Fluid dynamics governs many phenomena on the Earth's surface and interior, from the emplacement of fluid magma, to the viscous deformation of mountain ranges on the longest time scales. Understanding these processes presents a challenge to traditional modelling techniques. However, simplifying models of the leading-order features of the flow can give insight into the dominant physical balances at play. In this dissertation I use theoretical analysis, numerical simulations, and laboratory experiments to address two geophysical processes: the formation of fold-thrust belts and the dynamics of shallow magmatic intrusions. Although geophysically distinct, these two problems both involve the interplay between viscous flow and elastic deformation and so inform the modelling of one another.

Fold-thrust belts are formed at convergent margins, where accretion of weak sediments to the front of the overriding plate results in continued flexural subsidence of the underthrusting plate. In this dissertation I build 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. The analysis shows that the evolution of fold-thrust belts can be dominated by either gravitational spreading or vertical thickening. I apply the model to the Makran accretionary prism and the Indo-Burman Ranges, and show that for the Makran flexure is important, while in the Indo-Burman Ranges the incoming sediment thickness has a first-order control on topography.

The propagation of shallow magmatic intrusions is governed by the interplay between elastically deforming sedimentary layers, the viscous flow of magma beneath, and the requirement to fracture at the front. In this dissertation I describe this process by extending the model for elastic-plated gravity currents to an axisymmetric geometry and show that adhesion (or fracture toughness) gives rise to two dynamical regimes of spreading; viscosity dominant spreading and adhesion dominant spreading. Experiments using clear, PDMS elastic sheets enable new, direct measurements of the vapour tip, and confirm the existence of spreading regimes controlled by viscosity and adhesion. I extend this laminar model of magma propagation to large mafic sills, which are thought to exhibit turbulent flow. Using a hybrid laminar-turbulent flow model I examine the transition to turbulence and show that volume fluxes several orders of magnitude above the average are required to reproduce the aspect ratios of large mafic sills measured in the field. Finally, I explore the role topographic gradients may play in driving magmatic intrusions by carrying out further experiments where the elastic sheets are inclined at an angle to the horizontal. Experimental observations show the formation of a transient head and a static tail structure with good first order comparisons to the deformation patterns of the Piton de la Fournaise flank sill intrusion.