# MHD Slip Flow of Casson Fluid along a Nonlinear Permeable Stretching Cylinder Saturated in a Porous Medium with Chemical Reaction, Viscous Dissipation, and Heat Generation/Absorption

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## Abstract

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## 1. Introduction

## 2. Mathematical Formulation

## 3. Results and Discussion

## 4. Conclusions

- The fluid velocity, temperature, and concentration are found to increase with $\gamma $.
- The magnitude of wall shear stress and mass transfer rate increase with the growth of $\beta $, whereas the heat transfer rate is enhanced.
- The effect of $M$ on fluid velocity is more pronounced when $K=0$ (nonporous medium).
- The temperature field is more influenced with increasing $Ec$ when $K\ne 0$.
- The velocity, temperature, and concentration distributions decrease when $S>0$, while the reverse trend is seen when $S<0$.
- The concentration boundary layer is observed to be thinner during destructive chemical reaction.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Crane, L.J. Flow past a stretching plate. Z. Angew. Math. Phys.
**1970**, 21, 645–647. [Google Scholar] [CrossRef] - Gupta, P.S.; Gupta, A.S. Heat and mass transfer on a stretching sheet with suction or blowing. Can. J. Chem. Eng.
**1977**, 55, 744–746. [Google Scholar] [CrossRef] - Chen, C.K.; Char, M.I. Heat transfer of a continuous, stretching surface with suction or blowing. J. Math. Anal. Appl.
**1988**, 135, 568–580. [Google Scholar] [CrossRef] - Gorla, I.; Sidawi, R.S.R. Free Convection on a Vertical Stretching Surface with Suction and Blowing. Appl. Sci. Res.
**1994**, 52, 247–257. [Google Scholar] [CrossRef] - Vajravelu, K. Viscous flow over a nonlinearly stretching sheet. Appl. Math. Comput.
**2001**, 124, 281–288. [Google Scholar] [CrossRef] - Vajravelu, K.; Cannon, J.R. Fluid flow over a nonlinearly stretching sheet. Appl. Math. Comput.
**2006**, 181, 609–618. [Google Scholar] [CrossRef] - Bachok, N.; Ishak, A. Flow and heat transfer over a stretching cylinder with prescribed surface heat flux. Malays. J. Math. Sci.
**2010**, 4, 159–169. [Google Scholar] - Hayat, T.; Saeed, Y.; Asad, S.; Alsaedi, A. Convective heat and mass transfer in flow by an inclined stretching cylinder. J. Mol. Liq.
**2016**, 220, 573–580. [Google Scholar] [CrossRef] - Majeed, A.; Zeeshan, A.; Alamri, S.Z.; Ellahi, R. Heat transfer analysis in ferromagnetic viscoelastic fluid flow over a stretching sheet with suction. Neur. Comp. Appl.
**2018**, 30, 1947–1955. [Google Scholar] [CrossRef] - Vyas, A.; Ranjan, P. Dissipative mhd boundary layer flow in a porous medium over a sheet stretching nonlinearly in the presence of radiation. Appl. Math. Sci.
**2010**, 4, 3133–3142. [Google Scholar] - Mukhopadhyay, S. MHD boundary layer flow along a stretching cylinder. Ain Shams Eng. J.
**2013**, 4, 317–324. [Google Scholar] [CrossRef] - Fathizadeh, M.; Madani, M.; Khan, Y.; Faraz, N.; Yildirim, A.; Tutkun, S. An effective modification of the homotopy perturbation method for MHD viscous flow over a stretching sheet. J. King Saud Univ. Sci.
**2013**, 25, 107–113. [Google Scholar] [CrossRef] - Akbar, N.S.; Ebaid, A.; Khan, Z.H. Numerical analysis of magnetic field effects on Eyring-Powell fluid flow towards a stretching sheet. J. Magn. Magn. Mater.
**2015**, 382, 355–358. [Google Scholar] [CrossRef] - Ellahi, R. The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: Analytical solutions. Appl. Math. Model.
**2013**, 37, 1451–1467. [Google Scholar] [CrossRef] - Fang, T.; Zhang, J.; Yao, S. Slip MHD viscous flow over a stretching sheet—An exact solution. Commun. Nonlinear Sci. Numer. Simul.
**2009**, 14, 3731–3737. [Google Scholar] [CrossRef] - Bhattacharyya, K.; Mukhopadhyay, S.; Layek, G.C. Slip effects on an unsteady boundary layer stagnation-point flow and heat transfer towards a stretching sheet. Chin. Phys. Lett.
**2011**, 28, 094702. [Google Scholar] [CrossRef] - Yazdi, M.H.; Abdullah, S.; Hashim, I.; Sopian, K. Slip MHD liquid flow and heat transfer over non-linear permeable stretching surface with chemical reaction. Int. J. Heat Mass Transf.
**2011**, 54, 3214–3225. [Google Scholar] [CrossRef] - Hayat, T.; Qasim, M.; Mesloub, S. MHD flow and heat transfer over permeable stretching sheet with slip conditions. Int. J. Numer. Methods Fluids
**2011**, 66, 963–975. [Google Scholar] [CrossRef] - Seini, I.Y.; Makinde, O.D. Boundary layer flow near stagnation-points on a vertical surface with slip in the presence of transverse magnetic field. Int. J. Numer. Methods Heat Fluid Flow
**2014**, 24, 643–653. [Google Scholar] [CrossRef] - Rahman, S.U.; Ellahi, R.; Nadeem, S.; Zia, Q.Z. Simultaneous effects of nanoparticles and slip on Jeffrey fluid through tapered artery with mild stenosis. J. Mol. Liq.
**2016**, 218, 484–493. [Google Scholar] [CrossRef] - Casson, N. A Flow Equation for Pigment-Oil Suspensions of the Printing Ink Type; Mill, C.C., Ed.; Rheol. Disperse Syst. Pergamon Press: Oxford, UK, 1959; pp. 84–104. [Google Scholar]
- Nandy, S.K. Analytical Solution of MHD Stagnation-Point Flow and Heat Transfer of Casson Fluid over a Stretching Sheet with. Thermodynamics
**2013**. [Google Scholar] [CrossRef] - Singh, S. Clinical significance of aspirin on blood flow through stenotic blood vessels. J. Biomim. Biomater. Tissue Eng.
**2011**, 10, 17–24. [Google Scholar] - Mukhopadhyay, S.; Bhattacharyya, K.; Hayat, T. Exact solutions for the flow of Casson fluid over a stretching surface with transpiration and heat transfer effects. Chin. Phys. B
**2013**, 22, 114701. [Google Scholar] [CrossRef] - Shawky, H.M. Magnetohydrodynamic Casson fluid flow with heat and mass transfer through a porous medium over a stretching sheet. J. Porous Media
**2012**, 15, 393–401. [Google Scholar] [CrossRef] - Mukhopadhyay, S. Casson fluid flow and heat transfer over a nonlinearly stretching surface. Chin. Phys. B
**2013**, 22, 074701. [Google Scholar] [CrossRef] - Medikare, M.; Joga, S.; Chidem, K.K. MHD Stagnation Point Flow of a Casson Fluid over a Nonlinearly Stretching Sheet with Viscous Dissipation. Am. J. Comput. Math.
**2016**, 37–48. [Google Scholar] [CrossRef] - Mythili, D.; Sivaraj, R. Influence of higher order chemical reaction and non-uniform heat source/sink on Casson fluid flow over a vertical cone and flat plate. J. Mol. Liq.
**2016**, 216, 466–475. [Google Scholar] [CrossRef] - Ullah, I.; Khan, I.; Shafie, S. Hydromagnetic Falkner-Skan flow of Casson fluid past a moving wedge with heat transfer. Alex. Eng. J.
**2016**, 55, 2139–2148. [Google Scholar] [CrossRef] - Imtiaz, M.; Hayat, T.; Alsaedi, A. Mixed convection flow of Casson nanofluid over a stretching cylinder with convective boundary conditions. Adv. Powder Technol.
**2016**, 27, 2245–2256. [Google Scholar] [CrossRef] - Cebeci, T.; Bradshaw, P. Physical and Computational Aspects of Convective Heat Transfer, 1st ed.; Springer: New York, NY, USA, 1988. [Google Scholar]

**Figure 2.**Effect of curvature parameter $\gamma $ on velocity for linear $(n=1)$ and nonlinear $n=10$ stretching parameter.

**Figure 3.**Effect of Casson fluid parameter $\beta $ on velocity profile for different values of suction/blowing parameter $S$.

**Figure 4.**Effect of nonlinear stretching parameter $n$ on velocity profile in the presence and absence of magnetic parameter $M$.

**Figure 5.**Effect of magnetic parameter $M$ on velocity profile in the presence and absence of porosity parameter $K$.

**Figure 6.**Effect of porosity parameter $K$ on velocity profile for different values of suction/blowing parameter $S$.

**Figure 7.**Effect of Grashof number $Gr$ on velocity profile in the presence and absence of magnetic parameter $M$.

**Figure 8.**Effect of mass Grashof number $Gm$ on velocity profile in the presence and absence of magnetic parameter $M$.

**Figure 9.**Effect of suction/blowing parameter $S$ on velocity profile for nonlinear stretching parameter $n$.

**Figure 10.**Effect of slip parameter $\delta $ on velocity profile in the presence and absence of porosity parameter $K$.

**Figure 11.**Effect of curvature parameter $\gamma $ on temperature profile for Newtonian fluid $\beta =\infty $ and Casson fluid $\beta =0.6$.

**Figure 12.**Effect of Casson fluid parameter $\beta $ on temperature profile for different values of suction/blowing parameter $S$.

**Figure 13.**Effect of nonlinear stretching parameter $n$ on temperature profile in the presence and absence of magnetic parameter $M$.

**Figure 14.**Effect of magnetic parameter $M$ on temperature profile for different values of suction/blowing parameter $S$.

**Figure 15.**Effect of porosity parameter $K$ on temperature profile in the presence and absence of slip parameter $\delta $.

**Figure 16.**Effect of suction/blowing parameter $S$ on temperature profile for different values of nonlinear stretching parameter $n$.

**Figure 17.**Effect of Prandtl number $\mathrm{Pr}$ on temperature profile in the presence and absence of Eckert number $Ec$.

**Figure 18.**Effect of radiation parameter ${R}_{d}$ on temperature profile for different values of suction/blowing parameter $S$.

**Figure 19.**Effect of Eckert number $Ec$ on temperature profile in the presence and absence of porosity parameter $K$.

**Figure 20.**Effect of heat generation/absorption parameter $\epsilon $ on temperature profile in the presence and absence of magnetic parameter $M$.

**Figure 21.**Effect of Biot number $B{i}_{1}$ on temperature profile in the presence and absence of porosity parameter $K$.

**Figure 22.**Effect of curvature parameter $\gamma $ on concentration profile for Newtonian fluid $\beta =\infty $ and Casson fluid $\beta =0.6$.

**Figure 23.**Effect of Casson parameter $\beta $ on concentration profile in the presence and absence of magnetic parameter $M$.

**Figure 24.**Effect of nonlinear stretching parameter $n$ on concentration profile for different values of suction/blowing parameter $S$.

**Figure 25.**Effect of magnetic parameter $M$ on concentration profile in the presence and absence of porosity parameter $K$.

**Figure 26.**Effect of porosity parameter $K$ on concentration profile for different values of suction/blowing parameter $S$.

**Figure 27.**Effect of slip parameter $\delta $ on concentration profile in the presence and absence of porosity parameter $K$.

**Figure 28.**Effect of suction/blowing parameter $S$ on concentration profile in the presence and absence of magnetic parameter $M$.

**Figure 29.**Effect of Schmidt number $Sc$ on concentration profile for Newtonian fluid $\beta =\infty $ and Casson fluid $\beta =0.6$.

**Figure 30.**Effect of chemical reaction parameter $R$ on concentration profile for different values of suction/blowing parameter $S$.

**Figure 31.**Effect of Biot number $B{i}_{1}$ on concentration profile in the presence and absence of magnetic parameter $M$.

**Figure 32.**Variation of skin friction coefficient for various values of Casson fluid parameter $\beta $, curvature parameter $\gamma $, and magnetic parameter $M$.

**Figure 33.**Variation of skin friction coefficient for various values of nonlinear stretching parameter $n$, porosity parameter $K$, and suction/blowing parameter $S$.

**Figure 34.**Variation of Nusselt number for various values of Casson parameter $\beta $, curvature parameter $\gamma $, and Eckert number $Ec$.

**Figure 35.**Variation of Sherwood number for various values of Casson fluid parameter $\beta $, curvature parameter $\gamma $, and chemical reaction parameter $R$.

**Table 1.**Comparison of skin friction coefficient ${f}^{\u2033}(0)$ for different values of $M$ with $\beta \to \infty $, $B{i}_{1}\to \infty $, $B{i}_{2}\to \infty $, $n=1$ and $\gamma =M=K=Gr=Gm=S=\delta ={R}_{d}=Ec=\epsilon =R=0$.

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**MDPI and ACS Style**

Ullah, I.; Alkanhal, T.A.; Shafie, S.; Nisar, K.S.; Khan, I.; Makinde, O.D.
MHD Slip Flow of Casson Fluid along a Nonlinear Permeable Stretching Cylinder Saturated in a Porous Medium with Chemical Reaction, Viscous Dissipation, and Heat Generation/Absorption. *Symmetry* **2019**, *11*, 531.
https://doi.org/10.3390/sym11040531

**AMA Style**

Ullah I, Alkanhal TA, Shafie S, Nisar KS, Khan I, Makinde OD.
MHD Slip Flow of Casson Fluid along a Nonlinear Permeable Stretching Cylinder Saturated in a Porous Medium with Chemical Reaction, Viscous Dissipation, and Heat Generation/Absorption. *Symmetry*. 2019; 11(4):531.
https://doi.org/10.3390/sym11040531

**Chicago/Turabian Style**

Ullah, Imran, Tawfeeq Abdullah Alkanhal, Sharidan Shafie, Kottakkaran Sooppy Nisar, Ilyas Khan, and Oluwole Daniel Makinde.
2019. "MHD Slip Flow of Casson Fluid along a Nonlinear Permeable Stretching Cylinder Saturated in a Porous Medium with Chemical Reaction, Viscous Dissipation, and Heat Generation/Absorption" *Symmetry* 11, no. 4: 531.
https://doi.org/10.3390/sym11040531