheet
2.3 Boundary-integral equation for an immerseduid sheet .
2.4 The discrete viscous thin sheet model
3 Subduction of a Newtonian sheet
3.1 Bending and stretching dissipation potentials of a Newtonian sheet
3.2 Numerical implementation
3.3 The Green’s function for sheet subduction
3.4 Numerical solutions and analysis
3.4.1 Vertical sheet
3.4.2 Horizontal sheet
3.4.3 Subduction of a bent sheet
4 Parallel code for three dimensional multi phaseow
4.1 Mathematical formulation
4.2 Numerical method
4.2.1 Interface treatment
4.2.2 Extended interface for parallel processing
4.2.3 Solution procedure in parallel computing
5 Simulation of viscous folding in diverging microchannels