# Mathematical Modeling of Mixed Convection Boundary Layer Flows over a Stretching Sheet with Viscous Dissipation in Presence of Suction and Injection

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations of the System

## 3. Formulation of the Governing Principle of Dissipative Processes (GPDP)

## 4. Method of Solution

## 5. Analysis of Results and Discussions

## 6. Conclusions

- The governing non linear partial differential equations are converted into coupled polynomial equations whose solutions are quite straight forward.
- The solutions have been computed for the non-dimensional skin friction, heat transfer, velocity and thermal boundary layers for various values of $Pr$, $\lambda $, H and $Ec$.
- The skin friction values increase with the increase in $a/c$ in the absence of buoyancy, suction and injection parameters.
- The heat transfer values increase with the increase in $Pr$ for both assisting and opposing flows.
- The skin friction values decrease with increasing $Pr$ for assisting flow and increase with the increase in $Pr$ for opposing flows during $a/c=1$, $\lambda =1$, $H=0$ and $Ec=0$.
- For small values of viscous dissipation, suction and injection, the buoyancy assisting flow increases the skin friction and opposing flow decreases the skin friction.
- During suction and injection, the heat transfer values increase with the increase in $a/c$ for both flows when $a/c>1$. When $a/c<1$, the heat transfer decreases with the decrease in $a/c$ for both flows.
- For small values of viscous dissipation, suction and injection, when $a/c>1$, the flow has a definite boundary layer structure and the flow has an inverted boundary layer when $a/c<1$.
- The advantage of the present variational principle is that the results have been arrived at directly without sacrificing the accuracy of the results.
- The comparison of the present solution with numerical solutions reveals that the accuracy of the present variational technique is acceptable for the applications of engineering and technology.
- The method of solution exhibited in this analysis has further advantage for obtaining analytic solution for the present problem.
- The amount of calculation time is also certainly less when compared with other numerical procedures.
- Hence it can be concluded that this variational principle is a unique approximate technique for solving various boundary layer flows and heat transfer problems.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Karwe, M.V.; Jaluria, Y. Numerical simulation of thermal transport associated with a continuously moving flat sheet in materials processing. ASME J. Heat Transf.
**1991**, 113, 612–619. [Google Scholar] [CrossRef] - Karwe, M.V.; Jaluria, Y. Fluid flow and mixed convection transport from a moving plate in rolling and extrusion processes. ASME J. Heat Transf.
**1998**, 110, 655–661. [Google Scholar] [CrossRef] - Sakiadis, B.C. Boundary layer behaviour on continuous solid surfaces I. Boundary layer equations for two dimensional and axisymmetric flow. AIChE J.
**1961**, 7, 26–28. [Google Scholar] [CrossRef] - Sakiadis, B.C. Boundary layer behaviour on continuous solid surfaces II. The boundary layer on a continuous flat surface. AIChE J.
**1961**, 7, 221–225. [Google Scholar] [CrossRef] - Tsou, F.K.; Sparrow, E.M.; Goldstein, R.J. Flow and heat transfer in the boundary layer on a continuous moving surface. Int. J. Heat Mass Transf.
**1967**, 10, 219–235. [Google Scholar] [CrossRef] - Crane, L.J. Flow past a stretching plate. Z. Angew. Math. Phys. ZAMP
**1970**, 21, 645–647. [Google Scholar] [CrossRef] - Brady, J.F.; Acrivos, A. Steady flow in a channel or tube with an accelerating surface velocity. An exact solution to the Navier-Stokes equations with reverse flow. J. Fluid Mech.
**1981**, 112, 127–150. [Google Scholar] [CrossRef] - Chiam, T.C. Stagnation-point flow towards a stretching plate. J. Phys. Soc. Jpn.
**1994**, 63, 2443–2444. [Google Scholar] [CrossRef] - Chamkha, A.J. Hydromagnetic three dimensional free convection on a vertical stretching sheet with heat generation or absorption. Int. J. Heat Fluid Flow
**1999**, 20, 84–92. [Google Scholar] [CrossRef] - Mahapatra, T.R.; Gupta, A.S. Heat transfer in stagnation point flow towards a stretching sheet. Heat Mass Transf.
**2002**, 38, 517–521. [Google Scholar] [CrossRef] - Mahapatra, T.R.; Gupta, A.S. Stagnation point flow of a viscoelastic fluid towards a stretching surface. Int. J. Non Linear Mech.
**2004**, 39, 811–820. [Google Scholar] [CrossRef] - Nazar, R.; Amin, N.; Filip, D.; Pop, I. Unsteady boundary layer flow in the region of the stagnation point on a stretching sheet. Int. J. Eng. Sci.
**2004**, 42, 1241–1253. [Google Scholar] [CrossRef] [Green Version] - Ishak, A.; Nazar, R.; Pop, I. Mixed convection boundary layers in the stagnation-point flow toward a stretching vertical sheet. Meccanica
**2006**, 41, 509–518. [Google Scholar] [CrossRef] [Green Version] - Ishak, A.; Nazar, R.; Pop, I. Mixed convection on the stagnation point flow towards a vertical continuously stretching sheet. J. Heat Transf.
**2007**, 129, 1087–1090. [Google Scholar] [CrossRef] - Patil, P.M.; Roy, S.; Chamkha, A.J. Mixed convection flow over a vertical power-law stretching sheet. Int. J. Numer. Methods Heat Fluid Flow
**2010**, 20, 445–458. [Google Scholar] [CrossRef] - Zaimi, K.; Ishak, A. Stagnation-point flow towards a stretching vertical sheet with slip effects. Mathematics
**2016**, 4, 27. [Google Scholar] [CrossRef] - Gyarmati, I. On the governing principle of dissipative processes and its extension to non-linear problems. Ann. Phys.
**1969**, 23, 353–378. [Google Scholar] [CrossRef] - Gyarmati, I. Non-Equilibrium Thermodynamics, Field Theory and Variational Principles; Springer: Berlin, Germany, 1970. [Google Scholar]
- Onsager, L. Reciprocal relations in irreversible processes-I. Phys. Rev.
**1931**, 37, 405. [Google Scholar] [CrossRef] - Onsager, L. Reciprocal relations in irreversible processes-II. Phys. Rev.
**1931**, 38, 2265. [Google Scholar] [CrossRef] [Green Version] - Chandrasekar, M.; Kasiviswanathan, M.S. Analysis of Heat and mass transfer on MHD flow of a Nanofluid pass a stretching sheet. Procedia Eng.
**2015**, 127, 493–500. [Google Scholar] [CrossRef] [Green Version] - Chandrasekar, M.; Kasiviswanathan, M.S. Magneto hydrodynamic flow with viscous dissipation effects in the presence of suction and injection. J. Theor. Appl. Mech.
**2015**, 53, 93–107. [Google Scholar] [CrossRef] [Green Version] - Chandrasekar, M.; Kasiviswanathan, M.S. Variational approach to MHD stagnation flow of nanofluid towards permeable stretching sheet. Int. J. Heat Technol.
**2018**, 36, 411–421. [Google Scholar] [CrossRef]

**Figure 3.**Local Nusselt number. (

**a**) Values of Nusselt number as a function of $\lambda $ for various values of $a/c$ when $Pr=6.8$, $Ec=0$ and $H=0$. (

**b**) Values of Nusselt number as a function of $\lambda $ for various values of $a/c$ when $Pr=6.8$, $Ec=0$ and $H=-0.5$. (

**c**) Values of Nusselt number as a function of $\lambda $ for various values of $a/c$ when $Pr=6.8$, $Ec=0$ and $H=0.5$. (

**d**) Values of Nusselt number as a function of $\lambda $ for various values of $Pr$ when $a/c=1$, $Ec=0$ and $H=0$.

**Figure 5.**Temperature profiles. (

**a**) Temperature profile as a function of $\eta $ for various values of $a/c$ when $Pr=6.8$, $\lambda =1$, $Ec=0$ and $H=0$. (

**b**) Temperature profile as a function of $\eta $ for various values of $a/c$ when $Pr=6.8$, $\lambda =1$, $Ec=0$ and $H=-0.5$. (

**c**) Temperature profile as a function of $\eta $ for various values of $a/c$ when $Pr=6.8$, $\lambda =1$, $Ec=0$ and $H=0.5$.

**Table 1.**Comparison of skin friction values (${\tau}_{w}$) for various values of $a/c$ when $\lambda =0$ and $H=0$ with known results.

$\mathit{a}/\mathit{c}$ | Mahapatra and Gupta [10] | Nazar [12] | Ishak [13] | Present Values |
---|---|---|---|---|

0.1 | $-0.9694$ | $-0.9694$ | $-0.9694$ | $-1.0020185155$ |

0.2 | $-0.9181$ | $-0.9181$ | $-0.9181$ | $-0.9415583926$ |

0.5 | $-0.6673$ | $-0.6673$ | $-0.6673$ | $-0.6750132565$ |

2.0 | 2.0175 | 2.0176 | 2.0175 | 2.0030860233 |

3.0 | 4.7293 | 4.7296 | 4.7294 | 4.6788471978 |

**Table 2.**Comparison of skin friction (${\tau}_{w}$) and heat transfer ($N{u}_{l}$) for assisting flow when $a/c=1$, $\lambda =1$, $Ec=0$ and $H=0$ for various $Pr$ with known results.

$\mathit{Pr}$ | Skin Friction (${\mathit{\tau}}_{\mathit{w}}$) | Heat Transfer (${\mathit{Nu}}_{\mathit{l}}$) | ||||
---|---|---|---|---|---|---|

Ishak [13] | Zaimi and Ishak [16] | Present Values | Ishak [13] | Zaimi and Ishak [16] | Present Values | |

0.72 | 0.3645 | 0.36449 | 0.3603630329 | 1.0931 | 1.09310 | 1.1465484808 |

6.8 | 0.1804 | 0.18041 | 0.2034171744 | 3.2902 | 3.28957 | 3.2776601900 |

10 | 0.15563 | 0.1677420992 | 3.98240 | 3.9747468140 | ||

20 | 0.1175 | 0.11750 | 0.1186115759 | 5.6230 | 5.62013 | 5.6211408516 |

30 | 0.09889 | 0.0968459461 | 6.87771 | 6.8844634292 | ||

40 | 0.0873 | 0.08724 | 0.0838710496 | 7.9463 | 7.93830 | 7.9494936288 |

50 | 0.07903 | 0.0750165473 | 8.87292 | 8.8878040696 | ||

60 | 0.0729 | 0.07284 | 0.0684804252 | 9.7327 | 9.71801 | 9.7361015508 |

70 | 0.06794 | 0.0634005541 | 10.49524 | 10.5161915945 | ||

80 | 0.0640 | 0.06394 | 0.0593057879 | 11.2413 | 11.21874 | 11.2422817008 |

90 | 0.06059 | 0.0559140330 | 11.89831 | 11.9242404422 | ||

100 | 0.0578 | 0.05772 | 0.0530447092 | 12.5726 | 12.54109 | 12.5692530540 |

**Table 3.**Comparison of skin friction (${\tau}_{w}$) and heat transfer ($N{u}_{l}$) for opposing flow when $a/c=1$, $\lambda =1$, $Ec=0$ and $H=0$ for various $Pr$ with known results.

$\mathit{Pr}$ | Skin Friction (${\mathit{\tau}}_{\mathit{w}}$) | Heat Transfer (${\mathit{Nu}}_{\mathit{l}}$) | ||||
---|---|---|---|---|---|---|

Ishak [13] | Zaimi and Ishak [16] | Present Values | Ishak [13] | Zaimi and Ishak [16] | Present Values | |

0.72 | $-0.3852$ | $-0.38518$ | $-0.3603630329$ | 1.0293 | 1.02925 | 1.1465484808 |

6.8 | $-0.1832$ | $-0.18323$ | $-0.2034171744$ | 3.2466 | 3.24608 | 3.2776601900 |

10 | $-0.15747$ | $-0.1677420992$ | 3.94370 | 3.9747468140 | ||

20 | $-0.1183$ | $-0.11831$ | $-0.1186115759$ | 5.5923 | 5.58959 | 5.6211408516 |

30 | $-0.09938$ | $-0.0968459461$ | 6.85149 | 6.8844634292 | ||

40 | $-0.0876$ | $-0.08758$ | $-0.0838710496$ | 7.9227 | 7.91489 | 7.9494936287 |

50 | $-0.07929$ | $-0.0750165473$ | 8.85153 | 8.8878040696 | ||

60 | $-0.0731$ | $-0.07304$ | $-0.0684804252$ | 9.7126 | 9.69818 | 9.7361015508 |

70 | $-0.06810$ | $-0.0634005541$ | 10.47665 | 10.5161915945 | ||

80 | $-0.0642$ | $-0.06407$ | $-0.0593057879$ | 11.2235 | 11.20117 | 11.2422817008 |

90 | $-0.06070$ | $-0.0559140330$ | 11.88161 | 11.9242404422 | ||

100 | $-0.0579$ | $-0.05782$ | $-0.0530447092$ | 12.5564 | 12.52515 | 12.5692530540 |

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**MDPI and ACS Style**

Muthukumaran, C.; Bathrinathan, K.
Mathematical Modeling of Mixed Convection Boundary Layer Flows over a Stretching Sheet with Viscous Dissipation in Presence of Suction and Injection. *Symmetry* **2020**, *12*, 1754.
https://doi.org/10.3390/sym12111754

**AMA Style**

Muthukumaran C, Bathrinathan K.
Mathematical Modeling of Mixed Convection Boundary Layer Flows over a Stretching Sheet with Viscous Dissipation in Presence of Suction and Injection. *Symmetry*. 2020; 12(11):1754.
https://doi.org/10.3390/sym12111754

**Chicago/Turabian Style**

Muthukumaran, Chandrasekar, and Kalidoss Bathrinathan.
2020. "Mathematical Modeling of Mixed Convection Boundary Layer Flows over a Stretching Sheet with Viscous Dissipation in Presence of Suction and Injection" *Symmetry* 12, no. 11: 1754.
https://doi.org/10.3390/sym12111754