A numerical model is developed to examine the effects of thermal radiation on unsteady mixed convection flow of a viscous dissipating incompressible micropolar fluid adjacent to a heated vertical stretching surface in the presence of the buoyancy force and heat generation/absorption. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The model contains nonlinear coupled partial differential equations which have been converted into ordinary differential equation by using the similarity transformations. The dimensionless governing equations for this investigation are solved by Runge-Kutta-Fehlberg fourth fifth-order method with shooting technique. Numerical solutions are then obtained and investigated in detail for different interesting parameters such as the local skin-friction coefficient, wall couple stress, and Nusselt number as well as other parametric values such as the velocity, angular velocity, and temperature.

The micro polar fluids are those which contain microconstituents that can undergo rotation, the presence of which can affect the hydrodynamics of the flow. The classical Navier-Stokes theory does not describe the flow properties of micropolar fluids, for example, colloidal suspension, polymeric fluids, liquid crystals, fluids with additives, suspension solutions, animal’s blood, human blood, body fluids, biofluids, and fluids containing certain additives. Eringen [

Unsteady mixed convection flow plays an important role in chemical engineering, turbomachinery, aerospace technology, geophysics, and so forth; Zueco et al. [

The heat transfer in the fluid flow due to a stretching sheet has attracted considerable attention during the last few decades due to its various applications in many industrial and engineering processes such as hot rolling, wire drawing, glass-fiber and paper production, drawing of plastic films, metal and polymer extrusion, and metal spinning. The pioneering work in this area was first made by Crane [

Influence of thermal radiation on flow and heat transfer study is much more important in different industries. The heat transfer and temperature profile of a fluid flow over different geometries can be affected significantly at high temperature. Mohamed and Abo-Dahab [

The effect of heat generation on heat transfer is an important issue in view of various physical problems. Ziabakhsh et al. [

The purpose of the present work is to study the effects of thermal radiation on mixed convection flow of a micropolar fluid through an unsteady stretching surface with viscous dissipation and heat generation/absorption. This problem is important in the processing of chemical engineering fluids including polymeric suspensions, lubricant manufacture, and so forth. The nonlinearity of basic equations associated with their inherent mathematical difficulties has led us to use numerical method. Thus the transformed dimensionless pertinent equations are solved numerically by using the Runge-Kutta-Fehlberg fourth fifth-order method along with shooting technique. The velocity, angular velocity, and temperature profiles are shown and the influences of the micropolar parameter, the thermal radiation parameter, the unsteadiness parameter, and the buoyancy parameter on the flow and heat transfer characteristics are discussed in detail. To the best of author’s knowledge such a study does not appear in the scientific literature.

We consider a two-dimensional unsteady mixed convection boundary layer flow of a viscous incompressible micropolar fluid over an elastic, vertical, and impermeable stretching sheet which emerges vertically in the upward direction from a narrow slot with velocity:

Physical model and coordinate system.

The appropriate boundary conditions for the problem are

Using Roseland’s approximation, the radiative heat flux

It is worth mentioning that, for

The most important physical quantities for the problem are the local skin-friction coefficient

The nonlinear differential equations (

The domain of the problem is discretized and the boundary conditions for

In order to validate the numerical results obtained, we compare our results with those reported by Abd El-Aziz [

Comparison of

| | Abd El-Aziz [ | Present results |
---|---|---|---|

0 | 0 | 1.00000000 | 1.00000000 |

0 | 1 | 1.08727816 | 1.08727818 |

0 | 2 | 1.14233927 | 1.14233928 |

0 | 3 | 1.18529030 | 1.18529032 |

1 | 0 | 1.68199254 | 1.68199255 |

1 | 1 | 1.70391279 | 1.70391281 |

Figures

Velocity profiles for various values of

Microrotation profiles for various values of

Temperature profiles for various values of

Figures

Velocity profiles for various values of

Microrotation profiles for various values of

Temperature profiles for various values of

Figure

Temperature profiles for various values of

Figures

Velocity profiles for various values of

Microrotation profiles for various values of

Temperature profiles for various values of

Figures

Velocity profiles for various values of

Microrotation profiles for various values of

Temperature profiles for various values of

The effects of the unsteadiness parameter

Computed values of

| | | | | | |
---|---|---|---|---|---|---|

0 | 0 | 0.2 | 0.5 | −0.91248 | 0.027431 | 0.36217 |

0.2 | 0 | 0.2 | 0.5 | −0.97460 | 0.024750 | 0.45577 |

0.4 | 0 | 0.2 | 0.5 | −1.03540 | 0.022970 | 0.53131 |

0.2 | −0.2 | 0.2 | 0.5 | −1.16166 | 0.034717 | 0.38816 |

0.2 | −0.1 | 0.2 | 0.5 | −1.04170 | 0.026600 | 0.44245 |

0.2 | 0.05 | 0.2 | 0.5 | −0.94316 | 0.023959 | 0.46128 |

0.2 | 0.1 | 0.2 | 0.5 | −0.91260 | 0.023225 | 0.46628 |

0 | 0 | 0 | 0.5 | −1.08705 | 0 | 0.243760 |

0 | 0 | 0.1 | 0.5 | −1.04570 | 0.026701 | 0.240340 |

0 | 0 | 0.3 | 0.5 | −0.97566 | 0.072558 | 0.232590 |

0 | 0 | 0.2 | 0.2 | −0.91248 | 0.027431 | 0.251050 |

0 | 0 | 0.2 | 1 | −0.91248 | 0.027431 | 0.475040 |

The present work deals with the numerical analysis of thermal radiation effects of a mixed convection flow over an unsteady stretching surface. Fluid is a micropolar fluid in the presence of viscous dissipation and heat generation/absorption. The relevant nonlinear partial differential equations were transformed to a set of ordinary differential equations and then are solved numerically using the Runge-Kutta-Fehlberg fourth fifth-order method along with shooting technique. Conclusions drawn from the numerical results are as follows:

Temperature reduces with increase in the values of the unsteadiness parameter, buoyancy parameter, and thermal radiation parameter.

The skin-friction is enhanced with an increase in the values of the micropolar parameter and buoyancy parameter, while it decreases with the increase of unsteadiness parameter.

The increasing value of the micropolar parameter is to increase the couple stress, whereas increasing unsteadiness and buoyancy parameter decreases the couple stress.

The rate of heat transfer increases with the unsteadiness parameter, buoyancy parameter, and thermal radiation parameter; however rate of heat transfer decreases with increasing micropolar parameter and thermal radiation parameter.

Unsteadiness parameter

Constant [m^{−1}]

Microinertia density parameter

Constant [m^{−1}

Coefficient of local skin-friction

Specific heat at constant pressure [J kg^{−1} K^{−1}]

Eckert number

Heat generation/absorption parameter

Acceleration due to gravity [m s^{−2}]

Microinertia density [m^{2}]

Micropolar parameter

Thermal conductivity of the fluid [W m^{−1} K^{−1}]

Absorption coefficient

Coefficient of local wall couple stress

Component of microrotation [rad s^{−1}]

Nusselt number

Prandtl number

Radiative heat flux

Thermal radiation parameter

Local Reynolds number

Temperature [K]

Velocities in ^{−1}]

Axial and perpendicular coordinates [m].

Stream function

Dynamic viscosity

Stefan–Boltzmann constant

Volumetric coefficient of the thermal expansion [K^{−1}]

Spin-gradient viscosity [N s]

Nondimensional distance

Vortex viscosity

Kinematic viscosity [m^{2} s^{−1}]

Fluid density [kg m^{−3}]

Nondimensional temperature

Heat source/sink coefficient

Spin gradient viscosity parameter

Mixed convection or buoyancy parameter.

Free stream condition

Condition at the wall of stretching sheet.

Derivative with respect to

The authors declare that there is no conflict of interests regarding the publication of this paper. The authors also confirm that the mentioned received funding in Acknowledgments did not lead to any conflict of interests regarding the publication of this manuscript.

The authors wish to express their very sincere thanks to honorable referees for their valuable comments and suggestions for the improvement of the manuscript. The first author gratefully acknowledges the financial support of UGC, India, under F. 17-97/2008 (SA-I), for pursuing this work.