Instability of two-layer flows with viscosity and density stratification

The three-dimensional linear stability characteristics of two-layer immiscible fluid with a free surface are considered, wherein a Newtonian fluid contains a small amount of sediment particles, and overlies a lower fluid that contents hyperconcentration of sediment. Rheological experiments are carried out on three kinds of artificial sediments with different silt-kaolin mixing ratios to study the rheological properties of the lower fluid. The results show that the rheological properties of the lower fluid conform to the power-law model. Based on this, the modified N–S and Squire equations in each layer are derived for the gravity-driven flow, which are calculated by using the finite difference method. The effect of various dimensionless parameters, such as the viscosity ratio ( $$c_{m})$$ , the density ratio (s), the power-law index (n), and the thickness ratio ( $$h_{s})$$ on the instability characteristics of the flow is investigated. It is observed that increasing $$c_{m}$$ , s and $$h_{s}$$ is stabilizing for both long and short waves, which leads to an increase of the critical Reynolds number (Re $$_\mathrm{{{wcr}}})$$ , and a decrease of the bandwidth of the unstable wavenumbers. The disturbed velocities (u and w) are also decreased. Decreasing the shear-thinning tendency (increasing n) of the lower fluid is destabilizing for both long and short waves, which leads to a decrease of Re $$_\mathrm{{{wcr}}}$$ and an increase of the disturbed velocities.