This paper is concerned with the numerical simulation of a thermodynamically compatible viscoelastic shear-thinning fluid model, particularly well suited to describe the rheological response of blood, under physiological conditions. Numerical simulations are performed in two idealized three-dimensional geometries, a stenosis and a curved vessel, to investigate the combined effects of flow inertia, viscosity and viscoelasticity in these geometries. The aim of this work is to provide new insights into the modeling and simulation of homogeneous rheological models for blood and a basis for further developments in modeling and prediction.

We derive a biomembrane model consisting of a fluid enclosed by a lipid membrane. The membrane is characterized by its Canham-Helfrich energy (Willmore energy with area constraint) and acts as a boundary force on the Navier-Stokes system modeling an incompressible fluid. We give a concise description of the model and of the associated numerical scheme. We provide numerical simulations with emphasis on the comparisons between different types of flow: the geometric model which does not take into account the bulk fluid and the biomembrane model for two different regimes of parameters.

Atmospheric flow equations govern the time evolution of chemical concentrations in the atmosphere. When considering gas and particle phases, the underlying partial differential equations involve advection and diffusion operators, coagulation effects, and evaporation and condensation phenomena between the aerosol particles and the gas phase. Operator splitting techniques are generally used in global air quality models. When considering organic aerosol particles, the modeling of the thermodynamic equilibrium of each particle leads to the determination of the convex envelope of the energy function. Two strategies are proposed to address the computation of convex envelopes. The first one is based on a primal-dual interior-point method, while the second one relies on a variational formulation, an appropriate augmented Lagrangian, an Uzawa iterative algorithm, and finite element techniques. Numerical experiments are presented for chemical systems of atmospheric interest, in order to compute convex envelopes in various space dimensions.

Peristaltic pumping of fluid is a fundamental method of transport in many biological processes. In some instances, particles of appreciable size are transported along with the fluid, such as ovum transport in the oviduct or kidney stones in the ureter. In some of these biological settings, the fluid may be viscoelastic. In such a case, a nonlinear constitutive equation to describe the evolution of the viscoelastic contribution to the stress tensor must be included in the governing equations. Here we use an immersed boundary framework to study peristaltic transport of a macroscopic solid particle in a viscoelastic fluid governed by a Navier-Stokes/Oldroyd-B model. Numerical simulations of peristaltic pumping as a function of Weissenberg number are presented. We examine the spatial and temporal evolution of the polymer stress field, and also find that the viscoelasticity of the fluid does hamper the overall transport of the particle in the direction of the wave.

This paper deals with the evolution Fokker-Planck-Smoluchowski configurational probability diffusion equation for the FENE dumbbell model in dilute polymer solutions. We prove the exponential convergence in time of the solution of this equation to a corresponding steady-state solution, for arbitrary velocity gradients.

We present two-dimensional simulations of chemotactic self-propelled bacteria swimming in a viscous fluid. Self-propulsion is modelled by a couple of forces of same intensity and opposite direction applied on the rigid bacterial body and on an associated region in the fluid representing the flagellar bundle. The method for solving the fluid flow and the motion of the bacteria is based on a variational formulation written on the whole domain, strongly coupling the fluid and the rigid particle problems: rigid motion is enforced by penalizing the strain rate tensor on the rigid domain, while incompressibility is treated by duality. This model allows to achieve an accurate description of fluid motion and hydrodynamic interactions in moderate to concentrated active suspensions. A mesoscopic model is also used, in which the size of the bacteria is supposed to be much smaller than the elements of fluid: the perturbation of the fluid due to propulsion and motion of the swimmers is neglected, and the fluid is only subjected to the buoyant forcing induced by the presence of the bacteria, which are denser than the fluid. Although this model does not accurately take into account hydrodynamic interactions, it is able to reproduce complex collective dynamics observed in concentrated bacterial suspensions, such as bioconvection. From a mathematical point of view, both models lead to a minimization problem which is solved with a standard Finite Element Method. In order to ensure robustness, a projection algorithm is used to deal with contacts between particles. We also reproduce chemotactic behaviour driven by oxygen: an advection-diffusion equation on the oxygen concentration is solved in the fluid domain, with a source term accounting for oxygen consumption by the bacteria. The orientations of the individual bacteria are subjected to random changes, with a frequency that depends on the surrounding oxygen concentration, in order to favor the direction of the concentration gradient.

We propose a mixed formulation for non-isothermal Oldroyd–Stokes problem where the both extra stress and the heat flux’s vector are considered. Based on such a formulation, a dual mixed finite element is constructed and analyzed. This finite element method enables us to obtain precise approximations of the dual variable which are, for the non-isothermal fluid flow problems, the viscous and polymeric components of the extra-stress tensor, as well as the heat flux. Furthermore, it has properties analogous to the finite volume methods, namely, the local conservation of the momentum and the mass.

Waxy Crude Oils (WCO’s) are characterized by the presence of heavy paraffins in sufficiently large concentrations. They exhibit quite complex thermodynamical and rheological behaviour and present the peculiar property of giving rise to the formation of segregated wax deposits, when temperature falls down the so called WAT, or Wax Appearance Temperature. In extreme cases, segregated waxes may lead to pipeline occlusion due to deposition on cold walls. In this paper we review the mathematical models formulated to describe: (i) wax cystallization or thawing in cooling/heating cycles; (ii) the mechanisms of mass transport in saturated non-isothermal solutions; (iii) the experimental device used to measure wax solubility and wax diffusivity; (iv) wax deposition in pipelines carrying a warm, wax-saturated WCO through cold regions; (v) wax deposition accompanied by gelification during the cooling of a WCO under a thermal gradient.

Diffusion preserves the positivity of concentrations, therefore, multicomponent diffusion should be nonlinear if there exist non-diagonal terms. The vast variety of nonlinear multicomponent diffusion equations should be ordered and special tools are needed to provide the systematic construction of the nonlinear diffusion equations for multicomponent mixtures with significant interaction between components. We develop an approach to nonlinear multicomponent diffusion based on the idea of the reaction mechanism borrowed from chemical kinetics.

Chemical kinetics gave rise to very seminal tools for the modeling of processes. This is the stoichiometric algebra supplemented by the simple kinetic law. The results of this invention are now applied in many areas of science, from particle physics to sociology. In our work we extend the area of applications onto nonlinear multicomponent diffusion.

We demonstrate, how the mechanism based approach to multicomponent diffusion can be included into the general thermodynamic framework, and prove the corresponding dissipation inequalities. To satisfy thermodynamic restrictions, the kinetic law of an elementary process cannot have an arbitrary form. For the general kinetic law (the generalized Mass Action Law), additional conditions are proved. The cell–jump formalism gives an intuitively clear representation of the elementary transport processes and, at the same time, produces kinetic finite elements, a tool for numerical simulation.

A numerical model to compute the dynamics of glaciers is presented. Ice damage due to cracks or crevasses can be taken into account whenever needed. This model allows simulations of the past and future retreat of glaciers, the calving process or the break-off of hanging glaciers. All these phenomena are strongly affected by climate change.

Ice is assumed to behave as an incompressible fluid with nonlinear viscosity, so that the velocity and pressure in the ice domain satisfy a nonlinear Stokes problem. The shape of the ice domain is defined using the volume fraction of ice, that is one in the ice region and zero elsewhere. The volume fraction of ice satisfies a transport equation with a source term on the upper ice-air free surface accounting for ice accumulation or melting. If local effects due to ice damage must be taken into account, the damage function D is introduced, ranging between zero if no damage occurs and one. Then, the ice viscosity μ in the momentum equation must be replaced by (1 − D)μ. The damage function D satisfies a transport equation with nonlinear source terms to model cracks formation or healing.

A splitting scheme allows transport and diffusion phenomena to be decoupled. Two fixed grids are used. The transport equations are solved on an unstructured grid of small cubic cells, thus allowing numerical diffusion of the volume fraction of ice to be reduced as much as possible. The nonlinear Stokes problem is solved on an unstructured mesh of tetrahedrons, larger than the cells, using stabilized finite elements.

Two computations are presented at different time scales. First, the dynamics of Rhonegletscher, Swiss Alps, are investigated in 3D from 2007 to 2100 using several climatic scenarios and without considering ice damage. Second, ice damage is taken into account in order to reproduce the calving process of a 2D glacier tongue submerged by water.

Blood rheology is completely determined by its major corpuscles which are erythrocytes, or red blood cells (RBCs). That is why understanding and correct mathematical description of RBCs behavior in blood is a critical step in modelling the blood dynamics. Various phenomena provided by RBCs such as aggregation, deformation, shear-induced diffusion and non-uniform radial distribution affect the passage of blood through the vessels. Hence, they have to be taken into account while modelling the blood dynamics. Other important blood corpuscles are platelets, which are crucial for blood clotting. RBCs strongly affect the platelet transport in blood expelling them to the vessel walls and increasing their dispersion, which has to be considered in models of clotting. In this article we give a brief review of basic modern approaches in mathematical description of these phenomena, discuss their applicability to real flow conditions and propose further pathways for developing the theory of blood flow.

Various particle methods are widely used to model dynamics of complex media. In this work molecular dynamics and dissipative particles dynamics are applied to model blood flows composed of plasma and erythrocytes. The properties of the homogeneous particle fluid are studied. Capillary flows with erythrocytes are investigated.

We present the fiber-spring elastic model of the arterial wall with atherosclerotic plaque composed of a lipid pool and a fibrous cap. This model allows us to reproduce pressure to cross-sectional area relationship along the diseased vessel which is used in the network model of global blood circulation. Atherosclerosis attacks a region of systemic arterial network. Our approach allows us to examine the impact of the diseased region onto global haemodynamics.