The transient spreading of a viscous fluid beneath an elastic sheet adhered to the substrate is controlled by the dynamics at the tip where the divergence of viscous stresses necessitates the formation of a vapor tip separating the fluid front and fracture front. The model for elastic-plated currents is extended for an axisymmetric geometry with analysis showing that adhesion gives rise to the possibility of static, elastic droplets and to two dynamical regimes of spreading; viscosity dominant spreading controlled by flow of viscous fluid into the vapor tip, and adhesion dominant spreading. Constant flux experiments using clear, PDMS elastic sheets enable new, direct measurements of the vapor tip and confirm the existence of spreading regimes controlled by viscosity and adhesion. The theory and experiments thereby provide an important test coupling the dynamics of flow with elastic deformation and have implications in fluid-driven fracturing of elastic media more generally.

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The formation of fold-thrust belts at convergent margins is a dynamic process. Accretion of weak sediments to the front of the overriding plate results in crustal thickening and continued flexural subsidence of the underthrusting plate. Fold-thrust belts are often treated as a Coulomb wedge having self-similar geometries with a critical taper, and either a rigid or isostatically compensated base. In this paper we build upon this work by developing a new dynamic model to investigate both the role of the thickness and material properties of the incoming sediment, and the flexure in the underthrusting plate in controlling the behaviour and evolution of fold-thrust belts. Our analysis shows that the evolution of fold-thrust belts can be dominated 2 T. V. Ball, C. E. Penney, J. A. Neufeld, A. C. Copley by either gravitational spreading or vertical thickening, depending on the relative importance of sediment flux, material properties and flexure. We apply our model to the Makran accretionary prism and the Indo-Burman Ranges, and show that for the Makran flexure must be considered in order to explain the dip of the sedimentbasement interface from seismic reflection profiles. In the Indo-Burman Ranges, we show that incoming sediment thickness has a first-order control on the variations in the characteristics of the topography from north to south of the Shillong Plateau.

A wide range of initial-value problems in fluid mechanics in particular, and in the physical sciences in general, are described by nonlinear partial differential equations. Recourse must often be made to numerical solutions, but a powerful, well-established technique is to solve the problem in terms of similarity variables. A disadvantage of the similarity solution is that it is almost always independent of any specific initial conditions, with the solution to the full differential equation approaching the similarity solution for times$t\gg t_{\ast }$, for some$t_{\ast }$. But what is$t_{\ast }$? In this paper we consider the situation of viscous gravity currents and obtain useful formulae for the time of approach,$\unicode[STIX]{x1D70F}(p)$, for a number of different initial shapes, where$p$is the percentage disagreement between the radius of the current as determined by the full numerical solution of the governing partial differential equation and the similarity solution normalised by the similarity solution. We show that for any initial shape of volume$V,\unicode[STIX]{x1D70F}\propto 1/(\unicode[STIX]{x1D6FD}V^{1/3}\unicode[STIX]{x1D6FE}_{0}^{8/3}p)$(as$p\downarrow 0$), where$\unicode[STIX]{x1D6FD}=g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/(3\unicode[STIX]{x1D707})$, with$g$representing the acceleration due to gravity,$\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$the density difference between the gravity current and the ambient,$\unicode[STIX]{x1D707}$the dynamic viscosity of the fluid that makes up the gravity current and$\unicode[STIX]{x1D6FE}_{0}$the initial aspect ratio. This framework can used in many other situations, including where it is not an initial condition (in time) that is studied but one valid for specified values at a special spatial coordinate.

An asymptotic model is constructed to describe the bending of thin sheets, or plates, of viscoplastic fluid described by the Herschel–Bulkley constitutive law, which incorporates the von Mises yield condition and a nonlinear viscous stress. The model reduces to a number of previous ones from plasticity theory and viscous fluid mechanics in various limits. It is characterized by a yield criterion proposed by Ilyushin which compactly combines the effect of the bending moment and in-plane stress tensors through three particular invariants. The model is used to explore the bending of loaded flat plates, the deflection of impulsively driven circular plates, and the tension-controlled deflection of loaded beams.