Identification of spatially distributed rheological parameters for viscoplastic gravity currents

Abstract

The rheology of a fluid is often difficult to measure in situ and in real time. This work investigates whether it is possible to recover rheological information from direct field observations. We present a closed-form analytical solution for inverse problem of identification of the mobility coefficient in gravity-driven fluid flowed down an inclined plane. The lubrication approximation is used to model the three-dimensional thin film of fluid and the governing equation is solved numerically by the explicit finite difference method to calculate the fluid thickness. We show in this study that using very straightforward measurements of fluid thickness at only a few points and only two successive instants, the mobility coefficient of fluid flow can be estimated accurately. Then as an important application in Bingham fluids we show that the knowledge of the mobility coefficient together with velocity measurement at only one point and instant enable us to estimate temperature-dependent viscosity and yield stress of the Bingham fluid. Unlike other methods, the proposed method allows the local identification of the rheology which is particularly useful when the rheological parameters are functions of the local temperature and/or composition. The proposed inverse analysis is a non-iterative approach with negligible computational cost since our analysis revealed closed form formulae relating the rheological parameters to the local flow depth, its derivatives, and the local free surface velocity.

Keywords

• Gravity-driven flow
• Lubrication approximation
• Bingham fluid
• Inverse problem
• Finite difference method
• Analytical solution