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

We derive a biomembrane model consisting of a fluid enclosed by a lipid membrane. Themembrane is characterized by its Canham-Helfrich energy (Willmore energy with areaconstraint) and acts as a boundary force on the Navier-Stokes system modeling anincompressible fluid. We give a concise description of the model and of the associatednumerical scheme. We provide numerical simulations with emphasis on the comparisonsbetween different types of flow: the geometric model which does not take into account thebulk fluid and the biomembrane model for two different regimes of parameters.

Atmospheric flow equations govern the time evolution of chemical concentrations in theatmosphere. When considering gas and particle phases, the underlying partial differentialequations involve advection and diffusion operators, coagulation effects, and evaporationand condensation phenomena between the aerosol particles and the gas phase. Operatorsplitting techniques are generally used in global air quality models. When consideringorganic aerosol particles, the modeling of the thermodynamic equilibrium of each particleleads to the determination of the convex envelope of the energy function. Two strategiesare proposed to address the computation of convex envelopes. The first one is based on aprimal-dual interior-point method, while the second one relies on a variationalformulation, an appropriate augmented Lagrangian, an Uzawa iterative algorithm, and finiteelement techniques. Numerical experiments are presented for chemical systems ofatmospheric interest, in order to compute convex envelopes in various spacedimensions.

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

This paper deals with the evolution Fokker-Planck-Smoluchowski configurationalprobability diffusion equation for the FENE dumbbell model in dilute polymer solutions. Weprove the exponential convergence in time of the solution of this equation to acorresponding steady-state solution, for arbitrary velocity gradients.

We present two-dimensional simulations of chemotactic self-propelled bacteria swimming ina viscous fluid. Self-propulsion is modelled by a couple of forces of same intensity andopposite direction applied on the rigid bacterial body and on an associated region in thefluid representing the flagellar bundle. The method for solving the fluid flow and themotion 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 bypenalizing the strain rate tensor on the rigid domain, while incompressibility is treatedby duality. This model allows to achieve an accurate description of fluid motion andhydrodynamic interactions in moderate to concentrated active suspensions. A mesoscopicmodel is also used, in which the size of the bacteria is supposed to be much smaller thanthe elements of fluid: the perturbation of the fluid due to propulsion and motion of theswimmers is neglected, and the fluid is only subjected to the buoyant forcing induced bythe presence of the bacteria, which are denser than the fluid. Although this model doesnot accurately take into account hydrodynamic interactions, it is able to reproducecomplex collective dynamics observed in concentrated bacterial suspensions, such asbioconvection. From a mathematical point of view, both models lead to a minimizationproblem which is solved with a standard Finite Element Method. In order to ensurerobustness, a projection algorithm is used to deal with contacts between particles. Wealso reproduce chemotactic behaviour driven by oxygen: an advection-diffusion equation onthe oxygen concentration is solved in the fluid domain, with a source term accounting foroxygen consumption by the bacteria. The orientations of the individual bacteria aresubjected to random changes, with a frequency that depends on the surrounding oxygenconcentration, in order to favor the direction of the concentration gradient.

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

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

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

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

We demonstrate, how the mechanism based approach to multicomponent diffusion can beincluded into the general thermodynamic framework, and prove the corresponding dissipationinequalities. To satisfy thermodynamic restrictions, the kinetic law of an elementaryprocess cannot have an arbitrary form. For the general kinetic law (the generalized MassAction Law), additional conditions are proved. The cell–jump formalism gives anintuitively clear representation of the elementary transport processes and, at the sametime, produces kinetic finite elements, a tool for numerical simulation.

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

Ice is assumed to behave as an incompressible fluid with nonlinear viscosity, so that thevelocity and pressure in the ice domain satisfy a nonlinear Stokes problem. The shape ofthe ice domain is defined using the volume fraction of ice, that is one in the ice regionand zero elsewhere. The volume fraction of ice satisfies a transport equation with asource 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 functionD 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 Dsatisfies a transport equation with nonlinear source terms to model cracks formation orhealing.

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

Two computations are presented at different time scales. First, the dynamics ofRhonegletscher, Swiss Alps, are investigated in 3D from 2007 to 2100 using severalclimatic scenarios and without considering ice damage. Second, ice damage is taken intoaccount in order to reproduce the calving process of a 2D glacier tongue submerged bywater.

Blood rheology is completely determined by its major corpuscles which are erythrocytes,or red blood cells (RBCs). That is why understanding and correct mathematical descriptionof RBCs behavior in blood is a critical step in modelling the blood dynamics. Variousphenomena provided by RBCs such as aggregation, deformation, shear-induced diffusion andnon-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 importantblood corpuscles are platelets, which are crucial for blood clotting. RBCs strongly affectthe platelet transport in blood expelling them to the vessel walls and increasing theirdispersion, which has to be considered in models of clotting. In this article we give abrief review of basic modern approaches in mathematical description of these phenomena,discuss their applicability to real flow conditions and propose further pathways fordeveloping the theory of blood flow.

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

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