Indirect measurement of $$\mu (I)$$ μ ( I ) relation from finite granular avalanche down an inclined narrow reservoir of smooth bed

Abstract

This work examines the algebraic \(\mu -I\) relation proposed for steady uniform dry granular flows via unsteady granular avalanche experiments of finite nearly identical dry glass spheres down an inclined narrow reservoir of smooth bed. Lateral high-speed digital imaging permits particle tracking velocimetry with which we can evaluate bulk local instantaneous volume fraction and velocity components to conduct a quasi-two-dimensional control volume analysis of streamwise momentum assuming an internal shear stress based on the \(\mu -I\) rheology, a hydrostatic normal stress and a Coulomb yielding condition at lateral walls. Hence, the desired \(\mu \) is a function of flow dynamics and a wall friction coefficient \(\mu _w\). Complementary sliding table experiments were conducted to estimate an upper bound of \(\mu _w=0.17\) which was used with a chosen nonzero lower bound \(\mu _w=0.05\) to extract possible range of \(\mu \) at a local instantaneous inertial number I. The so-obtained local instantaneous \(\mu -I\) data conform to the non-linear monotonically increasing trend proposed for steady inertial flows above a crossover value \(I_c\approx 0.03\). Below \(I_c\), a peculiar segment of decaying \(\mu \) with I was revealed agreeing to the rheology tests in quasi-static regime.