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Effects of radiation on MHD free convection of a Casson fluid from a horizontal circular cylinder with partial slip in non-Darcy porous medium with viscous dissipation
- Gilbert Makanda^{1}Email author,
- Sachin Shaw^{1} and
- Precious Sibanda^{1}
https://doi.org/10.1186/s13661-015-0333-5
© Makanda et al.; licensee Springer. 2015
Received: 4 November 2014
Accepted: 14 April 2015
Published: 6 May 2015
Abstract
In the present study, the effects of radiation on MHD free convection from a cylinder with partial slip in a Casson fluid in non-Darcy porous medium is investigated. The surface of the cylinder is heated under constant surface temperature with partial slip. Partial slip factors are considered on the surface for both velocity and temperature. The boundary layer equations are normalized into a system of non-similar partial differential equations and are then solved using a bi-variate quasilinearization method (BQLM). The boundary layer velocity and temperature profiles are computed for different values of the physical parameters. Increasing the Forchheimer parameter decreases the temperature profiles. The decrease of the velocity profiles with the increase in magnetic parameter is more enhanced in the presence of the velocity slip factor. Increasing the Eckert number increases the temperature profiles in both suction and blowing cases. This study considers the unique problem of the effect of transpiration in a Casson fluid in the presence of radiation, a magnetic field, and viscous dissipation. The results obtained in this study are compared with other numerical methods and were found to be in excellent agreement.
Keywords
- Casson fluid
- partial slip
- viscous dissipation
- bi-variate quasilinearization method
1 Introduction
The flow of non-Newtonian fluids is applied in many situations in industry such as processing of materials and chemical engineering. These fluids show different characteristics from the Newtonian fluids which cannot be fully represented by the Navier-Stokes equations. To represent these non-Newtonian fluids some modifications to the Navier-Stokes equations are used and these are seen in many research works which studied viscoelastic and micropolar fluids [1, 2]. These fluids are categorized as viscoelastic, thixotropic, and power-law fluids. The constitutive equations of such fluids cannot fully represent the actual behavior of these fluids. These fluids include contaminated lubricants, molten metal, synovial fluids, etc.
The study of radiation effects on MHD free convection of a Casson fluid in porous medium with partial slip is an important aspect due to its practical application in the design of automatic cooking machines and the design of internal engine parts in mechanical engineering. Other examples arise in petroleum production, multiphase mixtures, pharmaceutical formulations, coal in water, paints, lubricants, jams, sewage, soup, blood. There has been a significant improvement in the study of non-Newtonian fluids in which many different situations have been considered. Studies in a Casson fluid include work by among others Mukhopadhyay et al. [3] and Nadeem et al. [4].
Several research workers have studied radiation effects, these include Kameswaran et al. [5] who studied radiation effects on hydromagnetic Newtonian liquid flow due to an exponentially stretching sheet. They studied radiation effects in the presence of a magnetic field, advancing the studies of radiation effects in Newtonian fluids. Shateyi and Marewo [6] investigated numerical analysis of MHD stagnation point flow of a Casson fluid, they considered thermal radiation in their work. Chamkha et al. [7] studied thermal radiation effects on MHD forced convection flow adjacent to a non-isothermal wedge in the presence of a heat source or sink. Pramanik [8] studied Casson fluid flow and heat transfer past an exponentially porous stretching surface in the presence of thermal radiation.
The study of fluid flow in the presence of a magnetic fluid has also been performed by many authors, among others Ece [9], who investigated free convection flow about a cone under mixed thermal boundary conditions and a magnetic field. Narayana et al. [10] studied free magnetohydrodynamic flow and convection from a vertical spinning cone with cross diffusion effects. Nadeem et al. [11] studied numerically MHD boundary layer flow of a Maxwell fluid past a stretching sheet in the presence of nanoparticles. Chen [12] investigated the combined heat and mass transfer in MHD free convection from a vertical surface with ohmic heating and viscous dissipation.
The study of fluid flow past a cylindrical geometry was performed by among others Anwa et al. [13], who studied mixed convection boundary layer flow of a viscoelastic fluid over a horizontal cylinder. Deka et al. [14] investigated transient free convection flow past an accelerated vertical cylinder in a rotating cylinder. Ribeiro et al. [15] studied viscoelastic flow past a confined cylinder with three dimensional effects and stability. Patel and Chhabra [16] studied steady flow of Bingham plastic fluids past an elliptical cylinder.
Studies in porous media and viscous dissipation have been carried out by among others, Awad et al. [17] who studied natural convection of viscoelastic fluid from a cone embedded in a porous medium with viscous dissipation. Awad et al. [17] investigated convection from an inverted cone in a porous medium with cross diffusion effects. Hayat et al. [18] studied heat and mass transfer for Soret and Dufour effects on mixed convection boundary layer flow over a stretching vertical surface in a porous medium filled with a viscoelastic fluid. Cheng [19] studied Soret and Dufour effects on free convection boundary layer over a vertical cylinder in a saturated porous medium. Chamkha and Rashad [20] investigated natural convection from a vertical permeable cone in nanofluid saturated porous media for uniform heat and nanoparticles volume fraction fluxes.
In this study we investigate the effects of radiation in MHD free convection of a Casson fluid from a horizontal circular cylinder with partial slip in non-Darcy porous medium with viscous dissipation. The surface of the cylinder is perforated in which we have the effects of transpiration which acts transversely in the direction ξ. The force which causes transpiration is sometimes called the Forchheimer drag force term \(-\xi\Lambda f^{\prime2}\), which appears in the momentum equation (12). This term is also associated with the geometry of the porous medium. In this work we extended the work of Ramachandra et al. [1] in which we introduced magnetic field, radiation effects and viscous dissipation. We also solved the system of the resulting partial differential equations by the bi-variate quasilinearization method (BQLM). The method is described in detail in Motsa et al. [21]. We study the effects of the magnetic field, radiation. and the Eckert number on velocity and temperature profiles for different values of the Casson parameter β, the local inertia (Forchheimer parameter) Λ, and the Darcian drag force coefficient \(\Lambda_{1}\). The results of this work are validated by comparison with other methods, which are the successive linearization method (SLM) and MATLAB’s routine ‘bvp4c’.
2 Mathematical formulation
3 Solution method
4 Results and discussion
Comparison of the values of the skin friction coefficient and heat transfer coefficient obtained by BQLM against the SLM and bvp4c for \(\pmb{\beta \rightarrow\infty}\) , \(\pmb{\Lambda=\Lambda_{1}=M=\xi =K=f_{w}= S_{f}= S_{T}=0}\)
Pr | SLM | bvp4c | Present | |||
---|---|---|---|---|---|---|
\(\boldsymbol{f''(0)}\) | \(\boldsymbol{-\theta'(0)}\) | \(\boldsymbol{f''(0)}\) | \(\boldsymbol{-\theta'(0)}\) | \(\boldsymbol{f''(0)}\) | \(\boldsymbol{-\theta'(0)}\) | |
1 | 0.87100777 | 0.42143140 | 0.81700776 | 0.42143144 | 0.81700776 | 0.42143144 |
10 | 0.54471433 | 0.88046306 | 0.54471423 | 0.88046307 | 0.54471422 | 0.88046307 |
Figure 2 shows the influence of Casson parameter β on the velocity and the temperature profiles at different values of the Forchheimer parameter Λ in the case of velocity profiles (see Figure 2(a)), and thermal slip factor \(S_{T}\) in the case of temperature profiles (see Figure 2(b)). Increasing the Casson parameter increases the velocity profiles close to the boundary, as \(\beta\rightarrow\infty\), \(\frac{1}{\beta}\rightarrow0\), the fluid becomes Newtonian. The same result is noted in Ramachandra et al. [1]. The difference in these results from those of Ramachandra et al. [1] is that there is a reverse effect that is noted further away from the boundary. This is due to the presence of the velocity slip factor which tends to assist flow at the boundary. Increasing the Forchheimer parameter reduce velocity profiles, this is caused by the transpiration effect taking place at the surface of the circle. Increasing the Casson parameter β reduces temperature profiles (see [1–3]). In the case of no thermal slip factor, higher temperature profiles are noticed and lower temperature profiles are noticed in the presence of the thermal slip factor, as shown in Figure 2(b).
5 Conclusion
The study presented in this analysis of effects of radiation on MHD free convection of a Casson fluid from a horizontal circular cylinder with partial slip in non-Darcy porous medium with viscous dissipation provides numerical solutions for the boundary layer flow and heat transfer. The coupled nonlinear governing partial differential equations were solved using the bi-variate quasilinearization method (BQLM) and validated by the successive linearization method (SLM) and MATLAB’s ‘bvp4c’. This paper also describes the BQLM numerical method which uses collocation methods in both directions η (direction of increasing boundary layer thickness) and ξ (radial transpiration direction). The most important results are those reflected in the presence of a magnetic field and viscous dissipation in a Casson fluid, which were never reported before.
Declarations
Acknowledgements
The authors would like to thank the University of KwaZulu-Natal, School of Mathematics, Statistics and Computer Science for the funding and support in the development of the paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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