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Inverse rheometry and basal properties inference for pseudoplastic geophysical flows. (English) Zbl 1408.76015
Summary: The present work addresses the question of performing inverse rheometry and basal properties inference for pseudoplastic gravity-driven free-surface flows at low Reynolds’ number. The modeling of these flows involves several parameters, such as the rheological ones or the state of the basal boundary (modeling an interface between the base and the fluid). The issues of inverse rheometry are addressed in a general laboratory flow context using surface velocity data. The inverse characterization of the basal boundary is proposed in a geophysical flow context where the parameters involved in the empirical effective sliding law are particularly difficult to estimate. Using an accurate direct and inverse model based on the adjoint method combined with an original efficient solver, sensitivity analyses and parameter identification are performed for a wide range of flow regimes, defined by the degree of slip and the non-linearity of the viscous sliding law considered at the bottom. The first result is the numerical assessment of the passive aspect of the viscosity singularity inherent to a power-law pseudoplastic (shear-thinning) description in terms of surface velocities. From this result, identification of the two parameters of the constitutive law, namely the power-law exponent and the consistency, are performed. These numerical experiments provide, on the one hand, a very robust identification of the power-law exponent, even for very noisy surface velocity observations and on the other hand, a strong equifinality problem on the identification of the consistency. This parameter has a minor influence on the flow, in terms of surface velocities. Typically for temperature-dependent geophysical fluids, a law describing a priori its spatial variability is then sufficient (e.g., based on a temperature vertical profile). This study then focuses on the basal properties interacting with the fluid rheology. An accurate joint identification of the scalar valued triple $$(n, m; \beta)$$ (respectively the rheological exponent, the non linear friction exponent and the friction coefficient) is achieved for any degree of slip, allowing to completely infer the flow regime. Next, in a geophysical flow context, identifications of a spatially varying friction coefficient are performed for various perturbed bedrock topography. The (2D-vertical) results demonstrate a severely ill-posed problem that allows to compute a given set of surface velocity data with different topography/friction pairs.

MSC:
 76A05 Non-Newtonian fluids 76M25 Other numerical methods (fluid mechanics) (MSC2010) 65N21 Numerical methods for inverse problems for boundary value problems involving PDEs 80A22 Stefan problems, phase changes, etc. 86A60 Geological problems