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Scaling analyses describe the evolution of an extrusion of viscous or plastic fluid in the presence of surface cooling and solidification, under the assumption that the flows consist of two components: an isothermal interior and a surface crust. The ‘crust’ is a complex thermal and rheological boundary layer which we model using a viscous, plastic or brittle rheology. These models are thought to be relevant to some types of lava flows and address the effects of cooling on their morphology and rate of advance of the flow front. They show that effects of crust strength will dominate over both viscous and yield stresses in the interior when the ratio of crust thickness to flow length exceeds the ratio of effective yield stress of the crust to basal shear stress exerted on the bulk of the flow, a condition that appears likely to be met by many lava flows and small outgrowths on large flows. Similarity solutions are compared with measurements on the spreading of extrusions of wax beneath cold water in the laboratory, where the extruded liquid is viscous but develops a solid crust. Crust strength provides the dominant retarding force for the wax flows in cases where surface solidification occurs rapidly compared with lateral advection. These conditions give flows topped by sheets of solid that buckle or rift apart, or extrusions that enlarge by small bulbous outgrowths (analogous to ‘pillows’ on submarine lavas and ‘toes’ on some sub-aerial basalt flows). A comparison of the models with data for the growth of lava domes in the craters of Mount St Helens and Soufrière of St Vincent volcanoes reveals that spreading of those domes was not controlled by stresses in the flow interior. Instead the data are consistent with a balance between gravity and yield stresses in a thin crustal layer over most of the growth period.