We show that a micropolar fluid model successfully describes granular flows on a slope. Micropolar fluid is the fluid with internal structures, and the coupling between the rotation of each particle and the macroscopic velocity field is taken into account; it is a hydrodynamical framework suitable for granular systems which consists of particles with macroscopic size. It is demonstrated that th...

Abstract. Motivated by lubrication problems, we consider a micropolar fluid flow in a 2D domain with a rough and free boundary. We assume that the thickness and the roughness are both of order 0 < ε << 1. We prove the existence and uniqueness of a solution of this problem for any value of ε and we establish some a priori estimates. Then we use the two-scale convergence technique to derive the l...

A modification of the homotopy analysis method (HAM) for solving a system of nonlinear boundary value problems (BVPs) in semi–infinite domain, micropolar flow due to a linearly stretching of porous sheet, is proposed. This method is based on operational matrix of exponential Chebyshev functions to construct the derivative and product of the unknown function in the matrix form. In addition, by u...

An assessment is made here of the role played by the micropolar continuum theory on the cracked Brazilian disc test used for determining rock fracture toughness. By analytically solving the corresponding mixed boundary-value problems and employing singular-perturbation arguments, we provide closed-form expressions for the energy release rate and the corresponding stress-intensity factors for bo...

We prove lower estimates for space periodic solutions (u,w)(t) of the micropolar equations in their maximal interval [0,T ∗) provided that T ∗ < ∞. For example, we show for 0 < δ < 1 that ∥(u,w)(t)∥Ḣs(T3) is at least of the order (T ∗− t)−(δs)/(1+2δ) for s ≥ 1/2+δ. Moreover, we prove the inequality ∥(̂u, ŵ)(t)∥l1(Z3) ≥ C(T ∗ − t)−1/2, which yields the blow–up rate (T ∗ − t)−s/3 for ∥(u,w)(t)∥Ḣs(...