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We study the partial regularity of weak solution to a certain class of non-Newtonian fluids (1.1) – (1.3) , the existence of weak solution to such system had been provided by D. Pierre and H. Norbert in [40] . In particular, we prove that the regularity point in Q T is an open set with full measure, and we obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In this paper we confined ourselves to considered the case p = 2 and space dimension d = 2 . [ABSTRACT FROM AUTHOR]

We present a hybrid particle-mesh method for numerically solving the hydrodynamic equations of incompressible active polar viscous gels. These equations model the dynamics of polar active agents, embedded in a viscous medium, in which stresses are induced through constant consumption of energy. The numerical method is based on Lagrangian particles and staggered Cartesian finite-difference meshes. We show that the method is second-order and first-order accurate with respect to grid and time-step sizes, respectively. Using the present method, we simulate the hydrodynamics in rectangular geometries, of a passive liquid crystal, of an active polar film and of active gels with topological defects in polarization. We show the emergence of spontaneous flow due to Fréedericksz transition, and transformation in the nature of topological defects by tuning the activity of the system. [ABSTRACT FROM AUTHOR]

Abstract: This article studies the pullback asymptotic behavior of solutions for a non-autonomous homogeneous two-phase flow model in a two-dimensional domain. We prove the existence of pullback attractors in (the velocity has the -regularity) and in (the velocity has the -regularity). Then we verify the regularity of the pullback attractors by proving that , which implies the pullback asymptotic smoothing effect of the model in the sense that the solutions eventually become more regular than the initial data. The method used in this article is similar to the one used in Zhao and Zhou (2007) in the case of the non-autonomous incompressible non-Newtonian fluid in a two-dimensional domain. Let us mention that the nonlinearity involved in the model considered in this article is stronger than the one in the two-dimensional non-Newtonian flow studied in Zhao and Zhou (2007) . [Copyright &y& Elsevier]

Abstract: It is suggested that a distinct population of Mars-sized planetary embryos can form by cold accretion in the outer part of the terrestrial zone. A possible radial structure for such embryos consists of an outer silicate-rich layer and an undifferentiated solid protocore. The protocore is divided in a cold undifferentiated central region, a layer of iron, and an outer shell of undifferentiated material. This structure is gravitationally unstable. We investigate the influence of the rheology of the silicate-rich material and of the thermal effects on the protocore destabilization and on the iron core formation. We use a 2D thermomechanical numerical model of a self-gravitating planetary body. The planetary surface is treated as a free surface by imposing a massless weak medium. Our calculations include a non-Newtonian temperature-, pressure-, and strain rate-dependent viscoplastic rheology and thermal contributions from shear heating. We explored the influence of rheology by varying the activation volume, friction angle, and the maximum yield strength of silicate-rich layers. We distinguished three different core formation regimes: exposure mode, fragmentation mode, and transition mode. Like in previous models with Newtonian rheology, the core experiences large deviations from the spherical shape and is temporarily exposed at the surface in the exposure mode. In contrast to the Newtonian models the destruction of the protocores in the fragmentation modes is driven by (i) the spontaneous strain localization along planetary-scale shear zones forming inside the protocore, and/or (ii) descending localized iron diapirs/sheets penetrating the protocore. Feedback from energy dissipation notably modifies the thermal structure of the deforming planetary body. In particular it causes a temperature increase of up to several hundred Kelvin (i) around the moving and deforming protocore and (ii) along planetary scale rupture zones that form inside the protocore. If the protocore is large and has a high viscosity, a large fraction of the dissipated heat is used to increase the temperature of iron. [Copyright &y& Elsevier]

Abstract: An analysis based on the Galerkin method is developed to examine the behaviour of a nonlinear bipolar viscous fluid mathematically modelled by stochastic non-Newtonian fluid motion equations. Existence and uniqueness of solutions to the stochastic equations are derived. [Copyright &y& Elsevier]

Abstract: We provide a new method to prove and improve the Chemin–Masmoudi criterion for viscoelastic systems of Oldroyd type in [J.Y. Chemin, N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal. 33 (1) (2001) 84–112] in two space dimensions. Our method is much easier than the one based on the well-known losing a priori estimate and is expected to be easily adopted to other problems involving the losing a priori estimate. [Copyright &y& Elsevier]