scholarly journals Mathematical models for blood flow in elastic vessels: Theory and numerical analysis

2021 ◽  
Author(s):  
◽  
Qian Li
Keyword(s):  
Blood Flow ◽  
Fluid Flow ◽  
Finite Difference Methods ◽  
General System ◽  
Boussinesq System ◽  
Alternative Derivation ◽  
Flow Problems ◽  
Boussinesq Systems ◽  
Difference Methods ◽  
Model Equations

<p>In this thesis we study model equations that describe the propagation of pulsatile flow in elastic vessels. Since dealing with the Navier-Stokes equations is a very difficult task, we derive new asymptotic weakly non-linear and weakly-dispersive Boussinesq systems. Properties of the these systems, such as the well-posedness, and existence of travelling waves are being explored. Finally, we discretize some of the new model equations using finite difference methods and we demonstrate their applicability to blood flow problems. First we introduce the basic equations that describe f luid flow in elastic vessels and previously derived systems. We also review previously derived model equations for fluid flow in elastic tubes. We start with the description of the equations of motion of elastic vessel. Then wederive asymptotically Boussinesq systems for fluid flow in elastic vessels. Because these systems are weakly non-linear and weakly dispersive we expect then to have solitary waves as special solutions. We explore some possibilities by construction analytical solutions. After that we continue the derivation of the previous chapter. We derive a general system where the horizontal velocity is evaluated at any distance from the center of the tube. Special emphasis is paid on the case of constant radius vessels. We also derive unidirectional models and obtain the dissipative Boussinesq system by taking the viscosity effects into account. There is also an alternative derivation of the general system when considering the equations of potential flow. We show that the two different derivations lead to the same system. The alternative derivation is based on asymptotic series expansions. Then we develop finite difference methods for the numerical solution of the BBM equation and for the classical Boussinesq system studied in the previous chapters. Finally, we demonstrate the application of the new models to blood flow problems. By performing several numerical simulations.</p>

scholarly journals Mathematical models for blood flow in elastic vessels: Theory and numerical analysis

2021 ◽  
Author(s):  
◽  
Qian Li
Keyword(s):  
Blood Flow ◽  
Fluid Flow ◽  
Finite Difference Methods ◽  
General System ◽  
Boussinesq System ◽  
Alternative Derivation ◽  
Flow Problems ◽  
Boussinesq Systems ◽  
Difference Methods ◽  
Model Equations

<p>In this thesis we study model equations that describe the propagation of pulsatile flow in elastic vessels. Since dealing with the Navier-Stokes equations is a very difficult task, we derive new asymptotic weakly non-linear and weakly-dispersive Boussinesq systems. Properties of the these systems, such as the well-posedness, and existence of travelling waves are being explored. Finally, we discretize some of the new model equations using finite difference methods and we demonstrate their applicability to blood flow problems. First we introduce the basic equations that describe f luid flow in elastic vessels and previously derived systems. We also review previously derived model equations for fluid flow in elastic tubes. We start with the description of the equations of motion of elastic vessel. Then wederive asymptotically Boussinesq systems for fluid flow in elastic vessels. Because these systems are weakly non-linear and weakly dispersive we expect then to have solitary waves as special solutions. We explore some possibilities by construction analytical solutions. After that we continue the derivation of the previous chapter. We derive a general system where the horizontal velocity is evaluated at any distance from the center of the tube. Special emphasis is paid on the case of constant radius vessels. We also derive unidirectional models and obtain the dissipative Boussinesq system by taking the viscosity effects into account. There is also an alternative derivation of the general system when considering the equations of potential flow. We show that the two different derivations lead to the same system. The alternative derivation is based on asymptotic series expansions. Then we develop finite difference methods for the numerical solution of the BBM equation and for the classical Boussinesq system studied in the previous chapters. Finally, we demonstrate the application of the new models to blood flow problems. By performing several numerical simulations.</p>

Extension of lattice Boltzmann flux solver for simulation of compressible multi-component flows

2018 ◽  
Vol 32 (12n13) ◽  
pp. 1840001 ◽  
Author(s):  
Li-Ming Yang ◽  
Chang Shu ◽  
Wen-Ming Yang ◽  
Yan Wang
Keyword(s):  
Fluid Flow ◽  
Euler Equations ◽  
Lattice Boltzmann ◽  
Passive Scalar ◽  
Two Phase ◽  
Material Interfaces ◽  
Flow Problems ◽  
Numerical Fluxes ◽  
Model Equations ◽  
Interface Equations

The lattice Boltzmann flux solver (LBFS), which was presented by Shu and his coworkers for solving compressible fluid flow problems, is extended to simulate compressible multi-component flows in this work. To solve the two-phase gas–liquid problems, the model equations with stiffened gas equation of state are adopted. In this model, two additional non-conservative equations are introduced to represent the material interfaces, apart from the classical Euler equations. We first convert the interface equations into the full conservative form by applying the mass equation. After that, we calculate the numerical fluxes of the classical Euler equations by the existing LBFS and the numerical fluxes of the interface equations by the passive scalar approach. Once all the numerical fluxes at the cell interface are obtained, the conservative variables at cell centers can be updated by marching the equations in time and the material interfaces can be identified via the distributions of the additional variables. The numerical accuracy and stability of present scheme are validated by its application to several compressible multi-component fluid flow problems.

Equilibrium characteristics of nearly normal turbulence

1970 ◽  
Vol 41 (1) ◽  
pp. 179-188 ◽  
Author(s):  
W. C. Meecham
Keyword(s):  
Fluid Flow ◽  
Incompressible Fluid ◽  
Linear Model ◽  
Gaussian Processes ◽  
Incompressible Fluid Flow ◽  
Hermite Expansion ◽  
Burgers Model ◽  
Flow Problems ◽  
Non Linear ◽  
Model Equations

We discuss some consequences of assuming that two different non-linear model equations, and real turbulence are nearly Gaussian. It is supposed when necessary that the process is driven and it is supposed that the processes have become statistically stationary. These problems are discussed from the viewpoint of the Wiener–Hermite expansion for non-linear, nearly Gaussian processes. Expected equilibria forms are related to corresponding expressions obtained from the zero-fourth-cumulant assumption. The spectrum for Burgers’ model and for incompressible fluid flow problems is found from this viewpoint to beE∼k−2. The kinematical properties leading to such spectra are discussed. It is noted, as has been remarked earlier, that this spectrum is characteristic of flows with near discontinuities. A conjecture is offered concerning how these discontinuities are related to Gaussianity.

DISSIPATIVE DISCRETIZATION METHODS FOR APPROXIMATIONS TO THE BOLTZMANN EQUATION

2001 ◽  
Vol 11 (01) ◽  
pp. 133-148 ◽  
Author(s):  
CHRISTIAN RINGHOFER
Keyword(s):  
Boltzmann Equation ◽  
Discrete System ◽  
Finite Difference Methods ◽  
Collision Operator ◽  
Entropy Condition ◽  
Discretization Methods ◽  
The Boltzmann Equation ◽  
Galerkin Approximations ◽  
Difference Methods ◽  
Model Equations

This paper deals with the spatial discretization of partial differential equations arising from Galerkin approximations to the Boltzmann equation, which preserves the entropy properties of the original collision operator. A general condition on finite difference methods is derived, which guarantees that the discrete system satisfies the appropriate equivalent of the entropy condition. As an application of this concept, entropy producing difference methods for the hydrodynamic model equations and for spherical harmonics expansions are presented.

Simplified Finite-Element Models for Reservoir Flow Problems

1979 ◽  
Vol 19 (05) ◽  
pp. 333-343 ◽  
Author(s):  
Vilgeir Dalen
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Finite Difference ◽  
Variational Methods ◽  
Finite Element Methods ◽  
Hyperbolic Equations ◽  
Finite Difference Methods ◽  
Flow Problems ◽  
Reservoir Flow ◽  
Difference Methods

Abstract This paper summarizes some research that was conducted to construct finite-element models for reservoir flow problems. The models are based on Galerkin's method, but the method is applied in an unorthodox manner to simplify calculation of coefficients and to improve stability. Specifically, techniques of compatibility relaxation, capacity lumping, and upstream mobility weighting are used, and schemes are obtained that seem to combine the simplicity and high stability of conventional finite-difference models with the generality and modeling flexibility of finite-element methods.The development of a model for single-phase gas flow and a two-phase oil/water model is described. Numerical examples are included to illustrate the usefulness of finite elements. In particular, the triangular element with linear interpolation is shown to be an attractive alternative to the standard five-point, finite-difference approximation for two-dimensional analysis. Introduction During the past decades, finite-element methods have been developed to a high level of sophistication and have gained wide popularity within several branches of engineering science. In some fields, such methods have replaced to some extent the older finite-difference methods in engineering practice because they have been regarded as a more convenient tool for numerical analysis. An increasing interest in finite-element methods, or variational methods in general, also may be noticed in the field of numerical reservoir simulation, but so far no definitive breakthrough has occurred in this field.One reason for this probably is the complexity of reservoir flow problems. Reservoir flow equations in most cases are nonlinear, and for multiphase flow, they are usually found on the borderline between parabolic and hyperbolic equations. For such parabolic and hyperbolic equations. For such problems, the dissimilarities between problems, the dissimilarities between finite-difference and finite-element methods are much more pronounced than for linear problems of the elliptic type. This means that the finite-element method may not be looked upon as easily as an extension or generalization of finite-difference methods. Second, one can question whether all the advantages that are gained in other instances by using finite elements may be realized at all.Applications of variational methods to single-phase flow problems, or diffusion-type problems in general, have been studied extensively. The merits of finite elements for such problems are apparently well established, at least as far as linear problems are concerned.The literature on variational methods in multiphase flow is comparatively sparse, and so far the results are inconclusive regarding the relative advantages of variational methods and finite- difference methods in this field. In summary, variational methods offer the potential advantages of (1) easy implementation of higher-order approximations, (2) a more proper treatment of variable coefficients, and (3) greater modeling flexibility.Previously, attention was focused on Aspects 1 and 2. Several authors used cubic Hermitian basis functions. SPEJ P. 333

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