GRAVITY-DRIVEN SLOW CREEPING FLOW OF A THERMOVISCOUS BODY AT ELEVATED TEMPERATURES

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Abstract
The flow of a thermoviscous body at elevated temperature under gravity is described by the mass, momentum, and energy balances of an incompressible, homogeneous, heat conducting, nonlinearly viscous fluid, in which the shear response includes a strongly temperature‐dependent rate factor. A scale analysis and coordinate stretching, appropriate for flows down an inclined surface, reflect the properties that depth‐to‐length ratios are different in both the “downhill” direction and perpendicular to it and that the flow is essentially from higher altitudes to lower ones. The normalized energy equation shows that, for the applications considered, in‐plane and out‐of plane (transverse) advections are important and that transverse diffusion and dissipation are both significant. Analogously, the stress‐deviator‐stretching relationship exhibits a conspicuous temperature dependence. Hence, there is strong ther-momechanical coupling. The small aspect ratio parameters of the nondimensional equations permit deduction of perturbation solutions. The leading‐order relations are reduced to (1) explicit relationships for the stresses. (2) quadrature formulas involving the temperature field and the stress fields for the in‐plane velocity components, (3) a two‐dimensional advection diffusion equation for the surface profile, and (4) a three‐dimensional advection‐reaction diffusion equation for the temperature field that incorporates the unknown surface slope in coefficients and boundary conditions. Depending on the sliding law, valid at the basal surface, equations may be singular at the grounding line. For regular conditions numerical solution procedures for unsteady flow are discussed and isothermal solutions are constructed. Next in order, to incorporate further thermodynamic processes, such as variation of moisture content or micromechanical properties, the effects of internal variables on the field equations are discussed. Finally, an alternative scaling is briefly touched upon and potential use of the model is discussed.

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