Numerical Simulation of Unsteady Viscous Flows Using an Implicit Projection Method

Pentaris, A.; Tsangaris, S.

doi:10.1002/(sici)1097-0363(19961115)23:9<897::aid-fld461>3.0.co;2-#

Cited by 5 publications

(6 citation statements)

References 27 publications

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“…The solution is independent of the adopted initial conditions. The credibility of the fluid flow methodology has already been tested and validated in various studies [19,24].…”

Section: Fluidmentioning

confidence: 99%

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Fluid flow of incompressible viscous fluid through a non-linear elastic tube

Lazopoulos

Tsangaris

2008

Arch Appl Mech

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The study of viscous flow in tubes with deformable walls is of specific interest in industry and biomedical technology and in understanding various phenomena in medicine and biology (atherosclerosis, artery replacement by a graft, etc) as well. The present work describes numerically the behavior of a viscous incompressible fluid through a tube with a non-linear elastic membrane insertion. The membrane insertion in the solid tube is composed by non-linear elastic material, following Fung's (Biomechanics: mechanical properties of living tissue, 2nd edn. Springer, New York, 1993) type strain-energy density function. The fluid is described through a Navier-Stokes code coupled with a system of non linear equations, governing the interaction with the membrane deformation. The objective of this work is the study of the deformation of a non-linear elastic membrane insertion interacting with the fluid flow. The case of the linear elastic material of the membrane is also considered. These two cases are compared and the results are evaluated. The advantages of considering membrane nonlinear elastic material are well established. Finally, the case of an axisymmetric elastic tube with variable stiffness along the tube and membrane sections is studied, trying to substitute the solid tube with a membrane of high stiffness, exhibiting more realistic response.Keywords Fluid flow · Elastic membrane · Nonlinear elastic material · Linear elastic material · Fluid flow membrane interaction A. Lazopoulos · S. Tsangaris (B) Fluids Section,

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“…The solution is independent of the adopted initial conditions. The credibility of the fluid flow methodology has already been tested and validated in various studies [19,24].…”

Section: Fluidmentioning

confidence: 99%

“…The Navier-Stokes and the continuity equations are used for the fluid study. The numerical algorithm is based on a projection method [19]. Likewise, the membrane is described by equilibrium equations for the axisymmetric case.…”

Section: Introductionmentioning

confidence: 99%

Fluid flow of incompressible viscous fluid through a non-linear elastic tube

Lazopoulos

Tsangaris

2008

Arch Appl Mech

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“…The temporal derivative in Eq. (3) is approximated via a generalized time differencing (Pentaris and Tsangaris, 1996):…”

Section: Time-marching Schemementioning

confidence: 99%

“…where A n =∂F n /∂Q n and B n =∂G n /∂Q n are the Jacobian matrices of the vectors F n and G n respectively (Pentaris and Tsangaris, 1996). Then the viscous fluxes V n , W n and C n are first split into two parts, one of which is a function of Q and Q ξ , and the other a function of Q and Q η : W are linearized implicitly using a Taylor series expansion: The substitution of the linear expressions of the flux vectors into the original non-linear equation for ΔQ n leads to a strongly coupled system of equations in both spatial directions.…”

Section: Time-marching Schemementioning

confidence: 99%

“…The mathematical formulation of the problem is presented briefly in section 2. As far as the fluid is concerned, the projection method is used and a Poisson's equation is adopted for the pressure gradients, while the numerical algorithm contains only momentum equations (Pentaris and Tsangaris, 1996). The membrane, accordingly, is described by equilibrium equations for the axisymmetric case.…”

Section: Introductionmentioning

confidence: 99%

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Flow of Incompressible Viscous Fluid Through a Tube with Non-Linear Elastic Membrane Insertion: Applications

Lazopoulos

Tsangaris

2008

Engineering Applications of Computational Fluid Mechanics

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ABSTRACT:The study of viscous flow in tubes with deformable walls is of specific interest to the biomedical technology industry as well as the medicine and biology professions in connection with atherosclerosis, artery replacement by a graft, clotting, etc. In line with this philosophy, this paper describes numerically the behavior of a viscous incompressible fluid through a tube with a non-linear elastic membrane insertion for various geometries of the tube and initial conditions. The emphasis is given to the applications. Six different test cases, with regard to the problem of atherosclerosis, are examined. The membrane insertion in the solid tube is studied for a linear as well as a non-linear material. The linear elastic material follows a linear elastic strain-energy density function. However, the deformation is large. The non-linear elastic material follows the type strain-energy density function of Fung (1993). The fluid is described through a Navier-Stokes code coupled with a system of non-linear equations that describe membrane deformation. The objective of this paper is to outline the similarities and differences between linear and non-linear membrane insertions in influencing fluid-membrane interaction in all six test cases. It is revealed that although the maximum deformed radius of the axisymmetric membrane is almost the same for all cases, the deformed shape of the membrane is quite different resulting in different regions of the circulation regions. In fact, the non-linear membrane exhibits a very restricted region of the circulation of the flow, which is important for understanding the cause of thrombosis.

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