Analysis of highly nonlinear oscillation systems using He’s max-min method and comparison with homotopy analysis and energy balance methods

In this paper, an analytical study of internal energy losses for the non-Darcy Poiseuille flow of silver-water nanofluid due to entropy generation in porous media is investigated. Spherical-shaped silver (Ag) nanosize particles with volume fraction 0.3%, 0.6%, and 0.9% are utilized. Four illustrative models are considered: (i) heat transfer irreversibility (HTI), (ii) fluid friction irreversibility (FFI), (iii) Joule dissipation irreversibility (JDI), and (iv) non-Darcy porous media irreversibility (NDI). The governing equations of continuity, momentum, energy, and entropy generation are simplified by taking long wavelength approximations on the channel walls. The results represent highly nonlinear coupled ordinary differential equations that are solved analytically with the help of the homotopy analysis method. It is shown that for minimum and maximum averaged entropy generation, 0.3% by vol and 0.9% by vol of nanoparticles, respectively, are observed. Also, a rise in entropy is evident due to an increase in pressure gradient. The current analysis provides an adequate theoretical estimate for low-cost purification of drinking water by silver nanoparticles in an industrial process.

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Homotopy Analysis of Carreau Fluid Flow Over a Stretching Cylinder

Journal of Advanced Research in Fluid Mechanics and Thermal Sciences ◽

10.37934/arfmts.88.2.8092 ◽

2021 ◽

Vol 88 (2) ◽

pp. 80-92

Author(s):

Lim Yeou Jiann ◽

Sharidan Shafie ◽

Ahmad Qushairi Mohamad ◽

Noraihan Afiqah Rawi

Keyword(s):

Nonlinear Differential Equations ◽

Velocity Profiles ◽

Weissenberg Number ◽

Stretching Cylinder ◽

Series Solutions ◽

Carreau Fluid ◽

Keller Box Method ◽

Highly Nonlinear ◽

Homotopy Analysis ◽

Curvature Parameter

Carreau fluid flows past a stretching cylinder is elucidated in the present study. The transformed self-similarity and dimensionless boundary layer equations are solved by using the Homotopy analysis method. A convergence study of the method is illustrated explicitly. Series solutions of the highly nonlinear differential equations are computed and it is very efficient in demonstrating the characteristic of the Carreau fluid. Validation of the series solutions is achieved via comparing with earlier published results. Those results are obtained by using the Keller-Box method. The effects of the Weissenberg number and curvature parameter on the velocity profiles are discussed by graphs and tabular. The velocity curves have shown different behavior in and for an increase of the Weissenberg number. Further, the curvature parameter K does increase the velocity profiles.

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HAM Solutions for Vortex-Induced Vibration of a Circular Cylinder

Volume 2: CFD and VIV ◽

10.1115/omae2014-24111 ◽

2014 ◽

Author(s):

Zhiliang Lin ◽

Longbin Tao

Keyword(s):

Circular Cylinder ◽

Practical Importance ◽

Nonlinear Problems ◽

Van Der Pol Oscillator ◽

Fluid Structure ◽

Vortex Induced Vibration ◽

Highly Nonlinear ◽

Homotopy Analysis ◽

Simple Linear Equation ◽

Wake Oscillation

The vortex-induced vibration (VIV) phenomenon is result of fluid-structure interaction which occurs in many engineering fields. The study of VIV of a circular cylinder is of practical importance (such as in marine cables and flexible risers in petroleum production). In this paper, one classical phenomenological VIV model — the motion of the cylinder is modeled by a simple linear equation, and the fluctuating nature of the vortex wake oscillation is modeled by a van der Pol oscillator, is analyzed. Firstly, the homotopy analysis method (HAM), a powerful technique for highly nonlinear problems, is developed to solve the coupled fluid-structure dynamical system with the convergence of the homotopy series solutions being demonstrated. Based on the HAM solutions, some properties of the fully nonlinear classical coupled VIV model are presented. All the results proved that the proposed HAM scheme has potential to be an effective analytic technique to study the VIV problems.

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Approximate analytical solution of the nonlinear system of differential equations having asymptotically stable equilibrium

Filomat ◽

10.2298/fil1709633t ◽

2017 ◽

Vol 31 (9) ◽

pp. 2633-2641 ◽

Cited By ~ 4

Author(s):

Mustafa Turkyilmazoglu

Keyword(s):

Differential Equations ◽

Stable Equilibrium ◽

Initial Conditions ◽

Analytic Solutions ◽

Auxiliary Function ◽

Asymptotically Stable ◽

Highly Nonlinear ◽

Homotopy Analysis ◽

Auxiliary Linear Operator ◽

Analytical Expressions

The present paper is concerned with the purely analytic solutions of the highly nonlinear systems of differential equations possessing an asymptotically stable equilibrium. A methodology combined with the homotopy analysis method is proposed. The methodology involves proper introduction of an auxiliary linear operator and an auxiliary function during the implementation of the homotopy method so that it can yield uniformly valid solutions, not affected from the existing parameters or initial conditions. The technique is applied to the systems particularly appearing in mathematical biology. The obtained explicit analytical expressions for the solution generate results that compare excellently with the numerically computed ones.

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Darcy-Forchheimer hybrid nanofluid flow over a stretching curved surface with heat and mass transfer

PLoS ONE ◽

10.1371/journal.pone.0249434 ◽

2021 ◽

Vol 16 (5) ◽

pp. e0249434

Author(s):

Anwar Saeed ◽

Wajdi Alghamdi ◽

Safyan Mukhtar ◽

Syed Imad Ali Shah ◽

Poom Kumam ◽

...

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Volume Fraction ◽

Curved Surface ◽

Hybrid Nanofluid ◽

Porous Space ◽

Highly Nonlinear ◽

Homotopy Analysis ◽

Curvature Parameter ◽

Swcnts And Mwcnts

The present article provides a detailed analysis of the Darcy Forchheimer flow of hybrid nanoliquid past an exponentially extending curved surface. In the porous space, the viscous fluid is expressed by Darcy-Forchheimer. The cylindrical shaped carbon nanotubes (SWCNTs and MWCNTs) and Fe3O4 (iron oxide) are used to synthesize hybrid nanofluid. At first, the appropriate similarity transformation is used to convert the modeled nonlinear coupled partial differential equations into nonlinear coupled ordinary differential equations. Then the resulting highly nonlinear coupled ordinary differential equations are analytically solved by the utilization of the “Homotopy analysis method” (HAM) method. The influence of sundry flow factors on velocity, temperature, and concentration profile are sketched and briefly discussed. The enhancement in both volume fraction parameter and curvature parameter k results in raises of the velocity profile. The uses of both Fe3O4 and CNTs nanoparticles are expressively improving the thermophysical properties of the base fluid. Apart from this, the numerical values of some physical quantities such as skin friction coefficients, local Nusselt number, and Sherwood number for the variation of the values of pertinent parameters are displayed in tabular forms. The obtained results show that the hybrid nanofluid enhances the heat transfer rate 2.21%, 2.1%, and 2.3% using the MWCNTs, SWCNTs, and Fe3O4 nanomaterials.

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Dynamical Origin of Low-Frequency Variability in a Highly Nonlinear Midlatitude Coupled Model

Journal of Climate ◽

10.1175/jcli3976.1 ◽

2006 ◽

Vol 19 (24) ◽

pp. 6391-6408 ◽

Cited By ~ 18

Author(s):

S. Kravtsov ◽

P. Berloff ◽

W. K. Dewar ◽

M. Ghil ◽

J. C. McWilliams

Keyword(s):

Nonlinear Oscillation ◽

Time Scale ◽

Low Frequency ◽

Coupled Model ◽

Zonal Flow ◽

Wave Mode ◽

Ocean Basin ◽

Coupled Mode ◽

Highly Nonlinear ◽

Channel Configuration

Abstract A novel mechanism of decadal midlatitude coupled variability, which crucially depends on the nonlinear dynamics of both the atmosphere and the ocean, is presented. The coupled model studied involves quasigeostrophic atmospheric and oceanic components, which communicate with each other via a constant-depth oceanic mixed layer. A series of coupled and uncoupled experiments show that the decadal coupled mode is active across parameter ranges that allow the bimodality of the atmospheric zonal flow to coexist with oceanic turbulence. The latter is most intense in the regions of inertial recirculation (IR). Bimodality is associated with the existence of two distinct anomalously persistent zonal-flow modes, which are characterized by different latitudes of the atmospheric jet stream. The IR reorganizations caused by transitions of the atmosphere from its high- to low-latitude state and vice versa create sea surface temperature anomalies that tend to induce transition to the opposite atmospheric state. The decadal–interdecadal time scale of the resulting oscillation is set by the IR adjustment; the latter depends most sensitively on the oceanic bottom drag. The period T of the nonlinear oscillation is 7–25 yr for the range of parameters explored, with the most realistic parameter values yielding T ≈ 20 yr. Aside from this nonlinear oscillation, an interannual Rossby wave mode is present in all coupled experiments. This coupled mode depends neither on atmospheric bimodality, nor on ocean eddy dynamics; it is analogous to the mode found previously in a channel configuration. Its time scale in the model with a closed ocean basin is set by cross-basin wave propagation and equals 3–5 yr for a basin width comparable with the North Atlantic.

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Serious Solutions for Unsteady Axisymmetric Flow over a Rotating Stretchable Disk with Deceleration

Symmetry ◽

10.3390/sym12010096 ◽

2020 ◽

Vol 12 (1) ◽

pp. 96 ◽

Cited By ~ 1

Author(s):

Sadiq

Keyword(s):

Differential Equations ◽

Numerical Solutions ◽

Similarity Transformation ◽

Nonlinear Partial Differential Equations ◽

Axisymmetric Flow ◽

Analysis Method ◽

Highly Nonlinear ◽

Homotopy Analysis ◽

Rotating Stretchable Disk ◽

Comparison Table

In this article, the author has examined the unsteady flow over a rotating stretchable disk with deceleration. The highly nonlinear partial differential equations of viscous fluid are simplified by existing similarity transformation. Reduced nonlinear ordinary differential equations are solved by homotopy analysis method (HAM). The convergence of HAM solutions is also obtained. A comparison table between analytical solutions and numerical solutions is also presented. Finally, the results for useful parameters, i.e., disk stretching parameters and unsteadiness parameters, are found.

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A New Method for Detecting the Time-Varying Nonlinear Damping in Nonlinear Oscillation Systems: Nonparametric Identification

Mathematical Problems in Engineering ◽

10.1155/2011/749309 ◽

2011 ◽

Vol 2011 ◽

pp. 1-12 ◽

Cited By ~ 7

Author(s):

T. S. Jang ◽

Hyoungsu Baek ◽

M. C. Kim ◽

B. Y. Moon

Keyword(s):

Nonlinear Oscillation ◽

Numerical Solutions ◽

Nonlinear Damping ◽

Original Method ◽

Nonparametric Identification ◽

Time Varying ◽

Oscillation System ◽

Damping Characteristics ◽

Highly Nonlinear ◽

Stabilization Technique

This paper presents an original method that can be used for identifying time-varying nonlinear damping characteristics of a nonlinear oscillation system. The method developed involves the nonparametric identification, in which only the system responses, namely, displacement and velocity need to be known for the identification. However, the method is concerned with a Volterra-type integral equation of the first kind, which leads to an instability of numerical solutions. That is, the solutions identified lack stability properties. In order to overcome the difficulty, a stabilization technique is applied to the identification process. A numerical example comprising a highly nonlinear system is examined to demonstrate the workability of the proposed method for the time-varying damping identification.

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On Extending the Quasilinearization Method to Higher Order Convergent Hybrid Schemes Using the Spectral Homotopy Analysis Method

Journal of Applied Mathematics ◽

10.1155/2013/879195 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 3

Author(s):

Sandile S. Motsa ◽

Precious Sibanda

Keyword(s):

Higher Order ◽

Nonlinear Boundary Value Problems ◽

Hybrid Schemes ◽

Iterative Schemes ◽

Flow Problems ◽

Highly Nonlinear ◽

Homotopy Analysis ◽

Layer Flow ◽

Type Boundary ◽

Significant Figures

We propose a sequence of highly accurate higher order convergent iterative schemes by embedding the quasilinearization algorithm within a spectral collocation method. The iterative schemes are simple to use and significantly reduce the time and number of iterations required to find solutions of highly nonlinear boundary value problems to any arbitrary level of accuracy. The accuracy and convergence properties of the proposed algorithms are tested numerically by solving three Falkner-Skan type boundary layer flow problems and comparing the results to the most accurate results currently available in the literature. We show, for instance, that precision of up to 29 significant figures can be attained with no more than 5 iterations of each algorithm.

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