Inferring rheology from free-surface observations

Abstract

We develop direct inversion methods for inferring the rheology of a fluid from observations of its shallow flow. First, the evolution equation for the free-surface flow of an inertia-less current with general constitutive law is derived. The relationship between the volume flux of fluid and the basal stress, $\tau _b$ , is encapsulated by a single function $F(\tau _b)$ , which depends only on the constitutive law. The inversion method consists of (i) determining the flux and basal stress from the free-surface evolution, (ii) comparing the flux with the basal stress to constrain $F$ and (iii) inferring the constitutive law from $F$ . Examples are presented for both steady and transient free-surface flows demonstrating that a wide range of constitutive laws can be directly obtained. For flows in which the free-surface velocity is known, we derive a different method, which circumvents the need to calculate the flux.

Two-layer gravity currents of generalized Newtonian fluids

“…The flux in the lower layer is rearranged as follows [20]: Ql=∫0hlul dz=∫0hlfalse(hl−zfalse)normal∂ulnormal∂z dz,where we have used no-slip at the base z=0. To write this flux in terms of the rheology of the lower layer, ϕlfalse(|τ|false), it is recast as an integral over the stress across the lower layer [20,22] with the substitution (from (2.15) b ) dz=−dτGl.We also use hl−z=false[τ−huGufalse]/Gl and the constitutive law (2.17) to obtain Ql=Gl−2[∫τIτBϕl(false|τfalse|)…”

“…At first sight, it may appear that the volume fluxes become singular as Gufalse→0 or Glfalse→0 but correctly taking limits furnishes finite expressions for Ql and Qu. Care is needed in the numerical method in order to handle small Gu or Gl (see appendix A of [22]).…”

“…The black lines show the initial interface positions and the interface evolution is calculated by integrating (2.28) numerically for each layer. Details of the numerical method are given in appendix A of [22]. The two layers are Bingham fluids with different yield stress and shear viscosity.…”

“…[20]. We show that the two-dimensional depth-integrated approach of [22] cannot generally be applied to two-layer flows because the direction of the shear stress varies across the lower layer. Motivated by till-lubricated glacial flow, in §3, we proceed to develop a general two-dimensional sliding law for the flow of the upper layer over a relatively thin lower layer.…”

“…This approach has the crucial advantage that the model does not require explicit knowledge of the velocity profile of the flow [20]. The model has been extended to a general formulation for two-dimensional flows, which enables the direct inference of a fluid's constitutive law from observations of its free surface [21,22]. Integrating over the stress rather than the velocity also has the advantage that for yield-stress fluids, the location of the yield surface does not need to be determined in order to calculate the volume flux.…”