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1

A revised model for the effect of nanoparticle mass flux on the thermal instability of a nanofluid layer

Alharbi Ozwah S., Abdullah Abdullah A.

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

488--499

EN

A revised model of the nanoparticle mass flux is introduced and used to study the thermal instability of the Rayleigh-Benard problem for a horizontal layer of nanofluid heated from below. The motion of nanoparticles is characterized by the effects of thermophoresis and Brownian diffusion. The nanofluid layer is confined between two rigid boundaries. Both boundaries are assumed to be impenetrable to nanoparticles with their distribution being determined from a conservation condition. The material properties of the nanofluid are allowed to depend on the local volume fraction of nanoparticles and are modelled by non-constant constitutive expressions developed by Kanafer and Vafai based on experimental data. The results show that the profile of the nanoparticle volume fraction is of exponential type in the steady-state solution. The resulting equations of the problem constitute an eigenvalue problem which is solved using the Chebyshev tau method. The critical values of the thermal Rayleigh number are calculated for several values of the parameters of the problem. Moreover, the critical eigenvalues obtained were real-valued, which indicates that the mode of instability is via a stationary mode.

2

Elastic solutions based on the Mori-Tanaka scheme for pressurized functionally graded cylinder

Eker Mehmet, Yarımpabuç Durmu, Yıldırım Ali, Çelebi Kerimcan

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 4

57--68

EN

In this paper, an elastic analysis of a thick-walled functionally graded cylinder subjected to internal pressure is examined. Material properties for the isotropic material are estimated to obey the Mori-Tanaka homogenization scheme through the thickness. The resulting two-point irregular boundary value problem is solved by the pseudospectral Chebyshev method that converts the boundary value problem to the system of equations, which can be solved by any appropriate decomposition method. Benchmark solutions are used to validate the method. The effect of the arbitrarily chosen volume fraction index is demonstrated for stress and displacement distributions. The effective stresses for different inner radius and volume fraction index are also discussed.

3

Commercial airplane trajectory optimization by a Chebyshev pseudospectral method.

Panasz P., Maroński R.

Archive of Mechanical Engineering

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2005

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Vol. LII, nr 1

5-19

EN

The paper presents application of direct pseudospectral Chebyshev method for solving a commercial airplane trajectory optimization problem. The method employs Nth-degree Lagrange polynominal approximations for the state and control variables with the values of these variables of the Chebyshev-Gauss-Lobatto (CGL) points as the expansion coefficients. This process is convented to a nonlinear programming problem (NLP) with the state and control values at the CGL points as unknown NLP parameters. The kinetic model of flight is formulated, where it is assumed that an airplane is a particle and the motion takes place in the vertical plane. The method is implemented in Matlab using sequential quadric programming algorithm (SQP) as an efficient solver. Sensitivity analyses are performed concerning the influence of the degree of discretization and the initial approximation on the solution. Three examples of optimized trajectories in presence of wind are shown.

PL

W pracy została zastosowana bezpośrednia, pseudospektralna metoda Czebyszewa do rozwiązywania zadania optymalizacji trajektorii lotu samolotu komunikacyjnego klasy Boeing 767. W metodzie tej zmienne stanu i sterujące obliczane są w punktach Czebyszewa-Gaussa-Lobatto jako współczynniki rozwinięcia w funkcje Lagrange'a. Są one traktowane jako parametry w zagadnieniu programowania nieliniowego z ograniczeniami. Przedstawiono model lotu, w którym samolot traktowany jest jak punkt materialny poruszający się w płaszczyźnie pionowej. Metoda została zaimplementowana w programie Matlab wykorzystując algorytm sekwencyjnego programowania kwadratowego. Wykonano analizy wrażliwości ze względu na stopień dyskretyzacji i przybliżenie.

4

Influence of approximation method on result of measurement

Swornowski P.

Archiwum Technologii Maszyn i Automatyzacji

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2001

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Vol. 21, Nr 2

113-119

EN

Determination of characteristics of error regarding CMM software has a significant metrological meaning, as with a small number of measuring points instability area of approximation or interpolation method occurs, what in consequence can be observed in a final result To assess an actual level of accuracy for the same input data four different approximation were tested according to Gauss, Tschebyscheff as well as inscribed and circumscribed circle.

PL

Wyznaczanie charakterystyki błędów posiadanego oprogramowania ma ogromne znaczenie metrologiczne, gdyż przy niewielkiej liczbie punktów pomiarowych znajdujemy się w obszarze niestabilności metod aproksymujących (lub interpolujących), co znajduje swoje odbicie w ostatecznym wyniku. Aby stwierdzić rzeczywisty poziom dokładności, przetestowano dla tych samych danych wejściowych cztery metody aproksymujące Gaussa, Czebyszewa oraz okręgu opisanego i wpisanego.