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This paper examines the effects of thermocapillarity and thermal radiation on the boundary layer flow and heat transfer in a thin film on an unsteady stretching sheet with nonuniform heat source/sink. The governing partial differential equations are converted into ordinary differential equations by a similarity transformation and then are solved by using the homotopy analysis method (HAM). The effects of the radiation parameter, the thermocapillarity number, and the temperature-dependent parameter in this study are discussed and presented graphically via velocity and temperature profiles.

The analysis of heat transfer of boundary layer flow with thermal radiation is important in electrical power generation, astrophysical flows, solar power technology, space vehicle reentry, and other industrial engineering processes.

Wang [

Wang [

Motivated by these studies, in this paper we extend the results of Wang [

The fluid flow, modeled as an unsteady, two-dimensional, incompressible viscous laminar flow on a horizontal thin elastic sheet, emerges from a narrow slot at the origin of a Cartesian coordinate system. Fluid motion and heat transfer arise in the stretching of the horizontal elastic sheet. It is assumed that the elastic sheet has internal heat generation/absorption and that the sheet temperature varies with the coordinate

In this section we apply HAM to solve system (

We note the presence of the auxiliary parameters

(a)

The residual errors for the solution using 15th-order HAM approximation for the case of

We solved (

In order to assess the accuracy of the numerical method, Table

Values of dimensionless film thickness

Wang [ | Abel et al. [ | Mahmoud [ | Present work | |||||

0.4 | 5.122490 | −1.307785 | 4.981455 | −1.134098 | — | — | 5.126821 | −1.040765 |

0.6 | 3.131250 | −1.195155 | 3.131710 | −1.195128 | — | — | 3.131668 | −1.193654 |

0.8 | 2.151990 | −1.245795 | 2.151990 | −1.245805 | 2.1519950 | −1.245810 | 2.151994 | −1.245793 |

1.0 | 1.543617 | −1.277762 | 1.543617 | −1.277769 | — | — | 1.543616 | −1.277768 |

1.2 | 1.127780 | −1.279177 | 1.127780 | −1.279171 | 1.1277815 | −1.279170 | 1.127780 | −1.279172 |

1.4 | 0.821032 | −1.233549 | 0.821033 | −1.233545 | — | — | 0.821032 | −1.233549 |

1.6 | 0.567173 | −1.114937 | 0.576176 | −1.114941 | — | — | 0.576173 | −1.114937 |

1.8 | 0.356389 | −0.867414 | 0.356390 | −0.867416 | — | — | 0.356389 | −0.867414 |

In Table

Values of free surface temperature

Mahmoud [ | Present work | ||

0 | 0.2 | 0.86681 | 0.866844 |

0.1 | 0.2 | 0.85184 | 0.854257 |

0.5 | 0.2 | 0.78815 | 0.810103 |

0.1 | 0 | 0.82693 | 0.829606 |

0.1 | 0.2 | 0.85184 | 0.854257 |

0.1 | 1.0 | 0.90594 | 0.907657 |

The effects of the radiation parameter

Variations of

0 | 1.008941 | 1.004460 | −1.201179 | 0.307714 | 1.841899 |

0.5 | 1.103655 | 1.050550 | −1.252414 | 0.396179 | 1.512100 |

1.0 | 1.173616 | 1.083335 | −1.289824 | 0.461740 | 1.299809 |

1.5 | 1.227240 | 1.107809 | −1.318863 | 0.513259 | 1.147157 |

2.0 | 1.270251 | 1.127054 | −1.341533 | 0.554268 | 1.031198 |

Table

Variations of

0 | 0.674094 | 0.821032 | −1.012774 | 0.618739 | 0.873492 |

0.5 | 0.969091 | 0.984424 | −1.172420 | 0.519700 | 1.140371 |

1.0 | 1.173616 | 1.083335 | −1.299824 | 0.461740 | 1.299809 |

1.5 | 1.350054 | 1.161918 | −1.382159 | 0.419534 | 1.425835 |

2.0 | 1.499933 | 1.224717 | −1.459757 | 0.388086 | 1.524753 |

The heat absorption sink

Variations of

−1.0 | 1.108464 | 1.052836 | −1.254129 | 0.399346 | 1.485058 |

−0.5 | 1.137235 | 1.066412 | −1.270027 | 0.426965 | 1.401209 |

0 | 1.170067 | 1.081696 | −1.287906 | 0.458354 | 1.309486 |

0.5 | 1.206136 | 1.099152 | −1.308352 | 0.494630 | 1.207863 |

1.0 | 1.253028 | 1.119387 | −1.332192 | 0.537395 | 1.093403 |

The effects of the different values of radiation parameter

Effects of the radiation parameter

Thermocapillarity produced an outward flow along the free surface. Figure

Effects of

Figure

Effects of

The unsteadiness parameter

Effects of

Effects of

The effects of thermal radiation and thermocapillarity in a thin liquid film on an unsteady stretching sheet with nonuniform heat source/sink was analyzed successfully by means of the homotopy analysis method (HAM). With the presence of internal heat generation/absorption, the radiation parameter plays a significant role in controlling the temperature of the fluid flow by enhancing the temperature of fluid flow and decreasing the dimensionless heat flux. The thermocapillarity enhances the velocity and the dimensionless heat flux while reducing the temperature and the surface shear stress.