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Approximate explicit solutions of Orr–Sommerfeld equations and heat transfer in a geometry with variable cross section by He’s methods and comparison with the ADM and DTM

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Abstract

In this paper two examples are studied; the Orr–Sommerfeld and nonlinear heat transfer equation. He’s variational iteration method (VIM) and Adomian’s decomposition method (ADM) are applied to the Orr–Sommerfeld equation. He’s VIM and differential transformation method (DTM) are applied to the nonlinear heat transfer equation. Comparison of different methods represents while the effect of the nonlinear term in the heat transfer equation is negligible, VIM and DTM lead approximately to the same answers; and when the kinematic viscosity in the Orr–Sommerfeld equation is small, VIM and ADM lead nearly to the same results.

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Abbreviations

\(A(x)\) :

Various surface (m\(^{2})\)

\(A_{0}\) :

Fin surface area (m\(^{2})\)

\(G_{0}\) :

Heat generation in beginning

\(G(x)\) :

Variable heat generation

HPM:

Homotopy perturbation method

VIM:

Variational iteration method

DTM:

Differential transformation method

\((t)\) :

Thermal conductivity (W/mk)

\(K_{0}\) :

Thermal conductivity in \(T\) = 0 (W/mk)

\(L\) :

Length of geometry (m)

\(T\) :

Temperature (K)

\(T_{0}\) :

Temperature in beginning (K)

\(T_{L}\) :

Temperature at the end (K)

Re:

Reynolds number

\(C\) :

Wave propagation velocity

\(V\) :

Undisturbed velocity profile

\(u\) :

Boundary-layer velocity profile

\(v\) :

Boundary-layer velocity profile

\(p\) :

Boundary-layer pressure

\(\Phi \) :

Perturbation stream function

\(\beta \) :

Coefficient of linear conductivity (1/k)

\(\nu \) :

Kinematic viscosity

\(\theta \) :

Dimension less temperature

\(\Psi \) :

Stream function

\(\alpha \) :

Disturbance wavelength

k:

Number of iteration

n:

Number of iteration

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Authors and Affiliations

  1. Mechanical Engineering Department, Mazandaran Institute of Technology, Babol, Iran

    R. Hasankhani Gavabari, A. Janbaz & D. D. Ganji

  2. Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran

    D. D. Ganji

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  1. R. Hasankhani Gavabari
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  2. A. Janbaz
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  3. D. D. Ganji
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Correspondence to D. D. Ganji.

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Gavabari, R.H., Janbaz, A. & Ganji, D.D. Approximate explicit solutions of Orr–Sommerfeld equations and heat transfer in a geometry with variable cross section by He’s methods and comparison with the ADM and DTM. Afr. Mat. 26, 1009–1023 (2015). https://doi.org/10.1007/s13370-014-0252-0

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