Emergent behaviors of a thermodynamic Cucker-Smale flock with a time-delay on a general digraph

We present emergent flocking dynamics of a thermodynamic Cucker-Smale (TCS) flock on a general digraph with spanning trees under the effect of communication time-delays. The TCS model describes a temporal evolution of mechanical and thermodynamic observables such as position, velocity and temperature of CS particles. In this paper, we study how variations in mechanical and thermodynamic variables can decay to zero along a time-independent network with position dependent weights from initial state configuration. For this, we provide a sufficient framework for a mechanical and thermodynamical flocking in terms of initial configuration, network topology, and system parameters. We also present several numerical examples and compare them with analytical results.     

New approach to the solvability of generalized Navier-Stokes equations

We propose a new approach to prove the existence of weak solutions to generalized (or modified) Navier-Stokes equations. We also consider systems presently well known from the theory of non-Newtonian electrorheological fluids.     

Global existence of strong solution for the Cucker–Smale–Navier–Stokes system

We present a global existence theory for strong solution to the Cucker–Smale–Navier–Stokes system in a periodic domain, when initial data is sufficiently small. To model interactions between flocking particles and an incompressible viscous fluid, we couple the kinetic Cucker–Smale model and the incompressible Navier–Stokes system using a drag force mechanism that is responsible for the global flocking between particles and fluids. We also revisit the emergence of time-asymptotic flocking via new functionals measuring local variances of particles and fluid around their local averages and the distance between local averages velocities. We show that the particle and fluid velocities are asymptotically aligned to the common velocity, when the viscosity of the incompressible fluid is sufficiently large compared to the sup-norm of the particles' local mass density.     

Flocking of Multi-particle Swarm with Group Coupling Structure and Measurement Delay

We investigate the flocking conditions of a group coupling system with time delays, in which the communication between particles includes inter-group and intra-group interactions, and the time delay comes from the theory of moving object observation. As an effective model, we introduce a system of nonlinear functional differential equations to describe its dynamic evolution mechanism. By constructing two differential inequalities on velocity and velocity fluctuation from a continuity argument, and using the Lyapunov functional approach, we present some sufficient conditions for the existence of asymptotic flocking solutions to the coupling system, in which an upper bound of the delay allowed by the system is quantitatively given to ensure the emergence of flocking behavior. All results are novel and can be illustrated by using some specific numerical simulations.     

Some Self-Similar Two-Phase Flows of Non-Newtonian Fluids Through a Porous Medium

Pascal This paper addresses the question of the rheological effects of non-Newtonian fluids in a flow system, in which a two-phase flow zone is coupled to a single-phase flow zone by a moving fluid interface. This flow system is involved in a technique for oil displacement in a porous medium, where a non-Newtonian displacing fluid (a polymer solution) is used to displace a non-Newtonian heavy oil. The self-similar solutions of the equations governing the dynamics of the moving interface, separating the displacing and displaced fluids, are obtained for the one-dimensional and plane radial flows. The effects associated with the presence of a two-phase flow zone, behind the moving interface, on the interface movement are analyzed. The existence of a pressure front ahead of the moving interface, moving with a finite velocity, is also shown. The relevance of this result to the propagation of pressure disturbances in a non-Newtonian fluid flowing through a porous medium is discussed with regard to interpretation of the transient pressure response in an injection well for polymer-solution floods.     

Flocking Dynamics of the Inertial Spin Model with a Multiplicative Communication Weight

In this paper, we study a flocking dynamics of the deterministic inertial spin (IS) model. The IS model was introduced for the collective dynamics of active particles with an internal angular momentum, or spin. When the generalized moment of inertia becomes negligible compared to spin dissipation (overdamped limit) and mutual communication weight is a function of a relative distance between interacting particles, the deterministic inertial spin model formally reduces to the Cucker–Smale (CS) model with constant speed constraint whose emergent dynamics has been extensively studied in the previous literature. We present several sufficient frameworks leading to the asymptotic mono-cluster flocking, in which spins and relative velocities tend to zero asymptotically. We also provide several numerical simulations for the decoupled and coupled inertial spin models to see the effect of the C–S velocity flocking and compare them with our analytical results.

     

Mathematical modeling for colloidal dispersion undergoing Brownian motion

An averaged motion approach for modeling Brownian dynamics for suspension systems of electrically charged particles in liquid is developed. The continuum model for the motion of particles consists of a system of integral equations coupled with a degenerate parabolic equation. Existence and uniqueness of global solution for the coupled system are established, and numerical results for the non-Newtonian viscosity of the mixture in terms of shear rate or Pechlet number are obtained. The model reveals some non-Newtonian properties such as the well-known shear thinning phenomenon for the viscosity of colloidal dispersions.     

一类非牛顿系统弱解的长时间性态

本文讨论一类非牛顿系统弱解的长时间性态,证明解在L2 范数下的衰减速率为(1 t)-n/4 达到和线性热方程一致.     

Global existence of weak and strong solutions to Cucker–Smale–Navier–Stokes equations in R2

We study the existence theory for the Cucker–Smale–Navier–Stokes (in short, CS–NS) equations in two dimensions. The CS–NS equations consist of Cucker–Smale flocking particles described by a Vlasov-type equation and incompressible Navier–Stokes equations. The interaction between the particles and fluid is governed by a drag force. In this study, we show the global existence of weak solutions for this system. We also prove the global existence and uniqueness of strong solutions. In contrast with the results of Bae et al. (2014) on the CS–NS equations considered in three dimensions, we do not require any smallness assumption on the initial data.     

非牛顿幂律流体球向不定常渗流

本文研究了弱压缩非牛顿幂律流体球向不定常渗流,导出了抛物型偏微分非线性方程.球向扩散方程是其特殊情况.用Laplace变换的方法,找到了线性化后方程的解析解和渐近解.用影响半径的概念和平均值方法求得了近似解.渐近解和近似解的结构是相似的,从而丰富了非牛顿流体一维不定常渗流的理论.     

Existence of weak solutions for a diffuse interface model of non-Newtonian two-phase flows

We consider a phase field model for the flow of two partly miscible incompressible, viscous fluids of non-Newtonian (power law) type. In the model it is assumed that the densities of the fluids are equal. We prove the existence of weak solutions for general initial data and arbitrarily large times with the aid of a parabolic Lipschitz truncation method, which preserves solenoidal velocity fields and was recently developed by Breit, Diening, and Schwarzacher.     

The Strong Solution for the Viscous Polytropic Fluids with Non-Newtonian Potential

The authors study an initial boundary value problem for the three-dimensional Navier-Stokes equations of viscous heat-conductive fluids with non-Newtonian potential in a bounded smooth domain. They prove the existence of unique local strong solutions for all initial data satisfying some compatibility conditions. The difficult of this type model is mainly that the equations are coupled with elliptic, parabolic and hyperbolic, and the vacuum of density causes also much trouble, that is, the initial density need not be positive and may vanish in an open set.     

A model for the evolution of sedimentation beds in the dynamic of a pipelined non-Newtonian fluid

In this work, the dynamic of particles in a non-Newtonian (Bingham) fluid is considered, developing a mathematical model (one nonlinear, free boundary, first-order PDE) for the sedimentation of particles in a cylindrical, axis symmetric flow. We proved existence and uniqueness (locally in some circumstances) of the solutions, and we tested numerically the fitting of the model with the real physical problem.     

Global existence and nonexistence of the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions

In this article, we deal with the global existence and nonexistence of solutions to the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions. The critical curve of Fujita type is conjectured with the aid of some new results, which extend the recent results of Zheng, Song, and Jiang [Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl. 298 (2004), pp. 308–324], Zhou and Mu [Critical curve for a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux, Nonlinear Anal. 68 (2008), pp. 1–11], and Zhou and Mu [Algebraic criteria for global existence or blow-up for a boundary coupled system of nonlinear diffusion equations, Appl. Anal. 86 (2007), pp. 1185–1197] to more general equations.     

Existence and uniqueness of solutions for a class of non-Newtonian fluids with vacuum and damping

In this paper, we proved local existence and uniqueness of solutions for a class of non-Newtonian fluids with vacuum and damping in one-dimensional bounded intervals. The main difficulty is due to the strong nonlinearity of the system and initial vacuum.     

Existence of global solutions to multidimensional equations for Bingham fluids

We consider equations describing the multidimensional motion of compressible viscous (non-Newtonian) Bingham-type fluids, i.e., fluids with multivalued function relating the stresses to the tensor of strain rates. We prove the global existence theorem in time and in the initial data for the first initial boundary-value problem corresponding to flows in a bounded domain in the class of “weak” generalized solutions. In this case, we admit an anisotropic relation between the stress and strain rate tensors and study admissible relations of this kind in detail.     

Numerical modelling of shear-thinning non-Newtonian flow in compliant vessels

We present recent results on numerical modelling of non-Newtonian flow in two-dimensional compliant vessels with application in hemodynamics. Two models of the shear-thinning non-Newtonian fluids are considered and compared with the Newtonian model. For the structure problem the generalized string equation for radial symmetric tubes is used and extended to a stenosed vessel. A global iterative approach is used to approximate the fluid-structure interaction. At the end we present numerical experiments for selected non-Newtonian models, comparisons with the Newtonian model and the hemodynamic wall parameters: the wall shear stress and the oscillatory stress index. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the exponential behaviour of stochastic evolution equations for non-Newtonian fluids

We investigate the exponential long-time behaviour of the stochastic evolution equations describing the motion of a non-Newtonian fluids excited by multiplicative noise. Some results on the exponential convergence in mean square and with probability one of the weak probabilistic solution to the stationary solutions are given. We also prove an interesting result related to the stabilization of these stochastic evolution equations.