Propagation of a lava flow is governed by slope topography, magma rheology, heat exchange with the atmosphere and the underlying terrain, and the rate of the eruption. Highly viscous crust is formed due to cooling and solidification of the uppermost layer of the flow. We consider here two numerical model problems for lava flows, both based on the fundamental physics of a hot fluid flow: a model problem, where thermal conditions (e.g. temperature and heat flow) at the lava surface are unknown a priori (a direct model problem), and a model problem, where the lava surface conditions are known and determined from observations(an inverse model problem). In both models, the lava viscosity depends on temperature and the volume fraction of crystals. By way of solving the direct model problem, we perform a parametric study of steady state lava flows to investigate the influence of the heat flux, viscosity, and effusion rate on the lava crust development. Numerical experiments show that a lava crust becomes thicker in the case of the nonlinear heat transfer compared to the case of a linear heat flow at the interface of lava with the atmosphere. ... mehrAlso, the crust thickens at lower lava effusion rates, while higher rates result in a rapid lava advection, slower cooling, and development of a thinner crust. Moreover, a lava crust becomes thicker with a higher coefficient of conductive heat transfer, or a higher lava viscosity, or the growth of effective emissivity of the lava surface. By way of solving the inverse model problem, we use an assimilation technique (that is, a method for an optimal combination of a numerical model of lava flows with observations) to propagate the temperature and heat flow, inferred from measurements at the interface between lava and the atmosphere, into the lava flow interior and to analyse the evolving lava crust. Results of thermal data assimilation illustrate that the physical parameters of lava flows, including the thickness of it crust, can be recovered from measured surface thermal data well enough at least for slow effusion rates.