The present numerical attempt deals with the analysis of natural convection flow in a trapezoidal cavity with triangular block embedded inside intended for various inclination angles ф. In this cavity, the bottom wall is non-uniformly heated, left and right (side) walls are cold and the top wall is well insulated. The triangular block or triangular solid body surface for different positions (LBC, RBC, LTC, RTC) inside the trapezoidal cavity is also non-uniformly heated (Thb). A Galerkin weighted residual finite element method is applied to solve the Navier-Stokes equations and energy balance equations and to find the results in the form of streamlines, isotherms (temperature), heat function, pressure and local Nusselt numbers and average Nusselt number for different parameters. Finite element method has also been used to solve the velocity and thermal fields. The fluid is concerned for various governing parameters, such as Rayleigh number, Ra (103≤Ra≤106), Prandtl number, Pr (0.026≤Pr≤1000) and various inclination angles ф (ф = 45°, 30° and 0°). It has been revealed that due to non-uniformly heated triangular block, heatlines by connecting cold and hot walls are found to be continuous lines and also perpendicular to the isothermal walls which occur for the conduction dominant regimes. Fluid flow and heat flow occur symmetrically. The local Nusselt number and average Nusselt number for the non-uniformly heated bottom wall of the trapezoidal cavity using the inside of the non-uniformly heated triangular block are illustrated by heatlines. It has been also found that heat transfer rates significantly depend on the tilt angles (ф), non-uniformly heated triangular block and aforementioned non-dimensional parameters.
A family of Trigonometrically Fitted Backward Differentiation Formula (TBDF) whose coefficients depend on the frequency and step size for periodic initial value problems is presented. The method is constructed based on collocation techniques. The primary method of TBDF is obtained from its continuous version while the additional methods are derived from the derivative of the continuous version which are combined and applied in block form as simultaneous numerical integrators. The stability properties of the method are discussed and numerical experiments show that the TBDF is an accurate numerical integrator.
In this paper, the Euler number e, the Euler sequence and its application in real life situations particularly in business world ( financial investment ) are discussed. A basic theorem concerning convergence of the Euler sequence in which some lemmas were considered in order to achieve the proof of the theorem is presented. The paper showed the application of the Euler sequence in continuous compounding and the development of a new formula (INTEREST FORMULA) in continuous compounding. To achieve this, the convergence of the Euler sequence to the Euler number e was established and finally, its application in continuous compounding and some striking examples are shown.
An efficient and accurate iterative scheme for the computation of the mean first passage times ( MFPTs) of ergodic Markov chains has been presented. Firstly, the computation problem of MFPTs is transformed into a set of linear equations. It has been proven that each of these equations is compatible and their minimal norm solutions constitute MFPTs. A new presentation of the MFPTs is also derived. Using linear least square algorithms, some numerical examples compared with the finite algorithm of Hunter  and iterative algorithm of J. Xu  are given. These results show that the new algorithm is suitable for large sparse systems.
Objectives: Nowadays invasive coronary angiogram is a gold standard clinical application to determine fractional flow reserve (FFR) over the world. The engineering tools computational fluid dynamics (CFD) enabled the simulation of coronary blood flow conditions in the idealized and patient-based coronary models. The CTA-based CFD simulation allowed us to produce a noninvasive assessment of the hemodynamic parameters, such as blood velocity magnitude, relative pressure difference and virtual fractional flow reserve (vFFR). These hemodynamic mechanisms provide the information on coronary stenosis conditions and predict the severity of coronary arterial lumen area which is responsible for the heart attack of the human patients.
Methodology: In the current study investigates the hemodynamic parameters by using idealized coronary artery prototypical model and patient-specific real coronary artery models. The patient-specific model reconstructed by following steps: CTA image acquisition, image segmentation, 3D reconstruction, smooth surface and computational mesh generation. The hemodynamics simulation derived by solving the Navier-Stokes equations for steady and pulsatile flow motions.
Results: The results reflect expected outcomes in both cases, for instance, a higher blood velocity in the coronary vessels tends to stretch the contrast agent gradient and a lower blood velocity magnitude tends to steepen the contrast agent gradient. The pressure difference and vFFR results allow to distinguish the unstenosed and stenosis arterial models.
Conclusion: The aim of the work was to set up a framework for an idealized and a patientspecific models reconstruction, geometric analysis and describing hemodynamic parameters. The parameters allowed to describe the intrinsic blood flow in a non-invasive procedure. The simulation results are feasible for clinical applications but there are some limitations in the approaches which need more studies to overcome them.