Abstract — Navier-Stokes models are of great usefulness in physics and applied sciences. In this paper, He’s polynomials approach is implemented for obtaining approximate and exact solutionsoftheNavier-Stokesmodel. These solutions are calculated in the form of series with easily computable components. This technique is showed to be very effective, efficient and reliable because it gives the exact solution ofthe solved problems with less computational work, without neglecting the level of accuracy. We therefore, recommend the extension and application of this novel method for solving problems arising in other aspect of applied sciences. Numerical computations, and graphics done in this work, are through Maple 18.
Abstract—The fluid equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton’s second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term - hence describing viscous flow. Due to specific of NS equations they could be transformed to full/partial inhomoge- neous parabolic differential equations: differential equations in respect of space variables and the full differential equation in respect of time variable and time dependent inhomogeneous part. Finally, orthogonal polynomials as the partial solutionsof obtained Helmholtz equations were used for derivation ofanalytical solution of incompressible fluid equations in 1D, 2D and 3D space for rectangular boundary. Solution in 3D space for any shaped boundary is expressed in term of 3D global solution of 3D Helmholtz equation accordantly.
In order to control the risk induced bythe movements of stock prices, options can be used for hedging assets and portfolios. With regard to theory of option pricing and valuation, Black and Scholes in 1973  proposed a classical formula for the prices of financial options. This is popularly referred to as Black-Scholes equation, which has been the hallmark of financial derivatives. The Black- Scholes model is a linear PDE based on some assumptions
Before proceeding let us clearly deﬁne what is meant byanalytical, exact and approximate solutions. An analytical solution is obtained when the governing boundary value problem is integrated using the methods of classical diﬀerential equations. The result is an algebraic expression giving the dependent variable(s) as a function(s) ofthe independent variable(s). An exact solution is obtained by integrating the governing boundary value problem numerically. The result is a tabulation ofthe dependent variable(s) as a function(s) ofthe independent variables(s). An approximate solution results when methods such as series expan- sion and the von Karman-Pohlhausen technique are used to solve the governing boundary value problem (see Schlichting [Schl60], p. 239).
Figure 2. Density raster plot in our 2d run Dth32 at a time of 180 seconds after the start ofthe slide. The reflective region representing the unchanging basement of La Palma is at left in black, the basalt fluid slide material is red, water is orange, and air is blue. Intermediate shades represent the mixing of fluids, in particular the turbidity currents mixing water and basalt are readily apparent. The water wave leads the bullnose ofthe slide material by a small amount; the forward-rushing slide material (with a velocity of 190 meters/second almost matching the wave velocity) continues to pump energy into the wave. The wave height at this time is 1500 meters, and the wavelength is roughly 60 km. This figure has a width of 50 km, representing less than half ofthe computational domain, which extends 120 km to the right.
We briefly describe the strategy ofthe proof of Theorem 4.1.1. In view of (4.1), it is enough to show that each λ > (m − 1) 2 k/4 lies in σ (M). To this end, we follow an approach inspired by a general result due to K.D. Elworthy and F-Y. Wang (ELWORTHY; WANG, 2004). However, Elworthy-Wang’s theorem is not sufficient to conclude, and we need to considerably refine the criterion in order to fit in the present setting. To construct the sequence as in Lemma 1.0.1, a key step is to couple the volume growth requirement (4.6) with a sharpened form ofthe monotonicity formula for minimal submanifolds, which improves on the classical ones in (SIMON, 1983; ANDERSON, 1982). Indeed, in Proposition 4.3.1 we describe three monotone quantities other than Θ(s), and we expect these to be useful beyond the purpose ofthe present paper. For example, in the very recent (GIMENO; MARKVOSEN, 2015) the authors discovered and used some ofthe relations in Proposition 4.3.1 to show interesting comparison results for the capacity and the first eigenvalue of minimal submanifolds.
There are some parts ofthes / l interface of non-faceted phase lamellae where instability develops or vanishes and branching of faceted phase lamellae is observed. This phenomenon decides on the existence of a whole spectrum of interlamellar spacings, O . From the thermodynamic viewpoint some regions ofthe system are in stationary state while others in state of rotation around it. Stationary state changes continually its localization. Thus, some respective regions oscillate from rotation (marginal stability for which excess entropy production vanishes) to stationary state. This is the fundamental assumption in the current analysis.
Presented paper contains evaluation of influence of selected parameters on sensitivity of a numerical modelof solidification. The investigated model is based on the heat conduction equation with a heat source and solved using the finite element method (FEM). Themodel is built with the use of enthalpy formulation for solidification and using an intermediate solid fraction growth model. Themodel sensitivity is studied with the use of Morris method, which is one of global sensitivity methods. Characteristic feature ofthe global methods is necessity to conduct a series of simulations applying the investigated model with appropriately chosen model parameters. The advantage of Morris method is possibility to reduce the number of necessary simulations. Results ofthe presented work allow to answer the question how generic sensitivity analysis results are, particularly if sensitivity analysis results depend only on model characteristics and not on things such as density ofthe finite element mesh or shape ofthe region. Results of this research allow to conclude that sensitivity analysis with use of Morris method depends only on characteristic ofthe investigated model.
dos dados em si não faz parte da biblioteca, podendo ser encontrada somente nos arquivos de exemplo que acompanham a distribuição da HigTree. O suporte à escrita de dados também era limitado pelo fato de somente a variante mais ineĄciente do VTK ter sido implementada, a saber: em texto, sem compressão, com arquivos independentes entre si. No contexto do presente trabalho, implementei dentro da HigTree um formato de saída mais eĄciente chamado eXtensible Data Model and Format (XDMF) (CLARKE; MARK, 2007), baseado em outros dois formatos de arquivo mais básicos, Extensible Markup Language (XML) (BRAY et al., 2008) e Hierarchical Data Format v. 5 (HDF5) (THE HDF GROUP, 1997-2018), que é mais eĄciente por utilizar codiĄcação binária na maior parte dos dados, utilizar compressão e evitar repetições ao fazer referência a arquivos escritos anteriormente. Este desenvolvimento é detalhado no capítulo 4.
is much lower than diffusion assumes [46–48]. Dispersal limitation becomes important when the number of discrete individuals is small , since random internal fluctuations can induce population extinction. Given discreteness and stochasticity, neither of which has a role in our cost-minimizing model, lattice-based results show that expected growth from rarity demands greater propagation, relative to mortality, as mean dispersal distance decreases [11,50]. We also assume that no explicit interspecific interactions affect the population during restoration. Species occupying the community to be restored may facilitate restoration; for example, trees may attract birds that disperse seeds of other tree species . Alternatively, resident species may resist the introduced species biotically [52,53]. Interspecific interactions will often affect the likelihood of restoration success, as well as the cost. Consequences of these interactions can sometimes be expressed abstractly through the introduced species’ positive equilibrium density; in other cases, successful restoration may demand quantification of these interactions.
Nas figuras 3.a-d têm-se os desenhos das funções peso para os orbitais atômicos de simetria s e p do átomo de Xe em siste- mas com todos os elétrons e com o uso de pseudopotencial (pp). Já nas figuras 3.a e 3.c observa-se as características corretas desejadas para a boa representação das funções peso, ou seja, contínuas e convergentes, tantos para os orbitais mais internos quanto para os de valência. Nas figuras 3.b e 3.d, ambos os orbitais mais externos apresentam uma descrição inadequada na região de valência (menores valores de ln(α )), isto indica a necessidade da
da sua simplicidade, reside no facto de que também pode ser usado para descrever fluidos dilatantes, a que correspondem valores de q tais que q > 2 e onde se incluem o gelo polar, a lava dos vulcões e a areia molhada, quando modelados como fluidos. Dada a sua analogia com a lei de Stokes, os fluidos modelados por (1) são designados por fluidos Newtonianos generalizados. O único inconveniente do modelo (1), é que se deve ter cuidado quando é usado para valores q > 2, uma vez que o modelo falha para tensões de corte muito grandes, quando a viscosidade deve, em última análise, aproximar-se de uma constante. De modo a rectificar esta situação, Sisko propôs, em 1958, o modelo seguinte para modelar o escoamento de algumas graxas comerciais,
Moreover, the mixing layer thickness ‘previously defined” seems to be constant for the three burners. However, this is only true for a general observation. In reality, an important decrease ofthe mixing coefficient is visible, especially between burner 4 and burner 5. This result is discussed with more details in the section ofthe mixing layer thickness. Moreover, it is noted that the seeded air width after injection decreases and the diameter of CRZ increases by increasing the number of injectors, which according to B.shi et al. (2014b), results in the decrease the mixing time in the burner (Eqs. (10) and (11)).
Here p (x, t) is the pressure field, and is the source of non-locality ofthe problem. Indeed, the continuity equa- tion for incompressible flows reads as a solenoidality prop- erty, ∇ ⋅ u = 0, and pressure is required to satisfy a Pois- son equation. Therefore, even if (2) is in principle evalu- ated locally at one single point, actually it contains a term which represents a contribution coming from a spatial inte- gral on the whole domain, as the propagation velocity of any disturbance is infinite. Despite this difficulty, incompress- ibity is a scheme widely used for the simplifications it brings about, and is usually abandoned only when compressibility effects cannot be neglected, most notably because the veloc- ities into play are not negligible with respect to the sound speed . In this latter case, the mass density varies. Also thermal effects can come into play and modify the param- eters, in which case also the evolution ofthe temperature field must be taken into account, along with a suitable equa- tion of state. Even more problematically, also viscosity can be different from a constant, and then the fluid under con- sideration is dubbed as non-Newtonian and described by a different equation.
The DGS, HVOF and plasma spraying techniques applied to production of coatings from powders offer practically unlimited abilities in the establishing composition of a coating. A control ofthe spraying parameters with the objective to obtain coatings with unique exploiting properties like resistance to abrasive wear, erosion, corrosion, high temperature corrosion or thermal shocks, as well as good adhesion and low porosity was the subject of number of works [1-6].
Here the divergence-free vector field v again denotes the approximate velocity from the previous Picard iteration. Note that when the “wind" function v is irrotational ( ∇ × ν = 0) Eq. 9-11 reduce to theStokes problem. It is not difficult to see that the linearizations 5-7 and 9-11, although both conservative (Olshanskii, 2002), are not mathematically equivalent. The momentum Eq. 9 is called the rotation form. We can see that no first-order terms in the velocities appear in 9 on the other hand, the velocities in the d scalar equations comprising 9 are now coupled due to the presence ofthe term w × u. The disappearance ofthe convective terms suggests that the rotation form 9 ofthe momentum equations may be advantageous over the standard form 5 from the linear solution point of view. This observation was first made by Olshanskii and his co-workers in 2002 (Olshanskii and Reusken, 2002). In their study, they showed the advantages ofthe rotation form over the standard convection form in several aspects. Benzi and Liu (2007), detailed discussion is provided for the preconditioned iterative methods oftheNavier-Stokes problems in rotation form.
Neste cap´ıtulo estudaremos a controlabilidade local exata para trajet´orias das equa¸c˜oes de Navier-Stokes com controle interno distribu´ıdo em conjuntos pequenos. Primeiramente demonstraremos uma desiguadade do tipo Calerman para o sistema de Navier-Stokes linearizado, `a qual nos permitir´a concluir a controlabilidade nula em qualquer tempo T > 0. Utilizando um teorema de fun¸c˜ao inversa e uma hip´otese adicional de regularidade sobre as trajet´orias, provaremos um resultado local con- cernente a controlabilidade exata para as trajet´orias do sistema de Navier-Stokes. Os resultados deste cap´ıtulo foram obtidos por E. Fern´andez-Cara, S. Guerrero, O. Yu. Imanuvilov e J. P. Puel em .
To illustrate the dependence ofthe evolution ofthe field per- turbations on its initial conditions and how this can generate isocurvature modes, we show in Figure 3, for three different sets of initial conditions, the evolution of ζ function, for three different values of α.
The immersed boundary (IB) methods are used to enforce boundary conditions on surfaces not aligned with the computational mesh in a numerical simulation. This methodology has been used as a practical approach to model ﬂow problems involving complex and/or moving bodies. Despite the great advantages ofthe immersed boundary methodology, it is shown in this work that some diﬃculties and challenges are posed when it is used to simulate the ﬂow past sharp geometries. In present work, two main objectives are proposed: ﬁrst, to assess the accuracy and eﬃciency of IB methods in simulations of ﬂows past immersed bodies with highly sharp corners or thin plates. Secondly, we implement a numerical method which is able to satisfy these ﬂow conditions. The study was composed of four stages: First, an extensive bibliographic review was conducted in order to know and understand the diﬀerent immersed boundary methods; in the second stage it was presented modiﬁcations in Multi-Direct Forcing method; further on, it was presented a local directional ghost cell approach. Finally, the methods are implemented and tested for a number of problems, the modiﬁed multi-direct forcing approach was validated for a uniform ﬂow past a circular cylinder, a sphere and an airfoil NACA0012. The local directional ghost cell approach was employed to calculate a Poiseuille ﬂow, an impulsively started ﬂow past a ﬂat plate and uniform ﬂow around a circular cylinder between two parallels walls.
A didactical approach is used in this work. The method ofthe boundary elements is applied to fluid problems, aiming also at introducing the methodology to new users. The computational implementation is based on the Kakuda and Tosaka (1988) reports. There, the boundary element method uses a reformulation ofthe unsteady NavierStokes equations in terms of velocity components only, by making use ofthe penalty function method, an approach successfully applied to flow analysis with finite element. The effectiveness of this method was illustrated by several numerical examples. Tosaka and Onishi (1985, 1986) proposed new integral representations for theNavierStokes equations for both steady and unsteady flow problems. The workability and validity ofthe methodology developed therein were shown with several numerical results for steady problems (Tosaka, Kakuda and Onishi (1985); Tosaka and Kakuda (1986); Tosaka (1986)).