Application of homotopy perturbation method to find an analytical solution for magnetohydrodynamic flows of viscoelastic fluids in converging/diverging channels

Shadloo, Mostafa Safdari; Kimiaeifar, Amin

doi:10.1243/09544062jmes2334

Cited by 31 publications

(21 citation statements)

References 45 publications

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“…Applying the Bubnov-Galerkin method and using the first eigenmode of the simply supported beam yields: (22) where (23) Table 1 shows the governing equations of motion for transversely vibrating usual beam, cubic and quintic nonlinear beams. To extend study and understanding on beam nonlinear frequency, this paper brings quintic nonlinearities into consideration.…”

Section: Resultsmentioning

confidence: 99%

“…There have been several classical approaches employed to solve the governing nonlinear differential equations to study the nonlinear vibrations including perturbation methods [18], He's Max-Min Approach (MMA) [2], He's Energy Balance Method [19], Combined Homotopy Variational Approach [20], Iteration perturbation method [21], Homotopy perturbation method (HPM) [22,23], Multistage Adomian Decomposition Method [24], Variational iteration method [3], Multiple scales method [25], Monotone iteration schemes [26], ADM-Padé technique [27], Navier and Levy-type solution [28], Hamiltonian approach [29], Parameter Perturbation Method [30], Differential Transform method [31], Laplace Transform method [32] . The application of new equivalent function for deadzone and preload nonlinearities on the dynamical behavior of beam vibration using PEM has been investigated by [6][7][8].…”

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confidence: 99%

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High precise analysis of lateral vibration of quintic nonlinear beam

Sedighi

Reza

2013

Lat. Am. j. solids struct.

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This article intends to achieve a new formulation of beam vibration with quintic nonlinearity, including exact expressions for the beam curvature. To attain a proper design of the beam structures, it is essential to realize how the beam vibrates in its transverse mode which in turn yields the natural frequency of the system. In this direction, new powerful analytical method called Parameter Expansion Method (PEM) is employed to obtain the exact solution of frequency-amplitude relationship. Afterwards, it is clearly shown that the first term in series expansions is sufficient to produce a highly accurate approximation of mentioned system. Finally, preciseness of the present analytic procedures is evaluated in contrast with numerical calculations methods, giving excellent results. Nonlinear vibration of beams is of substantial interest for engineers and has been studied, considerably. From the engineering outlook, structures like helicopter rotor blades, space craft antennae, flexible satellites, airplane wings, robot arms, high-rise buildings, long-span bridges, drill strings and vibratory drilling can be modelled as a beam-like member. The problem of the transversely vibrating beam was recently formulated in terms of the partial differential equation of motion by many researchers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] with different boundary conditions. Sedighi et al. [1] presented the advantages of recent modern analytical approaches applied on the governing equation of transversely vibrating cantilever beams. A comprehensive study on the analytical investigation of cubic nonlinear vibrating tapered beams has been conducted by Bayat et al. [2]. The analytical expression for geometrically non-linear vibration of clamped-clamped Euler-Bernoulli beams including cubic non-linear strain-displacement relationship has been obtained by Barari et al. [3]. The application of new analytical approaches on the dynamical behavior of beams vibration with different boundary conditions has been investigated by Sedighi et al. [5][6][7][8]. Closed form solution to free vibration of beams with mixed boundary conditions has been proposed by Motaghian et al. [9]. Nikkhah Bahrami et al.[10] used modified wave approach for calculation of natural frequencies and mode shapes of arbitrary non-uniform beams. Non-linear modal analysis of a rotating beams studied by [11,12]. When the vibration amplitudes are moderate or large, the geometric nonlinearity must be included.It is very important to provide an accurate analysis towards the understanding of the non-linear vibration characteristics of these structures. Most models dealing with nonlinear dynamics of flexible

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Section: Resultsmentioning

confidence: 99%

mentioning

confidence: 99%

High precise analysis of lateral vibration of quintic nonlinear beam

Sedighi

Reza

2013

Lat. Am. j. solids struct.

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“…There are several classical approaches employed to solve the governing nonlinear differential equations for studying nonlinear vibrations including strong parameters, such as the energy balance method (EBM) and variational approach (VA) [18], Laplace transform method [2], Hamiltonian approach (HA) [19], multiscale decomposition method [20], max-min approach (MMA) [21], iteration perturbation method [22], homotopy perturbation method (HPM) [23], multistage Adomian decomposition method [24], variational iteration method [3], multiple scales method [25], monotone iteration schemes [26], and Navier and Levy-type solutions [27,28]. Therefore, many different methods propose various ways to eliminate the small parameter.…”

Section: Introductionmentioning

confidence: 99%

Accurate investigation of lateral vibrations of a quintic nonlinear beam on an elastic foundation: Using an exact formulation of the beam curvature

Sedighi

Shirazi

2014

J Appl Mech Tech Phy

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This article attains a new formulation of beam vibrations on an elastic foundation with quintic nonlinearity, including exact expressions for the beam curvature. To achieve a proper design of the beam structures, it is essential to realize how the beam vibrates in its transverse mode, which, in turn, yields the natural frequency of the system. In this direction, a powerful analytical method called the parameter expansion method is employed to obtain the exact solution of the frequencyamplitude relationship. It is clearly shown that the first term in series expansions is sufficient to produce a highly accurate approximation of the above-mentioned system. Finally, the accuracy of the present analytic procedure is evaluated through comparisons with numerical calculations.

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“…There are also some other ways to control the flow separation and control the velocity and temperature profiles as well as the skin friction on the interface. Out of many one can mention the recent works by Shadloo et al [18,19] where they showed the effect of MHD and micropolar fluids, respectively, in order to control before mentioned flow properties.…”

Section: Introductionmentioning

confidence: 99%

Stagnation-Point Flow and Heat Transfer Towards a Shrinking Sheet with Suction in an Upper Convected Maxwell Fluid

Lok

Ishak

Pop

2013

Zeitschrift Für Naturforschung A

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A steady two-dimensional boundary-layer flow and heat transfer of an upper convected Maxwell fluid near a stagnation-point of a permeable shrinking sheet is studied numerically. The effects of elasticity, shrinking, and suction parameters on the flow and heat transfer characteristics are investigated. A similarity transformation reduces the governing equations to third-order nonlinear ordinary differential equations which are then solved numerically. For a fixed value of elastic parameter, it is found that dual solutions exist for some values of shrinking and suction parameters. The plotted streamlines show that for upper branch solutions, the effects of shrinking and suction are direct and obvious as the flow near the surface is seen to suck through the permeable sheet and drag to the origin of the sheet. However, aligned but reverse flow occurs for the case of lower branch solutions.

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Application of homotopy perturbation method to find an analytical solution for magnetohydrodynamic flows of viscoelastic fluids in converging/diverging channels

Cited by 31 publications

References 45 publications

High precise analysis of lateral vibration of quintic nonlinear beam

High precise analysis of lateral vibration of quintic nonlinear beam

Accurate investigation of lateral vibrations of a quintic nonlinear beam on an elastic foundation: Using an exact formulation of the beam curvature

Stagnation-Point Flow and Heat Transfer Towards a Shrinking Sheet with Suction in an Upper Convected Maxwell Fluid

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