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BackgroundIn literature, the effect of the inflow boundary condition was investigated by examining the impact of the waveform and the shape of the spatial profile of the inlet velocity on the cardiac hemodynamics. However, not much work has been reported on comparing the effect of the different combinations of the inlet/outlet boundary conditions on the quantification of the pressure field and flow distribution patterns in stenotic right coronary arteries.MethodNon-Newtonian models were used to simulate blood flow in a patient-specific stenotic right coronary artery and investigate the influence of different boundary conditions on the phasic variation and the spatial distribution patterns of blood flow. The 3D geometry of a diseased artery segment was reconstructed from a series of IVUS slices. Five different combinations of the inlet and the outlet boundary conditions were tested and compared.ResultsThe temporal distribution patterns and the magnitudes of the velocity, the wall shear stress (WSS), the pressure, the pressure drop (PD), and the spatial gradient of wall pressure (WPG) were different when boundary conditions were imposed using different pressure/velocity combinations at inlet/outlet. The maximum velocity magnitude in a cardiac cycle at the center of the inlet from models with imposed inlet pressure conditions was about 29% lower than that from models using fully developed inlet velocity data. Due to the fact that models with imposed pressure conditions led to blunt velocity profile, the maximum wall shear stress at inlet in a cardiac cycle from models with imposed inlet pressure conditions was about 29% higher than that from models with imposed inlet velocity boundary conditions. When the inlet boundary was imposed by a velocity waveform, the models with different outlet boundary conditions resulted in different temporal distribution patterns and magnitudes of the phasic variation of pressure. On the other hand, the type of different boundary conditions imposed at the inlet and the outlet did not have significant effect on the spatial distribution patterns of the PD, the WPG and the WSS on the lumen surface, regarding the locations of the maximum and the minimum of each quantity.ConclusionsThe observations from this study indicated that the ways how pressure and velocity boundary conditions are imposed in computational models have considerable impact on flow velocity and shear stress predictions. Accuracy of in vivo measurements of blood pressure and velocity is of great importance for reliable model predictions.

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This paper deals with the rigorous error analysis of finite element solutions of the two-and three-dimensional nonstationary compressible Navier-Stokes equations. The streamline diffusion technique is applied to the continuity equation to obtain a stabilized finite element scheme. It is proved that the numerical formulation has a unique solution and the solution is convergent. An a priori error estimate uniform in time is obtained. Introduction.In this paper, we analyze a fully discrete finite element method with streamline diffusion for the nonstationary compressible Navier-Stokes equations in two dimensions and three dimensions. This study is motivated by the great demand in developing stabilized numerical methods for compressible flows and analyzing numerical solutions.Various numerical methods have been developed for the computation of compressible viscous flows. Bristeau et al. constructed a finite difference-finite element algorithm with GMRES iterative methods for solving the nonlinear systems [4]. Shu et al. applied high-order essentially nonoscillatory finite difference scheme to the twodimensional (2D) and three-dimensional (3D) Euler and Navier-Stokes equations [23]. Fortin, Manouzi and Soulaimani developed the numerical methods using an upwinding methodology [9]. Bank et al. discussed how to enhance the numerical simulation of compressible Navier-Stokes equations via a posteriori error estimate analysis and mesh adaptive techniques [1]. However, there is a considerable lack in the complete analysis of the convergence, stability, and a priori error estimate for the numerical solutions of compressible Navier-Stokes equations. For the linearized compressible Stokes equations, several finite element formulations have been developed and analyzed for the steady problems on either a strip or a bounded domain via different techniques, such as penalization and regularization [15,16,17,20]. This paper is devoted to the rigorous error analysis of a stabilized finite element formulation for the 2D and 3D nonstationary compressible Navier-Stokes equations. In the computation of compressible flows, numerical results are often spoiled by nonphysical oscillations of pressure and density [9], especially for moderately large Mach numbers. This is because of the fact that the density transport equation is hyperbolic, which does not contain the natural dissipative term. Therefore, special attention must be paid when dealing with the continuity equation. In our finite element formulation, we apply the streamline diffusion technique to the continuity equation at each time level to stabilize the numerical scheme. The key of the streamline diffusion method,

A finite element formulation is developed for the two dimensional nonlinear time dependent compressible Navier-Stokes equations on a bounded domain. The existence and uniqueness of the solution to the numerical formulation is proved. An error estimate for the numerical solution is obtained.