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1

Thermal analysis of longitudinal a porous fin with temperature-dependent internal heat generation using the variation of parameters method

Güngör Osman, Arslantürk Cihat

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 3

5--16

EN

The present study investigates the thermal performance of longitudinal a porous fin with temperature-dependent internal heat generation. The Darcy model is utilized to obtain the differential form of the governing equation that solves the nonlinear temperature distribution equation using the method of variation of parameters. Although this method is applied to solve both linear and nonlinear differential equations, there exist rare applications of this method to solve nonlinear heat transfer problems. In the present study, we applied the method to estimate the thermal analysis of the porous fin exposed to convection. The heat generation is assumed as a function of temperature. The effects of the convection parameter Nc, internal heat generation ɛ, porosity Sh, and generation number G parameter on the dimensionless temperature distribution are discussed in detail. The accuracy of the variation of parameters method is verified through comparison with homotopy perturbation method and the Matlab bvp4c solver (NUM). The results have disclosed that the variation of parameters method can be used as a very effective and practical approach for further studies of the porous medium.

2

Numerical and analytical calculations of the free surface flow between two semi-infinite straights

Chedala Fatma Zohra, Amara Abdelkader, Meflah Mabrouk

Journal of Applied Mathematics and Computational Mechanics

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2020

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Vol. 19, nr 4

21--32

EN

The problem of two-dimensional flow with the free surface of the jet in a region between two semi-infinite straights intersections at point O is calculated analytically for each angle Beta and numerically for each of the various values of the Weber number and angle Beta. By assuming that the flow is potential, irrotational and that the fluid is incompressible and inviscid, and by taking account only the surface tension for a numerical method using the series truncation, and without the effect of gravity and surface tension for the analytic method utilize the hodograph transformation. The obtained results confirmed a good agreement between them when the Weber number tends to infinity, and the comparison of these surface shapes is illustrated.

3

A Fundamental Study on the Free Vibration of Geometrical Nonlinear Cantlever Beam using an Exact Solution and Experimental Investigation

Jamal-Omidi M., Shayanmehr M., Sazesh S.

Archive of Mechanical Engineering

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2018

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Vol. LXV, nr 1

65--82

EN

Two fundamental challenges in investigation of nonlinear behavior of cantilever beam are the reliability of developed theory in facing with the reality and selecting the proper assumptions for solving the theory-provided equation. In this study, one of the most applicable theory and assumption for analyzing the nonlinear behavior of the cantilever beam is examined analytically and experimentally. The theory is concerned with the slender inextensible cantilever beam with large deformation nonlinearity, and the assumption is using the first-mode discretization in dealing with the partial differential equation provided by the theory. In the analytical study, firstly the equation of motion is derived based on the theory of large deformable inextensible beam. Then, the partial differential equation of motion is discretized using the Galerkin method via the assumption of the first mode. An exact solution to the obtained nonlinear ordinary differential equation is developed, because the available semi analytical and approximated methods, due to their limitations, are not always sufficiently reliable. Finally, an experiment set-up is developed to measure the nonlinear frequency of oscillations of an aluminum beam within a domain of initial displacement. The results show that the proposed analytical method has excellent convergence with experimental data.

4

Hypothetical analysis for peristaltic transport of metallic nanoparticles in an inclined annulus with variable viscosity

Nadeem S., Sadaf H.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2016

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Vol. 64, nr 2

447--454

EN

The main objective of this article is to present a mathematical model for peristaltic transport in an inclined annulus. In this analysis, two-dimensional flow of a viscous nanofluid is observed in an inclined annulus with variable viscosity. Copper as nanoparticle with blood as its base fluid has been considered. The inner tube is unifom or rigid, while the outer tube takes a sinusoidal wave. Governing equations are solved under the well-known assumptions of low Reynolds number and long-wavelength. Exact solutions have been established for both velocity and nanoparticle temperature. The features of the peristaltic motion are explored by plotting graphs and discussed in detail.

5

Application of Rank Controlled Differential Quadrature Method for Solving an Infinite Steel Plate Cooling Problem

Żak P. L., Suchy J. S., Lelito J., Gracz B.

Archives of Foundry Engineering

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2013

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Vol. 13, iss. 3 spec.

201--206

EN

Rank Controlled Differential Quadrature method is a numerical method that allows to approximate the partial derivatives that appears in partial differential equations. Those equations with proper geometrical, physical, initial and boundary conditions make mathematical models of physical process. The heat transfer process is governed by Fourier–Kirchhoff equation, which is parabolic Partial Differential Equation. In this paper authors present the steel plate cooling problem. At the beginning of the process plate is heated up to 450 °C and is cooled to ambient temperature. The cooling of the plate is basic heat transfer problem. If the plates dimensions has proper proportions such problem may be described as one dimensional and solved exactly. The mathematical model and exact solution is given in the work. Authors apply the Rank Controlled Differential Quadrature to approximate derivatives in Fourier–Kirchhoff equation and in boundary conditions. After changing derivatives into quadrature formulation set of algebraic equations is obtained. Substituting thermo-physical parameters numerical model is obtained. The computer program was prepared to solve the problem numerically. Results of simulation are confronted with the exact ones. Error value at each time step as well as error value increase rate for examined numerical method is analyzed.

6

Flow analysis of viscoelastic fluid due to a shrinking sheet

Nandeppanavar M. M., Siddalingappa M. N., Kemparaju M. C.

International Journal of Applied Mechanics and Engineering

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2012

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Vol. 17, no. 1

249-257

EN

This work is concerned with the analysis of a two dimensional flow of a viscoelastic fluid (Walters' liquid B) induced by a shrinking sheet. The governing partial differential equations are first reduced to ordinary nonlinear differential equations by using the appropriate similarity transformation. An exponential solution is assumed to solve the considered boundary value problem. The effects of governing pertinent physical parameters on the flow characteristics are presented graphically and discussed.

7

Synteza parametrycznych liniowych oraz nieliniowych równań różniczkowych o znanych rozwiązaniach

Żuchowski A.

Przegląd Elektrotechniczny

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2008

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R. 84, nr 11

351-354

PL

Równania różniczkowe, zwłaszcza parametryczne liniowe lub nieliniowe o znanych rozwiązaniach są przydatne do testowania programów komputerowych służących do całkowania takich równań. Przedstawiono dwie metody ich syntezy.

EN

The linear and non-linear differential equations with known solutions, especially those parametric ones, are useful for testing of dedicated computer programs solving differential equations. The two methods for synthesis of differential equations applicable in solution accuracy tests are presented in the paper.

8

Properties of transient states in electrical circuits : transformation of F(s) in to the Maclaurin series f(t)

Piwowarczyk T.

Czasopismo Techniczne. Elektrotechnika

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2005

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R. 102, z. 5-E

45--52

EN

Transients in linear electrical circuits can be expressed as functions f(t), or as their Laplace transform F(s). The relation between them takes the form of a well known integral transform. The Laplace transform F(s) can also be computed from some finite subset of derivatives of f(t). In this approach the appropriate formulas take the form of the matrix transformation. It contains four arithmetic operations exclusively. Thus it is easy to be implemented in the assembly language. It also delivers a new interpretation of transients. A set of interesting theorems has been proved in the article. One of theorems states that some approximation of f(t) can contain all the information about a transient which is coded in a special way. This unexpected fact provides a basis for a new method of finding transients in linear electrical circuits.

PL

Stany nieustalone w liniowych obwodach elektrycznych mogą być wyrażone jako transformaty Laplace'a F(s) oryginału f(t). Jeżeli funkcja F(s) jest dana, to oryginał f(t) jest obliczany z zastosowaniem odwrotnej transformacji Laplace'a. W tym celu należy znać wszystkie zera mianownika wy.miernej funkcji F(s). Dla wyższych stopni funkcji F(s) rozkład na ułamki proste staje się poważnym problemem ze względu na twierdzenie Abela. W szczególności jest to zasadniczy problem, gdy elementy obwodu elektrycznego są podane w formie parametrycznej, tak jak to jest w przypadku syntezy obwodu. Możemy ominąć ten problem używając uogólnionych ciągów Fibonacciego. Co więcej, można do obliczeń użyć pewnego skończonego podciągu będącego częścią uogólnionego ciągu Fibonacciego bez straty informacji potrzebnej do odtworzenia dokładnego rozwiązania. Takie skończone podciągi tworzą podstawę nowej teorii rozwiązywania i analizowania stanów nieustalonych w obwodach elektrycznych.

9

The exact solution of nonlinear stress-free convection under the influence of magnetic fields

El-Kholy S. A.

Mechanics and Mechanical Engineering

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2004

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Vol. 7, nr 2

13-22

EN

This paper aims at transforming nonlinear equations (Navier-Stokes, energy and magnetic induction) in steady state into ordinary high order differential equations. The exact solution for it has been found using a suitable transformation. The second goal is studying the state conditions of heating from below in a horizontal layer and studying the influence of the magnetic field on the phenomenon of convection itself. This model paper has uncovered the properties of electricity-conducting fluid elements. These properties have been disclosed neither in theory nor in practice. This study has been applied on the stress-free boundaries. All of which have realistic manifestation, in nature. They study convection in geophysics and astrophysics. Results have been illustrated in three dimensions to generalize the study and use it practically.

10

Axisymmetric MHD oscillation of a compressible hollow jet

Radwan A. E., Eltaweel M. A.

Mechanics and Mechanical Engineering

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2004

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Vol. 7, nr 2

127-130

EN

The capillary oscillation of compressible hollow Jet under the action of the electromagnetic force for axisymmetric perturbation is discussed. Some published results are recovered as limiting cases on utilizing the dispersion relation of the present model. The capillary force has a destabilizing effect in a small range (0 ≤ χ < 1, where χ is the dimensionless longitudinal wavenumber) of perturbation. While it is stabilizing as 1 ≤ χ < ∞. The compressibility has a strong destabilizing tendency of the model for all wavenumber values χ ≠ 0. Each of the magnetic fields pervaded in the gas and liquid regions has a stabilizing effect for all short and. long wavelengths. As that effect is so strong, the capillary instability is suppressed and stability sets in. But in fact here the model is unstable due to the strong destabilizing effect of the compressibility.