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12
Numerical mathematics
, 2000
"... Abstract. In this paper we introduce some basic differential models for the description of blood flow in the circulatory system. We comment on their mathematical properties, their meaningfulness and their limitation to yield realistic and accurate numerical simulations, and their contribution for a ..."
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Abstract. In this paper we introduce some basic differential models for the description of blood flow in the circulatory system. We comment on their mathematical properties, their meaningfulness and their limitation to yield realistic and accurate numerical simulations, and their contribution for a better understanding of cardiovascular physiopathology. Mathematics Subject Classification (2000). 92C50,96C10,76Z05,74F10,65N30,65M60. Keywords. Cardiovascular mathematics; mathematical modeling; fluid dynamics; Navier– Stokes equations; numerical approximation; finite element method; differential equations. 1.
Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow
 SIAM J Applied Mathematics. Accepted
, 2006
"... Abstract. Fluidstructure interaction describing wave propagation in arteries driven by the pulsatile blood flow is a complex problem. Whenever possible, simplified models are called for. Onedimensional models are typically used in arterial sections that can be approximated by the cylindrical geome ..."
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Abstract. Fluidstructure interaction describing wave propagation in arteries driven by the pulsatile blood flow is a complex problem. Whenever possible, simplified models are called for. Onedimensional models are typically used in arterial sections that can be approximated by the cylindrical geometry allowing axially symmetric flows. Although a good first approximation to the underlying problem, the onedimensional model suffers from several drawbacks: the model is not closed (an ad hoc velocity profile needs to be prescribed to obtain a closed system) and the model equations are quasilinear hyperbolic (oversimplifying the viscous fluid dissipation), typically producing shock wave solutions not observed in healthy humans. In this manuscript we derived a simple, closed reduced model that accounts for the viscous fluid dissipation to the leading order. The resulting fluidstructure interaction system is of hyperbolicparabolic type. Arterial walls were modeled by a novel, linearly viscoelastic cylindrical Koiter shell model and the flow of blood by the incompressible, viscous Navier–Stokes equations. Kelvin–Voigttype viscoelasticity was used to capture the hysteresis behavior observed in the measurements of the arterial stressstrain response. Using the a priori estimates obtained from an energy inequality, together with the asymptotic analysis and ideas from homogenization theory for porous media flows, we derived an effective model which
SelfConsistent Effective Equations Modeling Blood Flow in MediumtoLarge Compliant Arteries
 SIAM J. Multiscale Analysis and Simulation
, 2005
"... Abstract. We study the flow of an incompressible viscous fluid through a long tube with compliant walls. The flow is governed by a given time dependent pressure head difference. The NavierStokes equations for an incompressible viscous fluid are used to model the flow, and the Navier equations for a ..."
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Abstract. We study the flow of an incompressible viscous fluid through a long tube with compliant walls. The flow is governed by a given time dependent pressure head difference. The NavierStokes equations for an incompressible viscous fluid are used to model the flow, and the Navier equations for a curved, linearly elastic membrane to model the wall. Employing the asymptotic techniques typically used in thin domains, we derive a set of effective equations that hold in mediumtolarge compliant vessels for laminar flow regimes. The main novelty is the derivation of the effective equations that do not assume any ad hoc closure, typically assumed in the derivation of onedimensional models. Using ideas from homogenization theory for porous media flows, we obtain a closed system of effective equations that are of Biot type with memory. Memory accounts for the wavelike phenomena in the problem. Although the equations are twodimensional, their simple structure enables a design of a numerical algorithm that has complexity of a onedimensional solver. Our numerical simulations show that our model captures twodimensional effects that cannot be captured using standard onedimensional methods. Key words. Blood flow, compliant arteries, fluidstructure interaction, effective equations. AMS subject classifications. 35Q30, 74K15, 76D27 1. Introduction. In
FLUIDSTRUCTURE INTERACTION IN A PRESTRESSED TUBE WITH THICK ELASTIC WALLS I: THE STATIONARY
"... Abstract. This is a study of the fluidstructure interaction between the stationary Stokes flow of an incompressible, Newtonian viscous fluid filling a threedimensional, linearly elastic, prestressed hollow tube. The main motivation comes from the study of blood flow in human arteries. Most literat ..."
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Abstract. This is a study of the fluidstructure interaction between the stationary Stokes flow of an incompressible, Newtonian viscous fluid filling a threedimensional, linearly elastic, prestressed hollow tube. The main motivation comes from the study of blood flow in human arteries. Most literature on fluidstructure interaction in blood flow utilizes thin structure models (shell or membrane) to describe the behavior of arterial walls. However, arterial walls are thick, threedimensional structures with the wall thickness comparable to the vessel inner radius. In addition, arteries in vivo exhibit residual stress: when cut along the radius, arteries spring open releasing the residual strain. This work focuses on the implications of the two phenomena on the solution of the fluidstructure interaction problem, in the parameter regime corresponding to the blood flow in mediumtolarge human arteries. In particular, it is assumed that the aspect ratio of the cylindrical structure ǫ = R/L is small. Using asymptotic analysis and ideas from homogenization theory for porous media flows, an effective, closed model is obtained in the limit as both the thickness of the vessel wall and the radius of the cylinder approach zero, simultaneously. The effective model satisfies the original threedimensional, axially symmetric problem to the ǫ 2accuracy. Several novel properties of the solution are obtained using this approach. A modification of the wellknown “Law of Laplace ” is derived, holding for thick elastic cylinders. A calculation of the effective longitudinal displacement is obtained, showing that the leadingorder longitudinal displacement is completely determined by the external loading. Finally, it is shown that the residual stress influences the solution only at the ǫorder. More precisely, it is shown that the only place where the residual stress influences the solution of this fluidstructure interaction problem is in the calculation of the ǫcorrection of the longitudinal displacement. 1. Introduction. The
Axisymmetric flow of a generalized Newtonian fluid in a straight pipe using a director theory approach
 Proceedings of the 8th WSEAS International Conference on Applied Mathematics, 2005
"... Abstract: The aim of this paper is to analyze the axisymmetric unsteady flow of an incompressible generalized Newtonian fluid in a straight rigid and impermeable tube with circular crosssection of constant radius. To study this problem, we use an approach based on the Cosserat theory (also called ..."
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Abstract: The aim of this paper is to analyze the axisymmetric unsteady flow of an incompressible generalized Newtonian fluid in a straight rigid and impermeable tube with circular crosssection of constant radius. To study this problem, we use an approach based on the Cosserat theory (also called director theory) related to fluid dynamics which reduces the exact threedimensional equations to a system depending only on time and on a single spatial variable. From this system we obtain for a flow without swirling motion the relationship between mean pressure gradient and volume flow rate over a finite section of the pipe for the specific case of the power law viscosity function. Moreover, we compare the 3D exact solution for steady volume flow rate with the corresponding solution obtained by the Cosserat theory using nine directors. Key–Words: Cosserat theory, nine directors, steady solution, axisymmetric motion, volume flow rate, power law viscosity function. 1
Homogenization Closure For A TwoDimensional Effective Model Describing FluidStructure Interaction in Blood Flow
"... We study the flow of a viscous incompressible fluid through a long and narrow elastic tube whose walls are modeled by the Navier equations for a curved, linearly elastic membrane. radial displacement ..."
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We study the flow of a viscous incompressible fluid through a long and narrow elastic tube whose walls are modeled by the Navier equations for a curved, linearly elastic membrane. radial displacement
Average Pressure Gradient of Swirling Flow Motion of a Viscoelastic Fluid in a Circular Straight Tube with Constant Radius
, 2009
"... Motived by the aim of modelling the behavior of swirling flow motion, we present a 1D hierarchical model for an RivlinEricksen fluid with complexity n = 2 flowing in a circular straight tube with constant radius. Integrating the equation of conservation of linear momentum over the tube crosssectio ..."
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Motived by the aim of modelling the behavior of swirling flow motion, we present a 1D hierarchical model for an RivlinEricksen fluid with complexity n = 2 flowing in a circular straight tube with constant radius. Integrating the equation of conservation of linear momentum over the tube crosssection, with the velocity field approximated by the Cosserat theory, we obtain a onedimensional system depending only on time and on a single spatial variable. The velocity field approximation satisfies both the incompressibility condition and the kinematic boundary condition exactly. From this new system, we derive the equation for the wall shear stress and the relationship between average pressure gradient, volume flow rate and swirling scalar function over a finite section of the tube. Also, we obtain the corresponding partial differential equation for the swirling scalar function.
Unsteady flow of OldroydB fluids in an uniform rectilinear pipe using
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