Instabilities of a dam-break wave of power-law fluids

Abstract

The paper theoretically investigates the stability properties of the dam-break wave of a fluid with power-law rheology. Assuming the long-wave approximation, a depth-averaged flow model is considered. The linear stability analysis of the wave is carried out to individuate the marginal stability conditions. To this aim, the multiple-scale technique is applied with reference to the kinematic wave solution, which formally limits the validity of the theoretical achievements to relatively long time scales. Both shear-thinning and shear-thickening fluids are considered. Similarly to the case with uniform conditions, the analysis indicates that stable conditions can be associated with a marginal value of the Froude number. However, differently from the uniform conditions, the marginal Froude number is shown to be a function not only of the power-law index but also of the streamwise gradient of the base flow velocity and of the disturbance wavelength. The critical Froude number is found to be larger than the corresponding one in uniform conditions. Numerical solutions of the full model confirmed the outcomes of the linear stability analysis for both shear-thinning and shear-thickening fluids.