In the paper, the authors establish two identities, which can be regarded as nonlinear differential equations, for the generating function of Eulerian polynomials, find two identities for the Stirling numbers of the second kind, and present two identities for Eulerian polynomials and higher order Eulerian polynomials, pose two open problems about summability of two finite sums involving the Stirling numbers of the second kind. Some of these conclusions meaningfully and significantly simplify several known results.

In this paper, a new class of q-Hermite based Frobenius type Eulerian polynomials is introduced by means of generating function and series representation. Several fundamental formulas and recurrence relations for these polynomials are derived via different generating methods. Furthermore, diverse correlations including the q-Apostol-Bernoulli polynomials, the q-Apostol-Euler poynoomials, the q-Apostol-Genocchi polynomials and the q-Stirling numbers of the second kind are also established by means of the their generating functions.

We introduce a new kind of extended Hermite-based Frobenius type Eulerian polynomials and then derive diverse explicit and implicit summation equations including some symmetric formulas by utilizing series manupulation method. Multifarious summation formulas and identities given earlier for some well known polynomials such as Eulerian polynomials and Frobenius type Eulerian polynomials are generalized.

In this paper a new monolithic Eulerian formulation in the framework of non-classical continuum is presented for the analysis of fluid-strucutre interaction problems. In this respect, Cosserat continuum theory taking into account the micro-rotational degrees of freedom of the particles is considered. Continuum description of the model and variational formulation of the governing flow dynamics for non-classical -fluid-structure interaction nCFSI is presented. The model is analyzed by computing a well known benchmark problem by Hecht and Pironneau [16]. The algorithmic description is presented and implemented using FreeFEM++. Code is validated with the benchmark solution of Turek and Hron [38] in case of flow around a flag attached with cylinder. New microstructral behavior of the solution is studied and numerical simulations and results are shown in the form of figures. Some interesting feature of the flow is observed and microstructural characteristics are discussed.

Purpose: To improve numerical simulation of the non-contact tonometry test by using Arbitrary Eulerian-Lagrangian deforming mesh in the coupling between computational fluid dynamics model of an air jet and finite element model of the human eye. Methods: Computational fluid dynamics model simulated impingement of the air puff and consisted of 25920 wedge6 elements and employed Spallart-Allmaras model to simulate capture turbulence of the air jet. The time span of the jet wais 30 ms and maximum Reynolds number

A critical challenge underpinning the adoption of Additive Manufacture (AM) as a technology is the postprocessing of manufactured components. For Selective Laser Sintering (SLS) this can involve the removal of powder from the interior of the component, often by vibrating the component to fluidise the powder to encourage drainage. In this paper we develop and validate a computational model of the flow of metal powder suitable for predicting powder removal from such AM components. The model is a continuum Eulerian multiphase model of the powder including models for the granular temperature; the effect of vibration can be included through appropriate wall boundaries for this granular temperature. We validate the individual sub-models appropriate for AM metal powders by comparison with in-house and literature experimental results, and then apply the full model to a more complex geometry typical of an AM Heat Exchanger. The model is shown to provide valuable and accurate results at a fraction of the computational cost of a particle-based model.

The computational model that is able to estimate the temperature distribution inside a solid specimen during the film boiling phase of immersion quenching (water entry) process has been presented in paper, and it is based on the prescribed initial temperatures of the solid specimen and the liquid quenchant. In addition, the turbulence effects have to be considered using the assumed turbulence kinetic energy value, i.e., the “frozen turbulence” approach, that remains constant thorough the simulation. The studied material is nickel alloy, Inconel 600, for which the extensive experimental data is available. The work has been carried out using ANSYS Fluent computational fluid dynamics software and the methods for solution of Stefan problem by Eulerian two fluid VOF model.. A satisfactory agreement between the experimental and the calculated data has been achieved, yielding thereby the computationally obtained data that fits to a great extent to the prescribed error band of ± 10 % during the estimated film boiling phase of the immersion quenching process itself.

Formal Calculation uses an auxiliary form to calculate various nested sums and provides results in three forms. In addition to computation, it is also a powerful tool for analysis, allowing one to study various numbers in a unified way. This article contains many results of two types of Stirling numbers, associated Stirling numbers, and Eulerian numbers, making a great generalization of Euler polynomials, Wilson's theorem, and Wolstenholme's theorem, showing that they are just special cases. Formal Calculation provides a novel method for obtaining combinatorial identities and analyzing q-binomial.This article has obtained a large number of results in q-analogues, including inversion formulas for q-binomial coefficients. This article also introduces a theorem on symmetry.

In recent years, there has been a growing interest in the preoperative modeling of fluid-structure interaction in the treatment of cerebral aneurysms. In this study, we investigate two cases involving laminar incompressible fluid flow interacting with hyperelastic materials (a vertical flap and aneurysm walls) under the effect of fluid flow. We present a finite element method (FEM) for these prototypical two-dimensional ($2D$) configurations, taking into account the complex flows and deformation of the variant's structure models. We use an Arbitrary Lagrangian-Eulerian (ALE) formulation in a continuum, fully monolithic coupled way, and discretize the fluid and solid domains using the quadratic LBB stable $P_2P_1$ finite element pair to approximate the displacement, velocity, and pressure spaces independently. The resulting discretized form of the nonlinear algebraic system is linearized using a variant of Newton's procedure, and the Jacobian matrices are approximated via the divided difference method. The resulting linear systems are solved using a direct solver MUMPS. We evaluate hydrodynamic forces such as drag and lift coefficients for nonlinear elastic material models, including Saint-Venant Kirchoff, Neo-Hookean, and Mooney-Rivlin (2 parameters) separately. To gain more physical insight into the problem, we verify the computed results by comparing the velocity, viscosity, and pressure fields. Our aim is to qualitatively analyze the changes in the mechanics of the wall behavior of the elasticity of the flap vs. the complexity of deformation, interface, details for prototypical flow situations, corresponding displacement, and deformation. We also provide a physical interpretation of blood flow magnitude and basic hemodynamic phenomena of wall shear stresses (WSS) in aneurysma and the range of deformation. Overall, this study presents a comprehensive approach to modeling fluid-structure interaction in cerebral aneurysms, which could help in the development of more effective treatment strategies.

Mathematical modeling is the most important tool for constructing the theory of different kinds of two-phase flows. The review is devoted to the analysis of the advent of the approach of mathematical modeling of two-phase flows, where solid particles act mainly as the dispersed phase. The main problems and features of the study of gas-solid two-phase flows are excluded. The main characteristics of gas flows with solid particles are given and the classification of two-phase flows is developed on their basis. The Lagrangian and Euler approaches to modeling the motion of a dispersed phase (particles) are described. Much attention is paid to the consideration of numerical simulation methods that describe descriptions of turbulent gas flow at different hierarchical levels (RANS, LES, DNS), different levels of description of interphase interactions (one-way coupling (OWC), two-way coupling (TWC) and four-way coupling (FWC)), as well as at different levels of interface resolution (partial-point (PP) and particle-resolved (PR)). Excluded are examples of some studies carried out on the basis of the identified approaches, as well as their exclusion for mathematical modeling of various classes of gas-solid two-phase flows.

The subject of this article is the dynamics of water in a soil pedostructure sample whose internal environment is subjected to a potential gradient created by the departure of water through surface evaporation. This work refers entirely to the results and conclusions of a fundamental theoretical study focused on the molecular thermodynamic equilibrium of the two aqueous phases of the soil pedostructure. The new concepts and descriptive variables of the hydro-thermodynamic equilibrium state of the soil medium, which have been established at the molecular level of the fluid phases of the pedostructure (water and air) in a previous article, are recalled here in the systemic paradigm of hydrostructural pedology. They allow access to the molecular description of water migration in the soil and go beyond the classical mono-scale description of soil water dynamics. We obtain a hydro-thermodynamic description of the soil's pedostructure at different hydro-functional scale levels including those relating to the water molecule and its atoms. The experimental results show a perfect agreement with the theory, validating at the same time the systemic approach which was the framework.