A semismooth Newton method for implicitly constituted non-Newtonian fluids and its application to the numerical approximation of Bingham flow

We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions–Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method.

We consider a system of evolutionary equations that is capable of describing certain viscoelastic effects in linearized yet nonlinear models of solid mechanics. The constitutive relation, involving the Cauchy stress, the small strain tensor and the symmetric velocity gradient, is given in an implicit form. For a large class of these implicit constitutive relations, we establish the existence and uniqueness of a global-in-time large-data weak solution. Then we focus on the class of so-called limiting strain models, i.e., models for which the magnitude of the strain tensor is known to remain small a priori, regardless of the magnitude of the Cauchy stress tensor. For this class of models, a new technical difficulty arises. The Cauchy stress is only an integrable function over its domain of definition, resulting in the underlying function spaces being nonreflexive and thus the weak compactness of bounded sequences of elements of these spaces is lost. Nevertheless, even for problems of this type we are able to provide a satisfactory existence theory, as long as the initial data have finite elastic energy and the boundary data fulfil natural compatibility conditions.

Fluid flows that do not have local equilibrium are characteristic of some of the new frontiers in engineering and technology, for example, high-speed high-altitude aerodynamics and the development of micrometre-sized fluid pumps, turbines and other devices. However, this area of fluid dynamics is poorly understood from both the experimental and simulation perspectives, which hampers the progress of these technologies. This paper reviews some of the recent developments in experimental techniques and modelling methods for non-equilibrium gas flows, examining their advantages and drawbacks. We also present new results from our computational investigations into both hypersonic and microsystem flows using two distinct numerical methodologies: the direct simulation Monte Carlo method and extended hydrodynamics. While the direct simulation approach produces excellent results and is used widely, extended hydrodynamics is not as well developed but is a promising candidate for future more complex simulations. Finally, we discuss some of the other situations where these simulation methods could be usefully applied, and look to the future of numerical tools for non-equilibrium flows.

On the microscopic interpretation of the coupling of the symmetric velocity gradient to the magnetization relaxation

On incorporating osmotic prestretch/prestress in image-driven finite element simulations of cartilage.

The equations of an incompressible, homogeneous fluid occupying a bounded domain in $\mathbb R^3$ are considered. &nbsp The stress tensor has a general polynomial dependence on the symmetric velocity gradient. The goal is to estimate the dimension of the global attractor in terms of relevant physical constants.

We consider a class of viscous fluids with a general monotone dependence of the viscous stress on the symmetric velocity gradient. We introduce the concept of dissipative solution to the associated initial boundary value problem inspired by the measure-valued solutions for the inviscid (Euler) system. We show the existence as well as the weak–strong uniqueness property in the class of dissipative solutions. Finally, the dissipative solution enjoying certain extra regularity coincides with a strong solution of the same problem.

Finite fractal dimension of the global attractor for a class of non-Newtonian fluids

The focus is on the description of the crystallographic texture and its evolution within a macro-scale model employing the Fourier coefficients of the orientation distribution function as internal variables. Assuming perfectly plastic single crystals, the overall structure of the corresponding evolution equations can be motivated. For initially isotropic polycrystals and processes with a symmetric velocity gradient, explicit representations for the initial evolution of the Fourier coefficients of even order can be given. The involved functions can be determined by comparison to a reference model or to experiments.

<p style='text-indent:20px;'>We introduce a new concept of <i>dissipative varifold solution</i> to models of two phase compressible viscous fluids. In contrast with the existing approach based on the Young measure description, the new formulation is variational combining the energy and momentum balance in a single inequality. We show the existence of dissipative varifold solutions for a large class of general viscous fluids with non–linear dependence of the viscous stress on the symmetric velocity gradient.</p>

We study the dynamics of an incompressible, homogeneous fluid of a power‐law type, with the stress tensor T = ν(1 + µ|Dv|)p−2Dv, where Dv is a symmetric velocity gradient. We consider the two‐dimensional problem with periodic boundary conditions and p ∈ (1, 2). Under these assumptions, we estimate the fractal dimension of the exponential attractor, using the so‐called method of 𝓁‐trajectories. Copyright © 2007 John Wiley & Sons, Ltd.

A mean-field Fokker-Planck equation approach to the dynamics of ferrofluids in the presence of a magnetic field and velocity gradients is proposed that incorporates magnetic dipole-dipole interactions of the colloidal particles. The model allows to study the combined effect of a magnetic field and dipolar interactions on the viscosity of the ferrofluid. It is found that dipolar interactions lead to additional non-Newtonian contributions to the stress tensor, which modify the behavior of the non-interacting system. The predictions of the present model are in qualitative agreement with experimental results, such as the enhancement as well as the different anisotropy of the magnetoviscous effect and the dependence on the symmetric velocity gradient.

Differentiability of the solution operator and the dimension of the attractor for certain power–law fluids

Rotation of axes for anisotropic metal in FEM simulations

Shear-dependent deformation of erythrocytes in rheology of human blood.