Stability of a Regularized Casson Flow down an Incline: Comparison with the Bingham Case

Abstract

In this paper, we study the two-dimensional linear stability of a regularized Casson fluid (i.e., a fluid whose constitutive equation is a regularization of the Casson obtained through the introduction of a smoothing parameter) flowing down an incline. The stability analysis has been performed theoretically by using the long-wave approximation method. The critical Reynolds number at which the instability arises depends on the material parameters, on the tilt angle as well as on the prescribed inlet discharge. In particular, the results show that the regularized Casson flow has stability characteristics different from the regularized Bingham. Indeed, for the regularized Casson flow an increase in the yield stress of the fluid induces a stabilizing effect, while for the Bingham case an increase in the yield stress entails flow destabilization.

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…”

“…is the Bingham number where τ 0 is the yield stress, and * = 2 H/U , so that → 0 provides the corresponding "exact" model [32,33,34,35]. In the sequel, the " * " are omitted to maintain the notation as light as possible.…”

“…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].…”