To elucidate the dynamics of free (buoyancy-driven) subduction of oceanic lithosphere, I study a model in which a 2-D sheet of viscous fluid with thickness h and viscosity gamma eta(1) subducts in an infinitely deep ambient fluid with viscosity eta(1). Numerical solutions for the sheet's evolution are obtained using the boundary-element method (BEM), starting from an initial configuration comprising a short 'protoslab' attached to a longer horizontal 'plate' that is free to move laterally beneath an impermeable traction-free surface. Interpretation of the solutions using thin viscous sheet theory shows that the fundamental length scale controlling the subduction is the 'bending length' l(b), defined at each instant as the length of the portion of the sheet's midsurface where the rate of change of curvature is significant. Geophysically speaking, l(b) is the sum of the lengths of the slab and of the region seaward of the trench where flexural bulging occurs. The bending length in turn enters into the definition of the sheet's dimensionless 'stiffness' S equivalent to gamma(h/l(b))3, which controls whether the sinking speed of the slab is determined by the viscosity of the sheet itself (S 1) or by that of the ambient fluid (S < 1). Motivated by laboratory observations of different modes of subduction (retreating versus advancing trench, folding versus no folding, etc.) in fluid layers with finite depth, I calculate numerically the dip theta(D) of the slab's leading end as a function of gamma and the normalized depth D/h to which it has penetrated. The contours of the function theta(D)(gamma, D/h) strongly resemble the intermode boundaries in the laboratory-based regime diagram of Schellart, supporting the hypothesis that the mode of subduction observed at long times in experiments is controlled by the dip of the slab's leading end when it reaches the bottom of the layer. In particular, the BEM solutions explain why trenches advance in the laboratory only when gamma lies in an intermediate range, and why they retreat when gamma is either smaller or larger than this. Application of the BEM model to Wu et al.'s compilation of the minimum curvature radii of subducted slabs suggests gamma is an element of [140, 510] for the Earth. This is too small to permit the laboratory-type 'trench advancing' mode, in agreement with the lack of tomographic evidence for slabs that are 'bent over backwards'.
