Newtonian and Non-Newtonian Fluids through Permeable Boundaries

We considered the situation where a container with a permeable boundary is immersed in a larger body of fluid of the same kind. In this paper, we found mathematical expressions at the permeable interface Γ of a domain Ω, where Ω⊂R3. Γ is defined as a smooth two-dimensional (at least class C2) manifold in Ω. The Sennet-Frenet formulas for curves without torsion were employed to find the expressions on the interface Γ. We modelled the flow of Newtonian as well as non-Newtonian fluids through permeable boundaries which results in nonhomogeneous dynamic and kinematic boundary conditions. The flow is assumed to flow through the boundary only in the direction of the outer normal n, where the tangential components are assumed to be zero. These conditions take into account certain assumptions made on the curvature of the boundary regarding the surface density and the shape of Ω; namely, that the curvature is constrained in a certain way. Stability of the rest state and uniqueness are proved for a special case where a “shear flow” is assumed

Existence Results for a Michaud Fractional, Nonlocal, and Randomly Position Structured Fragmentation Model

Until now, classical models of clusters’ fission remain unable to fully explain strange phenomena like the phenomenon of shattering (Ziff and McGrady, 1987) and the sudden appearance of infinitely many particles in some systems having initial finite number of particles. That is why there is a need to extend classical models to models with fractional derivative order and use new and various techniques to analyze them. In this paper, we prove the existence of strongly continuous solution operators for nonlocal fragmentation models with Michaud time derivative of fractional order (Samko et al., 1993). We focus on the case where the splitting rate is dependent on size and position and where new particles generating from fragmentation are distributed in space randomly according to some probability density. In the analysis, we make use of the substochastic semigroup theory, the subordination principle for differential equations of fractional order (Prüss, 1993, Bazhlekova, 2000), the analogy of Hille-Yosida theorem for fractional model (Prüss, 1993), and useful properties of Mittag-Leffler relaxation function (Berberan-Santos, 2005). We are then able to show that the solution operator to the full model is positive and contractive