Instability of two-layer flows with viscosity and density stratification

Abstract

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.

Appendix: Parameters in Eqs. (14) and (15)

$$\begin{aligned}&B{ }_{1}=h_{s} -1, \end{aligned}$$

(A.1)

$$\begin{aligned}&C_{1} =\frac{n}{n+1}\cdot \frac{Fr}{\mathrm{Re}\cdot \sin \theta }\cdot \left( {\frac{s\cdot \mathrm{Re}_{m} }{Fr}\cdot \sin \theta } \right) ^{\frac{1}{n}}\cdot \left[ {\left( {1-h_{s} } \right) ^{\frac{1}{n}+1}-1} \right] , \end{aligned}$$

(A.2)

$$\begin{aligned}&D_{1} =\frac{\cos \theta }{F_{r} }\cdot \left( {1-h_{s} } \right) , \end{aligned}$$

(A.3)

$$\begin{aligned}&B_{2} =-\frac{s\cdot \mathrm{Re}_{m} }{Fr}\sin \theta \cdot B_{1} =\frac{s\cdot \mathrm{Re}_{m} }{Fr}\sin \theta \cdot \left( {1-h_{s} } \right) , \end{aligned}$$

(A.4)

$$\begin{aligned}&C_{2} =\frac{Fr}{\sin \theta }\cdot \frac{1}{s\cdot \mathrm{Re}_{m} }\cdot \left( {\mathrm{Re}_{m} \frac{s\cdot \sin \theta }{Fr}h_{s} +B_{2} } \right) ^{\frac{1}{n}+1}\cdot \frac{n}{n+1}, \end{aligned}$$

(A.5)

$$\begin{aligned}&D_{2} =\frac{\cos \theta }{Fr}\left( {1-h_{s} } \right) . \end{aligned}$$

(A.6)