Long-wave instability of a regularized Bingham flow down an incline

Abstract

We investigate the linear stability of a flow down an incline when the fluid is modelled as a "mollified" Bingham material. We perform a theoretical analysis by using the long-wave approximation method. The results show the existence of a critical condition for the onset of instability which arises when the Reynolds number is above a critical threshold that depends on the tilt angle and on rheological parameters. The comparison of our findings with experimental studies is rather satisfactory.

Linear stability of viscoplastic flows down an incline

“…Here, we summarize the theoretical formulation of the problem, details are given in [32,33,34]. Figure 1 shows the flow domain, where we denote the tilt angle as θ ∈ (0, π/2), the length of the domain as L and the upper free surface (not a priori known) as y = h(x, t) with H = max{h}.…”

“…Journal of Physics: Conference Series 2701 (2024) 012071 more details are in [32,33,34]. We consider the chosen dimensionless regularization constitutive models…”

“…Our study focuses on the analysis of theoretical and numerical works regarding linear stability of viscoplastic flow down an inclined plane by using "regularized" models [32,33,34]. In particular, the theoretical study has been performed through the long-wave approximation approach in the case of Regularized Bingham and Casson-like materials [32,33].…”

“…Our study focuses on the analysis of theoretical and numerical works regarding linear stability of viscoplastic flow down an inclined plane by using "regularized" models [32,33,34]. In particular, the theoretical study has been performed through the long-wave approximation approach in the case of Regularized Bingham and Casson-like materials [32,33]. The numerical study has been performed by using the Papanastasiou model (a Bingham law regularization) through a spectral collocation method that makes use of Chebyshev polynomials [34].…”

“…The theoretical background on the mathematical formulation of the problem and the main features of a "regularized" fluid flowing down an inclined plane in Section 2 and Section 3, respectively. Then, Section 4, similarly to [32,34,36], summarizes the analysis of linear stability through both analytical and numerical approaches, i.e. the long-wave approximation and spectral collocation method, respectively.…”