Arghir Zarnescu

We analyze the dynamics of concentrated polymer solutions modeled by a 2D Smoluchowski equation. We describe the long-time behavior of the polymer suspensions in a fluid. When the flow influence is neglected the equation has a gradient structure. The presence of a simple flow introduces significant structural changes in the dynamics. We study the case of an externally imposed flow with homogeneous gradient. We show that the equation is still dissipative but new phenomena appear. The dynamics depend on both the concentration intensity and the structure of the flow. In certain limit cases the equation has a gradient structure, in an appropriate reference frame, and the solutions evolve to either a steady state or a tumbling wave. For small perturbations of the gradient structure we show that for small concentrations the solutions evolve in the long-time limit to a steady state. However for high concentrations there is a rigidity phenomenon for the tumbling wave.

This paper analytically explores a simplified model for the hydrodynamics of nematic liquid crystal colloids. We integrate a Stokes equation for the velocity field with a Ginzburg–Landau transported heat flow for the director field. The study focuses on a bounded spatial domain containing periodically distributed colloidal particles with no-anchoring conditions on the nematic liquid crystal. By progressively reducing the particle size to zero and simultaneously increasing the number of particles, we delve into the associated homogenization problem. Our analysis uncovers a form of decoupling where the velocity field asymptotically satisfies a Darcy equation, independent of the director, while the director follows a gradient flow, unaffected by the velocity field. One of the most intricate aspects of the homogenization process is the absence of an extension operator for the director field that preserves the uniform estimates related to the system’s energy. We address this challenge with a novel variation of the Aubin–Lions lemma, specifically adapted for homogenization problems.

In this paper we consider the 3D co-rotational Beris-Edwards system modeling the hydrodynamic motion of nematic liquid crystals in a thin strip. The system contains the incompressible Navier-Stokes, coupled with a parabolic system for matrix-valued functions, the $Q$-tensors. We show that under a suitable scaling, corresponding, in the Navier-Stokes part, to the hydrostatic scaling, one obtains in the limit a partly decoupled system. For the fluid part we obtain the Prandtl system while for the $Q$-tensors we obtain a non-standard system, involving fluids components and a non-standard combination of partly dissipative equations and algebraic constraints. We prove the convergence of the rescaled system and the well-posedness of the limit in Sobolev spaces.

In this paper, we study a mathematical model describing the movement of a colloidal particle in a fixed, bounded three dimensional container filled with a nematic liquid crystal fluid. The motion of the fluid is governed by the Beris–Edwards model for nematohydrodynamics equations, which couples the incompressible Navier–Stokes equations with a parabolic system. The dynamics of colloidal particle within the nematic liquid crystal is described by the conservation laws of linear and angular momentum. We prove the existence of global weak solutions for the coupled system.

This paper analytically explores a simplified model for the hydrodynamics of nematic liquid crystal colloids. We integrate a Stokes equation for the velocity field with a Ginzburg-Landau transported heat flow for the director field. The study focuses on a bounded spatial domain containing periodically distributed colloidal particles, which impose no-anchoring conditions on the nematic liquid crystal. By progressively reducing the particle size to zero and simultaneously increasing the number of particles, we delve into the associated homogenisation problem. Our analysis uncovers a form of decoupling where the velocity field asymptotically satisfies a Darcy equation, independent of the director, while the director follows a gradient flow, unaffected by the velocity field. One of the most intricate aspects of the homogenisation process is the absence of an extension operator for the director field that preserves the uniform estimates related to the system's energy. We address this challenge with a novel variation of the Aubin-Lions lemma, specifically adapted for homogenisation problems.

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We analyze the dynamics of concentrated polymer solutions modeled by a 2D Smoluchowski equation. We describe the long time behavior of the polymer suspensions in a fluid. \par When the flow influence is neglected the equation has a gradient structure. The presence of a simple flow introduces significant structural changes in the dynamics. We study the case of an externally imposed flow with homogeneous gradient. We show that the equation is still dissipative but new phenomena appear. The dynamics depend on both the concentration intensity and the structure of the flow. In certain {\it limit cases} the equation has a gradient structure, in an appropriate reference frame, and the solutions evolve to either a steady state or a tumbling wave. For small perturbations of the gradient structure we show that some features of the gradient dynamics survive: for small concentrations the solutions evolve in the long time limit to a steady state and for high concentrations there is a tumbling wave.