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 **solutions** **of** **the** **Navier**-**Stokes** **model**. 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 **of** **the** 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.

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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 **solutions** **of** obtained Helmholtz equations were used for derivation **of** **analytical** 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.

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In order to control **the** risk induced **by** **the** 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 [5] 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

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Before proceeding let us clearly deﬁne what is meant **by** **analytical**, 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**) **of** **the** independent variable(**s**). An exact solution is obtained **by** integrating **the** governing boundary value problem numerically. **The** result is a tabulation **of** **the** dependent variable(**s**) as a function(**s**) **of** **the** 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).

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Figure 2. Density raster plot in our 2d run Dth32 at a time **of** 180 seconds after **the** start **of** **the** 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 **of** **the** 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 **of** **the** computational domain, which extends 120 km to **the** right.

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We briefly describe **the** strategy **of** **the** 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 **of** **the** 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 **of** **the** present paper. For example, in **the** very recent (GIMENO; MARKVOSEN, 2015) **the** authors discovered and used some **of** **the** relations in Proposition 4.3.1 to show interesting comparison results for **the** capacity and **the** first eigenvalue **of** minimal submanifolds.

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There are some parts **of** **the** **s** / 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 **of** **the** 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.

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Presented paper contains evaluation **of** influence **of** selected parameters on sensitivity **of** a numerical **model** **of** solidification. **The** investigated **model** is based on **the** heat conduction equation with a heat source and solved using **the** finite element method (FEM). **The** **model** is built with **the** use **of** enthalpy formulation for solidification and using an intermediate solid fraction growth **model**. **The** **model** sensitivity is studied with **the** use **of** Morris method, which is one **of** global sensitivity methods. Characteristic feature **of** **the** 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 **of** **the** 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 **of** **the** finite element mesh or shape **of** **the** region. Results **of** this research allow to conclude that sensitivity analysis with use **of** Morris method depends only on characteristic **of** **the** investigated **model**.

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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.

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is much lower than diffusion assumes [46–48]. Dispersal limitation becomes important when **the** number **of** discrete individuals is small [49], 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 [51]. 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.

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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

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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,

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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 **of** **the** mixing coefficient is visible, especially between burner 4 and burner 5. This result is discussed with more details in **the** section **of** **the** 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)).

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Here p (x, t) is **the** pressure field, and is **the** source **of** non-locality **of** **the** 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 [13]. 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 **of** **the** 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.

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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 **the** **Stokes** 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 **of** **the** term w × u. **The** disappearance **of** **the** convective terms suggests that **the** rotation form 9 **of** **the** 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 **of** **the** rotation form over **the** standard convection form in several aspects. Benzi and Liu (2007), detailed discussion is provided for **the** preconditioned iterative methods **of** **the** **Navier**-**Stokes** problems in rotation form.

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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 [12].

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To illustrate **the** dependence **of** **the** evolution **of** **the** 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.

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A didactical approach is used in this work. **The** method **of** **the** 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 **of** **the** unsteady **Navier** **Stokes** equations in terms **of** velocity components only, **by** making use **of** **the** 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 **the** **Navier** **Stokes** equations for both steady and unsteady flow problems. **The** workability and validity **of** **the** methodology developed therein were shown with several numerical results for steady problems (Tosaka, Kakuda and Onishi (1985); Tosaka and Kakuda (1986); Tosaka (1986)).

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