Coupling 3D and 1D fluid-structure interaction models for blood flow simulations

Three-dimensional (3D) simulations of blood flow in medium to large vessels are now a common practice. These models consist of the 3D Navier-Stokes equations for incompressible Newtonian fluids coupled with a model for the vessel wall structure. However, it is still computationally unaffordable to simulate very large sections, let alone the whole, of the human circulatory system with fully 3D fluid-structure interaction models. Thus truncated 3D regions have to be considered. Reduced models, one-dimensional (1D) or zero-dimensional (0D), can be used to approximate the remaining parts of the cardiovascular system at a low computational cost. These models have a lower level of accuracy, since they describe the evolution of averaged quantities, nevertheless they provide useful information which can be fed to the more complex model. More precisely, the 1D models describe the wave propagation nature of blood flow and coupled with the 3D models can act also as absorbing boundary conditions. We consider in this work the coupling of a 3D fluid-structure interaction model with a 1D hyperbolic model. We study the stability of the coupling and present some numerical results. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A multiscale Darcy–Brinkman model for fluid flow in fractured porous media

The aim of this work is to present a reduced mathematical model for describing fluid flow in porous media featuring open channels or fractures. The Darcy’s law is assumed in the porous domain while the Stokes–Brinkman equations are considered in the fractures. We address the case of fractures whose thickness is very small compared to the characteristic diameter of the computational domain, and describe the fracture as if it were an interface between porous regions. We derive the corresponding interface model governing the fluid flow in the fracture and in the porous media, and establish the well-posedness of the coupled problem. Further, we introduce a finite element scheme for the approximation of the coupled problem, and discuss solution strategies. We conclude by showing the numerical results related to several test cases and compare the accuracy of the reduced model compared with the non-reduced one.     

A coupled system of ODEs and quasilinear hyperbolic PDEs arising in a multiscale blood flow model

We study a coupled system of ordinary differential equations and quasilinear hyperbolic partial differential equations that models a blood circulatory system in the human body. The mathematical system is a multiscale model in which a part of the system, where the flow can be regarded as Newtonian and homogeneous, and the vessels are long and large, is modeled by a set of hyperbolic PDEs in a one-spatial-dimensional network, and in the other part, where either vessels are too thin or the flow pattern is too complicated (such as in the heart), the flow is modeled as a lumped element by a set of ordinary differential equations as an analog of an electric circuit. The mathematical system consists of pairs of PDEs, one pair for each vessel, coupled at each junction through a system of ODEs. This model is a generalization of the widely studied models of arterial networks. We give a proof of the well-posedness of the initial-boundary value problem by showing that the classical solution exists, is unique, and depends continuously on initial, boundary and forcing functions and their derivatives.     

Two-fluid Casson model for pulsatile blood flow through stenosed arteries: A theoretical model

Pulsatile flow of blood through mild stenosed narrow arteries is analyzed by treating the blood in the core region as a Casson fluid and the plasma in the peripheral layer as a Newtonian fluid. Perturbation method is used to solve the coupled implicit system of non-linear differential equations. The expressions for velocity, wall shear stress, plug core radius, flow rate and resistance to flow are obtained. The effects of pulsatility, stenosis, peripheral layer and non-Newtonian behavior of blood on these flow quantities are discussed. It is found that the pressure drop, plug core radius, wall shear stress and resistance to flow increase with the increase of the yield stress or stenosis size while all other parameters held constant. The percentage of increase in the resistance to flow over the uniform diameter tube is considerably very low for the present two-fluid model compared with those of the single-fluid model.     

Transient Bernoulli flow in multi-port fluid devices with arbitrary geometry

We present a method for the solution of transient flow in a multi-port fluid device with arbitrary geometry. The method is applicable to fluid devices where the fluid motion is primarily inviscid throughout the volume, but locally near a device port some accommodation to viscous flow is introduced. The internal flow is characterized by an array of purely geometrical factors between ports, essentially a set of generalized impedances; the state variables elicited are the average volume flow rates through the device ports. The method creates a set of coupled non-linear time-dependent ordinary differential equations. The solution to this set of equations is much faster, typically by orders of magnitude, than a single run of a transient CFD model. We demonstrate our method with a simple example; we show that the results of the method agree well with a full CFD calculation.     

A numerical solution of a one-dimensional blood flow model—moving grid approach

We consider a one-dimensional blood flow model suitable for larger arteries. It consists of a hyperbolic system of two coupled nonlinear equations. The model has already been successfully used in practice. Its numerical solution is usually achieved by means of an explicit Taylor–Galerkin scheme. We have proposed a different approach. The system can be transformed to characteristic directions emphasizing the physical nature of the problem. We solved this system by using an operator splitting on a moving grid.     

Space-time discontinuous Galerkin method for the solution of fluid-structure interaction

The paper is concerned with the application of the space-time discontinuous Galerkin method (STDGM) to the numerical solution of the interaction of a compressible flow and an elastic structure. The flow is described by the system of compressible Navier-Stokes equations written in the conservative form. They are coupled with the dynamic elasticity system of equations describing the deformation of the elastic body, induced by the aerodynamical force on the interface between the gas and the elastic structure. The domain occupied by the fluid depends on time. It is taken into account in the Navier-Stokes equations rewritten with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. The resulting coupled system is discretized by the STDGM using piecewise polynomial approximations of the sought solution both in space and time. The developed method can be applied to the solution of the compressible flow for a wide range of Mach numbers and Reynolds numbers. For the simulation of elastic deformations two models are used: the linear elasticity model and the nonlinear neo-Hookean model. The main goal is to show the robustness and applicability of the method to the simulation of the air flow in a simplified model of human vocal tract and the flow induced vocal folds vibrations. It will also be shown that in this case the linear elasticity model is not adequate and it is necessary to apply the nonlinear model.     

An exact Riemann solver and a Godunov scheme for simulating highly transient mixed flows

The current research aims at deriving a one-dimensional numerical model for describing highly transient mixed flows. In particular, this paper focuses on the development and assessment of a unified numerical scheme adapted to describe free-surface flow, pressurized flow and mixed flow (characterized by the simultaneous occurrence of free-surface and pressurized flows). The methodology includes three steps. First, the authors derived a unified mathematical model based on the Preissmann slot model. Second, a first-order explicit finite volume Godunov-type scheme is used to solve the set of equations. Third, the numerical model is assessed by comparison with analytical, experimental and numerical results. The key results of the paper are the development of an original negative Preissmann slot for simulating sub-atmospheric pressurized flow and the derivation of an exact Riemann solver for the Saint-Venant equations coupled with the Preissmann slot.     

Coupled fluid-solid thermal interaction modeling for efficient transient simulation of biphasic water-steam energy systems

This paper proposes a fluid-solid coupled finite element formulation for the transient simulation of water-steam energy systems with phase change due to boiling and condensation. As it is commonly assumed in the study of thermal systems, the transient effects considered are exclusively originated by heat transfer processes. A homogeneous mixture model is adopted for the analysis of biphasic flow, resulting in a nonlinear transient advection-diffusion-reaction energy equation and an integral form for mass conservation in the fluid, coupled to the linear transient heat conduction equation for the solid. The conservation equations are approximated applying a stabilized Petrov-Galerkin FEM formulation, providing a set of coupled nonlinear equations for mass and energy conservation. This numerical model, combined with experimental heat transfer coefficients, provides a comprehensive simulation tool for the coupled analysis of boiling and condensation processes. For the treatment of enthalpy discontinuities traveling with the flow, a novel explicit-implicit time integration method based on Crank-Nicolson scheme is proposed, analyzing its accuracy and stability properties. To reduce problem size and enhance numerical efficiency, a modal superposition method with balanced truncation is applied to the solid equations. Finally, different example problems are solved to demonstrate the capabilities, flexibility and accuracy of the proposed formulation.     

Numerical simulations of water–gas flow in heterogeneous porous media with discontinuous capillary pressures by the concept of global pressure

We present an approach and numerical results for a new formulation modeling immiscible compressible two-phase flow in heterogeneous porous media with discontinuous capillary pressures. The main feature of this model is the introduction of a new global pressure, and it is fully equivalent to the original equations. The resulting equations are written in a fractional flow formulation and lead to a coupled degenerate system which consists of a nonlinear parabolic (the global pressure) equation and a nonlinear diffusion–convection one (the saturation equation) with nonlinear transmission conditions at the interfaces that separate different media. The resulting system is discretized using a vertex-centred finite volume method combined with pressure and flux interface conditions for the treatment of heterogeneities. An implicit Euler approach is used for time discretization. A Godunov-type method is used to treat the convection terms, and the diffusion terms are discretized by piecewise linear conforming finite elements. We present numerical simulations for three one-dimensional benchmark tests to demonstrate the ability of the method to approximate solutions of water–gas equations efficiently and accurately in nuclear underground waste disposal situations.     

Numerical simulation of the flow in semicircular canals with the method of fundamental solutions

We present a numerical model for the simulation of the flow in semicircular canals (SCCs). The governing equations for the flow are solved with the method of fundamental solutions (MFS), a mesh free method for boundary value problems. We describe the flow field in a SCC with utricle, and we find a vortex that had not yet been reported in literature. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling,analysis and simulation of case hardening of steel

A mathematical model for the gas carburizing in steel is presented. Carbon is dissolved in the surface layer of a low-carbon steel part at a temperature sufficient to render the steel austenitic, followed by quenching to form a martensitic microstructure. We have a nonlinear evolution equation for the temperature, coupled with a nonlinear evolution equation for the carbon concentration, both coupled with two ordinary differential equations to describe the phase fractions. We present mathematical results concerning the well-posedness of the model and finally present a simulation of the process using a finite element approximation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a coupled system of reaction–diffusion-transport equations arising from catalytic converter

This paper is concerned with two mathematical models which describe the transient behavior of a catalytic converter in automobile engineering. The first model consists of a coupled system of a heat-conduction equation and two integral equations while the second model involves only one integral equation. It is shown that for any nonnegative initial and boundary functions the three-equation model has a unique bounded global solution while the solution of the two-equation model blows up in finite time. The proof for the global existence and finite-time blow-up property of the solution is by the method of upper and lower solutions and its associated monotone iteration. This method can be used to develop computational algorithms for numerical solutions of the coupled systems.     

Novel analytical and numerical techniques for fractional temporal SEIR measles model

In this paper, a fractional temporal SEIR measles model is considered. The model consists of four coupled time fractional ordinary differential equations. The time-fractional derivative is defined in the Caputo sense. Firstly, we solve this model by solving an approximate model that linearizes the four time fractional ordinary differential equations (TFODE) at each time step. Secondly, we derive an analytical solution of the single TFODE. Then, we can obtain analytical solutions of the four coupled TFODE at each time step, respectively. Thirdly, a computationally effective fractional Predictor-Corrector method (FPCM) is proposed for simulating the single TFODE. And the error analysis for the fractional predictor-corrector method is also given. It can be shown that the fractional model provides an interesting technique to describe measles spreading dynamics. We conclude that the analytical and Predictor-Corrector schemes derived are easy to implement and can be extended to other fractional models. Fourthly, for demonstrating the accuracy of analytical solution for fractional decoupled measles model, we applied GMMP Scheme (Gorenflo-Mainardi-Moretti-Paradisi) to the original fractional equations. The comparison of the numerical simulations indicates that the solution of the decoupled and linearized system is close enough to the solution of the original system. And it also indicates that the linearizing technique is correct and effective.     

Homogenization of a three-phase flow model in porous media

We give homogenization results for an immiscible and incompressible three-phase flow model in a heterogeneous petroleum reservoir with periodic structure, including capillary effects. We consider a model which leads to a coupled system of partial differential equations which includes an elliptic equation and two nonlinear degenerate parabolic equations of convection–diffusion types. Using two-scale convergence, we get an homogenized model which governs the global behavior of the flow. The determination of effective properties require the numerical resolution of local problems in a standard cell.     

A micropolar fluid model of blood flow through a tapered artery with a stenosis

A nonlinear two‐dimensional micropolar fluid model for blood flow in a tapered artery with a single stenosis is considered. This model takes into account blood rheology in which blood consists of microelements suspended in plasma. The classical Navier–Stokes theory is inadequate to describe the microrotations or particles' spin of such suspension in a viscous medium. The governing equations involving unsteady nonlinear partial differential equations are solved using a finite difference scheme. A quantitative analysis performed through numerical computation shows that the axial velocity profile and the flow rate decrease and the wall shear stress increases once the artery is narrower in the presence of the polar effect. Furthermore, the taper angle certainly bears the potential to influence the velocity and the flow characteristics to considerable extent. Copyright © 2010 John Wiley & Sons, Ltd.     

A hybrid numerical model for multiphase fluid flow in a deformable porous medium

In this paper, a fully coupled finite volume-finite element model for a deforming porous medium interacting with the flow of two immiscible pore fluids is presented. The basic equations describing the system are derived based on the averaging theory. Applying the standard Galerkin finite element method to solve this system of partial differential equations does not conserve mass locally. A non-conservative method may cause some accuracy and stability problems. The control volume based finite element technique that satisfies local mass conservation of the flow equations can be an appropriate alternative. Full coupling of control volume based finite element and the standard finite element techniques to solve the multiphase flow and geomechanical equilibrium equations is the main goal of this paper. The accuracy and efficiency of the method are verified by studying several examples for which analytical or numerical solutions are available. The effect of mesh orientation is investigated by simulating a benchmark water-flooding problem. A representative example is also presented to demonstrate the capability of the model to simulate the behavior in heterogeneous porous media.     

Thermodynamic and statistical principles of the theory of elastoviscoplastic deformation and strengthening of materials

We propose a thermodynamic method and a statistical one for constructing the constitutive equations of elastoviscoplastic deformation and strengthening of materials. The thermodynamic method is based on the energy conservation law as well as the equations of entropy balance and entropy generation in the presence of self-equilibrated internal microstresses, which are characterized by coupled strengthening parameters. The general constitutive equations consist of the relations between thermodynamic flows and forces, which follow from nonnegativity of entropy generation and satisfy the generalized Onsager principle, as well as the thermoelasticity relations and the expression for entropy, which follow from the energy conservation law. The specific constitutive equations are obtained on the basis of representation of the energy dissipation rate as a sum of two constituents that describe translational and isotropic strengthening and are approximated by power and hyperbolic sine laws. Starting from the stochastic microstructural concepts, we construct the constitutive equations of elastoviscoplastic deformation and strengthening on the basis of the linear model of thermoelasticity and the nonlinear Maxwell model for spherical and deviatoric components of microstresses and microstrains, respectively. The solution of the problem of the effective properties and stress-strain state of a three-component material is constructed with the use of the combined Voigt–Reuss scheme and leads to constitutive equations coinciding, as to their form, with similar equations constructed by the thermodynamic method.     

Effect of hydrophobic patch on the modulation of electroosmotic flow and ion selectivity through nanochannel

In this article we made a systematic study of the electrokinetically driven flow through a long nanochannel patterned with charged hydrophobic patch embedded along the walls. The hydrophobic patch possess either similar or opposite charge to that of the hydrophilic portion of the channel wall. The hydrodynamic slip length is considered to be a function of the surface charge distributed along the patch. We consider the aqueous medium as binary symmetric electrolyte solution with constant property Newtonian fluid. We adopt a nonlinear model based on the Poisson-Nernst-Planck equations coupled with Navier–Stokes equations. The coupled set of governing equations are solved using a finite volume method. We use a higher order upwind based total variation diminishing (TVD) scheme to discretize the convective and electromigration terms (i.e., hyperbolic terms) and the central difference scheme is used to discretize the diffusion term. We have identified several interesting key features of the modulation of electroosmotic flow (EOF) through such a nanochannel. An enhancement or a reduction in average flow rate may be achieved depending on the polarity of the charges distributed along the hydrophobic or hydrophilic portion of the channel wall. We have also shown that the reversal in EOF may occur depending on the critical choice of the pertinent parameters. We have further studied the selectivity of mobile ions by regulating the charge properties of the channel walls.     

Global asymptotic stability for an age-structured model of hematopoietic stem cell dynamics

We investigate a system of two nonlinear age-structured partial differential equations describing the dynamics of proliferating and quiescent hematopoietic stem cell (HSC) populations. The method of characteristics reduces the age-structured model to a system of coupled delay differential and renewal difference equations with continuous time and distributed delay. By constructing a Lyapunov–Krasovskii functional, we give a necessary and sufficient condition for the global asymptotic stability of the trivial steady state, which describes the population dying out. We also give sufficient conditions for the existence of unbounded solutions, which describe the uncontrolled proliferation of HSC population. This study may be helpful in understanding the behavior of hematopoietic cells in some hematological disorders.