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The effect of second order slip condition on MHD nanofluid flow around a semi-circular cylinder

  • Jiahui Cao , Jing Zhu EMAIL logo , Xinhui Si and Botong Li
Published/Copyright: December 16, 2021
Zeitschrift für Naturforschung A
From the journal Zeitschrift für Naturforschung A Volume 77 Issue 4

Abstract

Steady forced convection of non-Newtonian nanofluids around a confined semi-circular cylinder subjected to a uniform magnetic field is carried out using ANSYS FLUENT. The numerical solution is obtained using the finite volume method. The user-defined scalar (UDS) is used for the first time to calculate the second order velocity slip boundary condition in semi-circular curved surface and the calculated results are compared with those of the first order velocity slip boundary condition. Besides, the effects of volume fraction, size, type of nanoparticles and magnetic field strength on heat transfer are studied. The present study displays that adding nanoparticles in non-Newtonian fluids significantly enhances heat transfer. In addition, it is observed that the heat transfer rate decreases first and then increases with the increase of Hartmann number. The effects of blocking rate on Nusselt number, wake length and heat transfer effect are shown in the form of graphs or tables.

Keywords: blocking rate; heat transfer; magnetic field; power-law nanofluids; second order velocity slip
Corresponding author: Jing Zhu, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China, E-mail:

Funding source: Fundamental Research Funds for the Central Universities of China

Award Identifier / Grant number: FRF-BR-18-008B

Acknowledgments

Project supported by National Natural Science Foundation of China (No. 11 801 029) and the Fundamental Research Funds for the Central Universities of China (No. FRF-BR-18-008B).

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix

The transformation relationship between the Cartesian coordinate system and the natural coordinate system is as follows:

(A.1) x = − t sin θ , x = s cos θ   y = t cos θ , y = s sin θ  

A vector U could be written as:

(A.2) U x i ⃗ + U y j ⃗

(A.3) U s p ⃗ + U t q ⃗

where p ⃗ and q ⃗ are unit vector natural coordinate.

According to the two-dimensional coordinate transformation formula, the relation between ( i ⃗ , j ⃗ ) and ( p ⃗ , q ⃗ ) is expressed as follows:

(A.4) i ⃗ = cos θ ⋅ p ⃗ − sin θ ⋅ q ⃗   j ⃗ = sin θ ⋅ p ⃗ − cos θ ⋅ q ⃗  

By transformation, U s can be expressed as:

(A.5) U s = U x cos θ + U y sin θ

According to Eqs. (A.1) and (A.5), ∂ U s ∂ t can be written as:

(A.6) ∂ U s ∂ t = − ∂ U x ∂ x cos θ + ∂ U y ∂ x sin θ   + ( U y cos θ − U x sin θ ) − 1 r cos θ sin θ + ∂ U x ∂ y cos θ + ∂ U y ∂ y sin θ   + ( U y cos θ − U x sin θ ) − 1 r sin θ cos θ

The partial derivatives ∂ θ ∂ x , ∂ θ ∂ y are obtained by the polar coordinates:

(A.7) ∂ θ ∂ x = − 1 r cos θ   ∂ θ ∂ y = − 1 r sin θ  

In line with Eq. (A.6), ∂ 2 U s ∂ t 2 can be simplified as:

(A.8) ∂ 2 U s ∂ t 2 = sin 2 ⁡ θ ∂ 2 U s ∂ x 2 − sin θ cos θ ∂ 2 U s ∂ x ∂ y − sin θ ⁡ cos θ ∂ 2 U s ∂ y ∂ x + ∂ 2 U s ∂ y 2 cos 2 ⁡ θ

The second partial derivatives are expressed as follows:

(A.9) ∂ 2 U s ∂ x 2 = ∂ ∂ x ∂ U s ∂ x = ∂ 2 U x ∂ x 2 cos θ + ∂ 2 U y ∂ x 2 sin θ + 2 ∂ U x ∂ x sin θ r cos θ − 2 r ∂ U y ∂ x − U x 1 r 2 cos 3 ⁡ θ

(A.10) ∂ 2 U s ∂ y 2 = ∂ ∂ y ∂ U s ∂ y = ∂ 2 U x ∂ y 2 cos ⁡ θ + ∂ 2 U y ∂ y 2 sin ⁡ θ − 2 ∂ U y ∂ y cos ⁡ θ r ⁡ sin ⁡ θ + 2 r ∂ U x ∂ y − U y 1 r 2 sin 3 ⁡ θ

(A.11) ∂ 2 U s ∂ x ∂ y = ∂ ∂ x ∂ U s ∂ y = ∂ 2 U x ∂ x ∂ y cos θ + ∂ 2 U y ∂ x ∂ y sin θ + ∂ U x ∂ y sin θ r cos θ − ∂ U y ∂ x cos θ r sin θ + 1 r ∂ U x ∂ x − 1 r ∂ U y ∂ y − U y 1 r 2 sin 2 ⁡ θ ⁡ cos θ

(A.12) ∂ 2 U s ∂ y ∂ x = ∂ ∂ y ∂ U s ∂ x = ∂ 2 U x ∂ y ∂ x cos θ + ∂ 2 U y ∂ y ∂ x sin θ + ∂ U x ∂ y sin θ r cos θ − ∂ U y ∂ x cos θ r sin θ + 1 r ∂ U x ∂ x − 1 r ∂ U y ∂ y − U x 1 r 2 cos 2 ⁡ θ ⁡ sin θ

Substituting Eqs. (A.9)–(A.12) into Eq. (A.8), the expression is shown below:

(A.13) ∂ 2 U s ∂ t 2 = sin 2 θ ⁡ cos θ ∂ 2 U x ∂ x 2 + cos 3 ⁡ θ ∂ 2 U x ∂ y 2 + sin 3 ⁡ θ ∂ 2 U y ∂ x 2 + cos 2 ⁡ θ ⁡ sin θ ∂ 2 U y ∂ y 2 − sin θ ⁡ cos 2 ⁡ θ ∂ 2 U x ∂ x ∂ y − sin 2 ⁡ θ ⁡ cos θ ∂ 2 U y ∂ x ∂ y − sin θ ⁡ cos 2 ⁡ θ ∂ 2 U x ∂ y ∂ x − sin 2 ⁡ θ ⁡ cos θ ∂ 2 U y ∂ y ∂ x + 2 s i n 3 θ r c o s θ − 2 r sin θ ⁡ cos θ ∂ U x ∂ x + 2 c o s 2 θ r − 2 s i n 2 θ r ∂ U x ∂ y + 2 c o s 2 θ r − 2 s i n 2 θ r ∂ U y ∂ x + − 2 c o s 3 θ r s i n θ + 2 r sin θ ⁡ cos θ ∂ U y ∂ y + − sin 2 ⁡ θ r 2 cos 3 ⁡ θ + 1 r 2 cos θ U x + − cos 2 ⁡ θ r 2 sin 3 ⁡ θ + 1 r 2 sin θ U y

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Received: 2021-07-12
Accepted: 2021-11-26
Published Online: 2021-12-16
Published in Print: 2022-04-26

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Dynamical Systems & Nonlinear Phenomena
  3. Investigation of the non-linear vibration behaviour and 3:1 internal resonance of the multi supported nanobeam
  4. Third harmonic generation of a relativistic self-focusing laser in plasma under exponential density ramp
  5. Flow and heat over a rotating disk subject to a uniform horizontal magnetic field
  6. Hydrodynamics
  7. The effect of surfactant on the drag and wall correction factor of a drop in a bounded medium
  8. The effect of second order slip condition on MHD nanofluid flow around a semi-circular cylinder
  9. Turbulent boundary layer heat transfer of CuO–water nanofluids on a continuously moving plate subject to convective boundary
  10. Thermodynamics & Statistical Physics
  11. A self-similar solution for shock waves in conducting rotating non-ideal dusty gas medium with monochromatic radiation and magnetic field
  12. Analysis of the second virial coefficient, and application to rare gas mixtures
https://doi.org/10.1515/zna-2021-0197
Keywords for this article
blocking rate; heat transfer; magnetic field; power-law nanofluids; second order velocity slip

Articles in the same Issue

  1. Frontmatter
  2. Dynamical Systems & Nonlinear Phenomena
  3. Investigation of the non-linear vibration behaviour and 3:1 internal resonance of the multi supported nanobeam
  4. Third harmonic generation of a relativistic self-focusing laser in plasma under exponential density ramp
  5. Flow and heat over a rotating disk subject to a uniform horizontal magnetic field
  6. Hydrodynamics
  7. The effect of surfactant on the drag and wall correction factor of a drop in a bounded medium
  8. The effect of second order slip condition on MHD nanofluid flow around a semi-circular cylinder
  9. Turbulent boundary layer heat transfer of CuO–water nanofluids on a continuously moving plate subject to convective boundary
  10. Thermodynamics & Statistical Physics
  11. A self-similar solution for shock waves in conducting rotating non-ideal dusty gas medium with monochromatic radiation and magnetic field
  12. Analysis of the second virial coefficient, and application to rare gas mixtures
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