Numerical simulation of effect of convection-diffusion on oxygen transport in microcirculation

The entire process of oxygen transport in microcirculation by developing a 3D porous media model is calculated numerically with coupled solid deformation-fluid seepage-convection and diffusion . The principal novelty of the model is that it takes into account volumetric deformation of both capillary and tissues resulting from capillary fluctuation. How solid deformation, fluid seepage, and convection-diffusion combine to affect oxygen transport is examined quantitatively: (1) Solid deformation is more significant in the middle of capillary, where the maximum value of volumetric deformation reaches about 0.5%. (2) Solid deformation has positive influence on the tissue fluid so that it flows more uniformly and causes oxygen to be transported more uniformly, and eventually impacts oxygen concentration by 0.1%-0.5%. (3) Convection-diffusion coupled deformation and seepage has a maximum (16%) and average (3%) increase in oxygen concentration, compared with pure molecular diffusion. Its more significant role is to allow oxygen to be transported more evenly.     

On the nonlinear dynamics of an ecoepidemic reaction–diffusion model

A reaction–diffusion ecoepidemic model of predator–prey type with a transmissible disease spreading among the predator species only is considered. The longtime behavior of solutions is analyzed and, in particular, absorbing sets in the phase space are determined. Conditions guaranteeing the non existence of non-constant equilibria have been found. Linear and non-linear stability conditions for biologically meaningful equilibria are determined.     

A modelling approach for the heterogeneous oxidation of elastomers

The influence of oxygen on elastomers, known as oxidation, is one of the most important ageing processes and becomes more and more important for nowadays applications. The interaction with thermal effects as well as antioxidants makes oxidation of polymers a complex process. Based on the polymer chosen and environmental conditions, the ageing processes may behave completely different. In a lot of cases the influence of oxygen is limited to the surface layer of the samples, commonly referred to as diffusion-limited oxidation. For the lifetime prediction of elastomer components, it is essential to have detailed knowledge about the absorption and diffusion behaviour of oxygen molecules during thermo-oxidative ageing and how they react with the elastomer. Experimental investigations on industrially used elastomeric materials are executed in order to develop and fit models, which shall be capable of predicting the permeation and consumption of oxygen as well as changes in the mechanical properties. The latter are of prime importance for technical applications of rubber components. Oxidation does not occur homogeneously over the entire elastomeric component. Hence, material models which include ageing effects have to be amplified in order to consider heterogeneous ageing, which highly depends on the ageing temperature. The influence of elevated temperatures upon accelerated ageing has to be critically analysed, and influences on the permeation and diffusion coefficient have to be taken into account. This work presents phenomenological models which describe the oxygen uptake and the diffusion into elastomers based on an improved understanding of ongoing chemical processes and diffusion limiting modifications. On the one side, oxygen uptake is modelled by means of Henry’s law in which solubility is a function of the temperature as well as the ageing progress. The latter is an irreversible process and described by an inner differential evolution equation. On the other side, further diffusion of oxygen into the material is described by a model based on Fick’s law, which is modified by a reaction term. The evolved diffusion-reaction equation depends on the ageing temperature as well as on the progress of ageing and is able to describe diffusion-limited oxidation.     

Behavior of detonation propagation in mixtures with concentration gradients

Behavior of detonation waves in mixtures with concentration gradients normal to the propagation direction was studied experimentally. Mixtures with various concentration gradients were formed by sliding the separation plate which divides a detonation chamber from a diffusion chamber in which a diffusion gas was initially introduced. A stoichiometric hydrogen–oxygen mixture was charged in the detonation chamber, while oxygen or nitrogen was filled in the diffusion gas chamber. Temporal concentration measurement was conducted by the infrared absorption method using ethane as alternate of oxygen. Smoked foil records show a deformation of regular diamond cells to parallelogram ones, which well corresponds to local mixture concentration. Schlieren photographs reveal the tilted wave front whose angle is consistent with the deflection angle of the detonation front obtained from trajectories of the triple point. The local deflection angle increases with increase in local concentration gradient. Calculation of wave trajectory based on the ray tracing theory predicts formation of the tilted wave front from an initial planar front.      

Mathematical Model for the Solid Electrolyte Fuel Cell Cathode

An analytically solvable mathematical model for the cathode of a solid polymer electrolyte fuel cell is proposed. The problem of diffusion in a multicomponent air-vapor mixture in a porous cathode and water transport due to hydrodynamic and electroosmotic forces is solved. The volt-ampere characteristic of the fuel cell is determined taking into account the polarization characteristics and finite conductivity of the electrolyte. An expression is obtained for the thickness of the electrochemical-reaction zone, which gives an estimate of the catalyst efficiency. It is shown that the finiteness of the rate of oxygen diffusion into the reaction zone limits the current density and the fuel cell efficiency. A comparison of the results with available theoretical and experimental data shows that the solutions obtained for the model coincide with the solutions for the more complex Bernardi and Verbrugge model.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 27–37, September–October, 2005.     

Vasculature based model for characterizing the oxygen transport in skin tissues – analogy to the Weinbaum-Jiji bioheat equation

Based on the conceptual three-layer microvascular structure of skin tissues proposed by Weinbaum et al. [20–25] and in analogy to the well known Weinbaum-Jiji (W-J) bioheat equation, a new oxygen transport model was established in this paper, which collectively included the contributions of the vascular geometry and the blood flow condition. The new one-dimensional three-layer oxygen transport model was then applied to predict the average oxygen concentration distribution in skin tissues and numerical solutions for the boundary value problem coupling the three layers were obtained. A simple expression for the tensor diffusivity (D_eff) of oxygen transport over the deep tissue layer was presented, which was orders of magnitude higher than the intrinsic diffusivity (D_t) in tissue without blood flow. Effects of blood flow velocity and vascular geometry to the oxygen transport were investigated. Calculations indicated that the vascular geometry had significant effects on oxygen transport. The oxygen exchange between the arteries and veins was relatively small for the deep tissue layer. Further, the average oxygen concentration gradient appears low in intermediate layer due to large capillary perfusion. The theoretical results were implemented to interpret some previous experimental results and a better understanding on the oxygen transport across the vascularized living tissues was obtained. The strategy proposed in this paper may provide a feasible way to comprehensively characterize the oxygen transport behaviors in living tissues with real and complex vasculature.     

Diffusive Effects on Recovery of Light Oil by Medium Temperature Oxidation

Volatile oil recovery by means of air injection is studied as a method to improve recovery from low permeable reservoirs. We consider the case in which the oil is directly combusted into small products, for which we use the term medium temperature oil combustion. The two-phase model considers evaporation, condensation and reaction with oxygen. In the absence of thermal, molecular and capillary diffusion, the relevant transport equations can be solved analytically. The solution consists of three waves, i.e., a thermal wave, a medium temperature oxidation (MTO) wave and a saturation wave separated by constant state regions. A striking feature is that evaporation occurs upstream of the combustion reaction in the MTO wave. The purpose of this paper is to show the effect of diffusion mechanisms on the MTO process. We used a finite element package (COMSOL) to obtain a numerical solution; the package uses fifth-order Lagrangian base functions, combined with a central difference scheme. This makes it possible to model situations at realistic diffusion coefficients. The qualitative behavior of the numerical solution is similar to the analytical solution. Molecular diffusion lowers the temperature of the MTO wave, but creates a small peak near the vaporization region. The effect of thermal diffusion smoothes the thermal wave and widens the MTO region. Capillary diffusion increases the temperature in the upstream part of the MTO region and decreases the efficiency of oil recovery. At increasing capillary diffusion the recovery by gas displacement gradually becomes higher, leaving less oil to be recovered by combustion. Consequently, the analytical solution with no diffusion and numerical solutions at a high capillary diffusion coefficient become different. Therefore high numerical diffusion, significant in numerical simulations especially in coarse gridded simulations, may conceal the importance of combustion in recovering oil.     

Diffusion–reaction of aluminum and oxygen in thermally grown Al2O3 oxide layers

The diffusion–reaction of aluminum (Al) and oxygen (O), to form thermally grown oxide (TGO) layers in thermal barrier coatings (TBCs), is studied through an analytical model. A nonsymmetrical radial basis function approach is used to numerically solve the mass balance equations that predict the TGO growth. Correct boundary conditions for the Al and O reactions are laid out using scaling arguments. The Damköhler number shows that the O–Al reaction is several orders of magnitude faster than diffusion. In addition, a comparison between aluminum and oxygen diffusivities indicates that TGO growth is governed by aluminum diffusion. The results are compared with experimental measurements on air plasma spray-deposited TBCs treated at 1,373 K with exposure times ranging from 1 to 1700 hours. We found that, for several time decades, the thickness of the thermally grown layer has power law dependence of time with an exponent of ½, following the diffusion control mechanism. At later times, however, the presence of other oxides and additional kinetics modify the diffusive exponent.     

Random Dynamics of Stochastic Reaction–Diffusion Systems with Additive Noise

For a typical autocatalytic stochastic reaction–diffusion system with additive noises, the multicomponent reversible Gray–Scott reaction–diffusion system on a two-dimensional bounded domain, the existence of a random attractor and its attracting regularity are proved through the sharp uniform estimates showing respectively the pullback absorbing, asymptotically compact, and flattening properties.     

Effect of Non-Stoichiometry and Difference between the Tensile and Compressive Moduli of Elasticity of Perovskite on the Diffusion Creep of a Thick-Walled Perovskite Cylinder

International Applied Mechanics - A physical model for describing the diffusion creep in perovskite-type material given oxygen non-stoichiometry and tensile–compressive asymmetry is...     

Modeling Concentration Distribution and Deformation During Convection-Enhanced Drug Delivery into Brain Tissue

Convection-enhanced drug delivery is a technique where a therapeutic agent is infused under positive pressure directly into the brain tissue. For predicting the final concentration distribution and optimizing infusion rate and catheter placement, numerical models can be of great help. However, despite advances in modeling this process, often the infused agent does not reach the targeted region prescribed in the modeling phase. In this study, patient-specific brain structure and parameters, obtained from diffusion tensor imaging (DTI), are implemented in a numerical model which describes the flow and transport in an elastic deformable matrix. To our knowledge, this is the first time that information from DTI is used in a numerical model which includes both transport of a therapeutic agent and tissue deformation. Fractional anisotropy (FA) is used to distinguish between gray and white matter and tortuosity to differentiate between inside and outside the brain tissue. One voxel in the DT-image is represented by one element of the numerical grid. The DT-images were in addition used to determine the orientation of the white matter fiber tracts and calibrate permeability and diffusion coefficients found in the literature. Values chosen for the porosity and Lamé parameters are also based on those found in the literature. Given realistic literature values, the calibration of the permeability and diffusion tensors are shown to be successful. Our result shows that preferential flow occur in direction of the white matter fiber tracts. The current model assumes linear deformation, corresponding to small porosity changes. But, because large porosity changes occur that may adversely affect drug transport, non-linear deformations should be included in the future.     

A perturbation method for mixed boundary-value problems in pressure transient testing

Data recorded during a well test is interpreted for formation parameters, such as permeability, by comparing the measured pressure transient with that predicted by a mathematical model of the system. In single-phase homogeneous situations, this model is based on the linear diffusion equation. Despite the simplicity of this equation, it is often difficult or laborious to construct exact solutions, in some cases because the relevant problem is of mixed boundary-value character. Here we describe an approximate calculational method which, with little effort, gives good results in a variety of well test problems of this type.We first prove a rather general perturbation theorem, and then apply it in three illustrative cases. The first example provides a test of our basic result for the case of a fully penetrating vertical fracture, for which an exact (though nonelementary) solution is known. The second example is that of partially penetrating or horizontal wells; it is shown that our general result can provide a mathematical basis for the so-called pressure-averaging technique in which the pressure-flux relationship is approximated by assuming that the flux is uniform along the well and then computing the spatial average of the wellbore pressure. Our third example is a new result for the steady-state pressure drop due to flow into a small circular hole on the surface of an impermeable cylinder. This example has relevance for the Schlumberger RFT tool, which withdraws fluid from the formation via a circular probe which penetrates the mudcake surrounding the borehole wall. Numerical results for the shape factor which represents the effect of the borehole curvature are provided.     

A porous media theory for characterization of membrane blood oxygenation devices

A porous media theory has been proposed to characterize oxygen transport processes associated with membrane blood oxygenation devices. For the first time, a rigorous mathematical procedure based a volume averaging procedure has been presented to derive a complete set of the governing equations for the blood flow field and oxygen concentration field. As a first step towards a complete three-dimensional numerical analysis, one-dimensional steady case is considered to model typical membrane blood oxygenator scenarios, and to validate the derived equations. The relative magnitudes of oxygen transport terms are made clear, introducing a dimensionless parameter which measures the distance the oxygen gas travels to dissolve in the blood as compared with the blood dispersion length. This dimensionless number is found so large that the oxygen diffusion term can be neglected in most cases. A simple linear relationship between the blood flow rate and total oxygen transfer rate is found for oxygenators with sufficiently large membrane surface areas. Comparison of the one-dimensional analytic results and available experimental data reveals the soundness of the present analysis.     

Analysis and Modeling of the Turbulent Diffusion of Turbulent Kinetic Energy in Natural Convection

Buoyant flows often contain regions with unstable and stable thermal stratification from which counter gradient turbulent fluxes are resulting, e.g. fluxes of heat or of any turbulence quantity. Basing on investigations in meteorology an improvement in the standard gradient-diffusion model for turbulent diffusion of turbulent kinetic energy is discussed. The two closure terms of the turbulent diffusion, the velocity-fluctuation triple correlation and the velocity-pressure fluctuation correlation, are investigated based on Direct Numerical Simulation (DNS) data for an internally heated fluid layer and for Rayleigh–Bénard convection. As a result it is decided to extend the standard gradient-diffusion model for the turbulent energy diffusion by modeling its closure terms separately. Coupling of two models leads to an extended RANS model for the turbulent energy diffusion. The involved closure term, the turbulent diffusion of heat flux, is studied based on its transport equation. This results in a buoyancy-extended version of the Daly and Harlow model. The models for all closure terms and for the turbulent energy diffusion are validated with the help of DNS data for internally heated fluid layers with Prandtl number Pr = 7 and for Rayleigh–Bénard convection with Pr = 0.71. It is found that the buoyancy-extended diffusion model which involves also a transport equation for the variance of the vertical velocity fluctuation gives improved turbulent energy diffusion data for the combined case with local stable and unstable stratification and that it allows for the required counter gradient energy flux.     

3D numerical study of tumor blood perfusion and oxygen transport during vascular normalization

The changes of blood perfusion and oxygen transport in tumors during tumor vascular normalization are studied with 3-dimensional mathematical modeling and numerical simulation. The models of tumor angiogenesis and vascular-disrupting are used to simulate "un-normalized" and "normalized" vasculatures. A new model combining tumor hemodynamics and oxygen transport is developed. In this model, the intravasculartransvascular-interstitial flow with red blood cell(RBC) delivery is tightly coupled, and the oxygen resource is produced by heterogeneous distribution of hematocrit from the flow simulation. The results show that both tumor blood perfusion and hematocrit in the vessels increase, and the hypoxia microenvironment in the tumor center is greatly improved during vascular normalization. The total oxygen content inside the tumor tissue increases by about 67%, 51%, and 95% for the three approaches of vascular normalization,respectively. The elevation of oxygen concentration in tumors can improve its metabolic environment, and consequently reduce malignancy of tumor cells. It can also enhance radiation and chemotherapeutics to tumors.     

Finite element simulation on the mechanical properties of MHS materials

Finite element simulations are carried out to examine the mechanical behavior of the metallic hollow sphere (MHS) material during their large plastic deformation and to estimate the energy absorbing capacity of these materials under uniaxial compression. A simplified model is proposed from experimental observations to describe the connection between the neighboring spheres, which greatly improves the computation efficiency. The effects of the governing physical and geometrical parameters are evaluated; whilst a special attention is paid to the plateau stress, which is directly related to the energy absorbing capacity. Finally, the empirical functions of the relative material density are proposed for the elastic modulus, yield strength and plateau stress for FCC packing arrangement of hollow spheres, showing a good agreement with the experimental results obtained in our previous study.     

Diffusion of magnetic nanoparticles in a multi-site injection process within a biological tissue during magnetic fluid hyperthermia using lattice Boltzmann method

In this study the lattice Boltzmann model (LBM) has been used to simulate diffusion of magnetic nanoparticles (MNPs) injected at multiple sites inside a biological tissue during magnetic fluid hyperthermia (MFH). To validate the numerical results, diffusion in infinite one and two dimensional domains have been compared with the analytical solutions. Agreement were excellent. Also diffusion of a water based nanofluid containing magnetite MNPs (ferrofluid) for mono and multi-site injection in the tissue has been studied. Moreover, the effects of ferrofluid injection volume as well as infusion flow rate of ferrofluid on the distribution of MNPs have been investigated.     

Diffusion separation of chemical elements on a catalytic surface

An asymptotic expansion of the solution, for large Schmidt numbers, of the system of equations of a chemically nonequilibrium multicomponent boundary layer on the catalytic surface of a blunt body [1] is used to obtain expressions for the diffusion fluxes of the reaction products and chemical elements and the heat flux as functions of the gradients of the reaction product concentrations, chemical element concentrations and enthalpy across the boundary layer. It is shown that when the body is exposed to a supersonic air flow, the diffusion separation of the chemical element oxygen depends importantly on the atom concentration at the outer edge of the boundary layer and the nature of the homogeneous and heterogeneous catalytic reactions. If the surface promotes the rapid recombination of oxygen atoms and is chemically neutral with respect to nitrogen atoms, then an excess of the chemical element oxygen is formed on the body. Otherwise we get an enhanced concentration of the element nitrogen. As distinct from the case of an ideally catalytic wall [2–4], on a surface possessing the property of catalytic selectivity the diffusion separation of chemical elements takes place even when only atoms are present at the outer edge of the boundary layer. On a chemically neutral surface diffusion separation may be caused by homogeneous recombination reactions between oxygen and nitrogen atoms if their rate constants are essentially different.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 115–121, July–August, 1988.     

Effect of mass diffusion on the damping ratio in micro-beam resonators

Present investigation is focused on studying the effect of mass diffusion on the quality factor of the micro-beam resonators. Equation of motion is obtained using Hamilton’s principle and also the equations of thermo-diffusive elastic damping are established using two dimensional non-Fourier heat conduction and non-Fickian mass diffusion models. Free vibration of a clamped–clamped micro-beam with isothermal boundary conditions at both ends, and also a cantilever micro-beam with adiabatic boundary condition assumption at the free end, is studied using Galerkin reduced order model formulation for the first mode of vibration. Mass diffusion effects on the damping ratio are studied for the various micro-beam thicknesses and temperatures and the obtained results are compared with the results of a model in which the mass diffusion effect is ignored. In addition to the classic critical thickness of thermoelastic damping, a new critical thickness concerning mass diffusion is introduced.