Downslope flow models of a Bingham liquid: Implications for lava flows

Abstract

It is widely recognized that lavas behave as Bingham liquids, which are characterized by a yield stress σϒ and a plastic viscosity η. We consider two models describing downslope flows of a Bingham liquid with different aspect ratios A (= flow height/flow width): model 1 with A ⪡ 1 and model 2 with A ≈ 1. Sufficiently uphill with respect to the front, such flows can be considered as laminar and locally isothermal. For both models, we obtain analytically the steady-state solution of the Navier-Stokes equations and the constitutive equation for a Bingham liquid. We study the flow height and velocity as functions of flow rate, rheological parameters and ground slope. It is found that such flows remain in the Newtonian regime at low yield stresses ( σϒ ⪅ 10 3 dyne/ cm 2), but the transition to the Bingham regime also depends on flow rate and occurs at higher values of σϒ for higher flow rates: for instance, a high aspect ratio flow (model 2) is still very close to the Newtonian regime at σϒ = 10 4 dyne/cm 2, if the flow rate is greater than 10 5 g/s. In the Bingham regime, flow heights are generally greater and flow velocities are smaller than in the Newtonian regime; moreover, flow heights are independent of flow rate, so that a change in flow rate results exclusively in a velocity change. After assuming a specific temperature dependence of σϒ and η between the solidus and the liquidus temperatures of an ideal Bingham liquid (1000°C and 1200 °C respectively), flow heights and velocities are examined as functions of temperature along the flow. Several effects observed in lava flows are predicted by these models and allow a more quantitative insight into the behaviour of lava flows.