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Temperature and Motion Analysis of a Rotational Maxwell Fluid with non-Constant Thermal Conductivity and non-Fourier Heat Transfer Model, Using Homotopy Perturbation Method

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Abstract

Our investigation numerically analyzed the motional behavior and temperature distribution of an upper-convected Maxwell fluid rotating on an expanding sheet in a 3D environment. To solve the governing PDEs, we used techniques to transform the main equations into ODEs. Then we applied an analytical and powerful method known as Homotopy Perturbation Method (HPM). The obtained outcomes were confirmed by the numerical fourth-fifth order Runge-Kutta method and previous results in the literature. Both ways of approving show the acceptable accuracy of our method. The study’s main results illustrate that the depth of the boundary layer depends on the rotation factor. By increasing the rotation factor, the boundary layer’s thickness declines; therefore, the non-Newtonian fluid’s velocities in all directions diminish. The next important parameter that was interpreted is the Deborah number, and we discovered an opposite relation between the Deborah number and the velocities in the horizontal and lateral directions. Prandtl number was another critical parameter that was investigated, so the thermal boundary layer’s thickness increases by raising the Prandtl number. The last parameter was thermal relaxation time, and by increasing that, the fluid temperature at every point rises dramatically.

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The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Authors and Affiliations

  1. Department of mechanical engineering, Babol Noshirvani University of Technology, Babol, Iran

    Maryam Fallah Najafabadi & Davood Domiri Ganji

  2. Department of mechanical engineering, Amirkabir University of Technology, Tehran, Iran

    Hossein Talebi Rostami

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  1. Maryam Fallah Najafabadi
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Contributions

MFN analyzed the results and worked on the code, and she created proper figures for the results. HTR worked on the code and prepared the English text for every section. DDG supervised the whole process.

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Correspondence to Hossein Talebi Rostami.

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Appendix

Appendix

PDEs must be converted to ODEs in order to be resolvable. For this act, some ways are beneficial, and one of them is to use similarity equations to convert the main equations of this article. Equations (7), (8), and (11) were converted to ordinary form; these are Eqs. (16)–(18) [49], [50]. The first step in selecting a similarity equation is to compute the layer thickness ψ on the plate. The thickness of the layer must have a relation as follows:

\(\psi =f\left( {u,v} \right)\)

According to the II theorem of dimensional analysis, the thickness of the layer can consider as:

\(\psi \sim \sqrt {\frac{u}{v}}\)

For applying the above proportion to the equation, we must use a constant number for a function. It should be noted that the velocity component in the x-direction on the plate is just a; therefore, we can use this parameter instead of u in the above proportion.

\(\psi \sim \sqrt {\frac{a}{v}}\)

The distance from the plate is in the z-direction, and if we use this distance in \(\psi \sim \sqrt {\frac{a}{v}}\) the dimensionless wall distance is introduced by:

\(\eta =\sqrt {\frac{a}{v}} z\)

By using the trial solutions, we have:

\({\text{ }}u=axf^{\prime}\left( \eta \right){\text{, }}\nu =axg\left( \eta \right){\text{, }}w= - \sqrt {a\nu } f\left( \eta \right){\text{, }}\theta =\frac{{{T_{temp}} - {T_a}}}{{{T_0} - {T_a}}}.\)

So, ordinary equations can be obtained. See Eqs. (16)–(18).

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Fallah Najafabadi, M., Talebi Rostami, H. & Domiri Ganji, D. Temperature and Motion Analysis of a Rotational Maxwell Fluid with non-Constant Thermal Conductivity and non-Fourier Heat Transfer Model, Using Homotopy Perturbation Method. Int. J. Appl. Comput. Math 9, 2 (2023). https://doi.org/10.1007/s40819-022-01465-4

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