^{1}

^{2}

^{3}

The effect of radiation on flow and heat transfer over a vertically oscillating porous flat plate embedded in porous medium with oscillating surface temperature is investigated. The analytic solutions of momentum and energy equations are obtained. The velocity and temperature profiles are computed. The frictional force at the plate due to viscosity of fluid is estimated in terms of non dimensional skin friction coefficient and heat convection at the plate is estimated in the form of Nusselt number. The effects of physical parameters Prandtl number Pr, Grashof number Gr, Suction parameter S and radiative parameter R on velocity and temperature profiles are analyzed through graphs. The effects of oscillation on the velocity and temperature profiles are shown through 3-D surface plot.

The radiative free convective flow has many important applications in countless industrial and environment processes e.g. heating and cooling chambers, fossil fuel combustion energy processes, evaporation from large open water reservoirs, astrophysical flows, solar power technology and space vehicle re-entry. The radiative heat transfer plays an important role in manufacturing industries for the design of reliable equipment. Nuclear power plants, gas turbines and various propulsion devices for aircraft, missiles and satellites are examples of such engineering applications.

Magneto hydrodynamic free convection flow past an infinite vertical plate oscillating in its own plane was first studied by Soundalgekar [

The aim of the present study is to investigate the radiative effect on flow and heat transfer over a vertically oscillating porous flat plate with oscillating surface temperature.

To study free convective flow and heat transfer through a vertical porous flate plate in the influence of radiative heat flux is considered (

variation along x-axis will be negligible as compared to the variation along y-axis so

In view of the physical description the governing equations are defined as follows:

The equation of continuity

Þv, is independent of y and

The momentum equation for the prescribed geometry is given by

Under usual Boussinesq’s approximation the Equation (2) becomes

The energy equation is given by

The associated boundary conditions are

where u the flow velocity component in the x-direction, ν the kinematic viscosity, g the acceleration due to gravity, β the volumetric expansion coefficient, α the thermal diffusivity,

The Roseland approximation for radiative heat flux [Brewster, 1992] is given by

where σ is Stefan ? Boltzmann constant and

Taking the Taylor series expansion of T^{4} and neglecting terms with higher powers, we have

Introducing following dimensionless quantities

Equations (3), (4) and the associated boundary conditions (5) reduces into

where,

The corresponding boundary conditions are

There is no loss of generality in omitting the asterisk from (6) to (8).

To solve the coupled nonlinear partial differential Equation (6) and (7) along with the boundary conditions (8), the solution for u and

where

Invoking the Equations (9) and (10) in the Equations (6) and (7) and equating harmonic and non harmonic terms, the set of ordinary differential equations are given as

where, primes denote derivative with respect to y

The corresponding boundary conditions are

The Equations (13) and (14) are ordinary differential equations with prescribed boundary conditions as given in (15) and (16), therefore their solutions are straight forward are given by

where

Now using expression of

In view of the boundary conditions associated with

where

Now substituting expression of

The skin-friction at the plate, which in the non-dimensional form is given by

and computed values are given in

The non dimensional coefficient of heat transfer defined by Nusselt number is obtained and given by

and computed values are given in

The velocity profiles for different parameter like Suction parameter, Grashof number, magnetic parameter, Darcy number, Prandtl number and radiation parameter are shown by Figures 2-7. Temperature profiles are also shown by Figures 8-10. In

S | Gr | M | Da | Pr | R | C_{f} |
---|---|---|---|---|---|---|

0.0 | 5 | 5 | 0.1 | 1 | 2 | 15.9195 |

0.2 | 5 | 5 | 0.1 | 1 | 2 | 16.2228 |

0.5 | 5 | 5 | 0.1 | 1 | 2 | 16.6869 |

−0.2 | 5 | 5 | 0.1 | 1 | 2 | 15.6222 |

−0.5 | 5 | 5 | 0.1 | 1 | 2 | 15.1854 |

0.5 | 2 | 5 | 0.1 | 1 | 2 | 3.9841 |

0.5 | 10 | 5 | 0.1 | 1 | 2 | 37.8582 |

0.5 | −2 | 5 | 0.1 | 1 | 2 | −12.9529 |

0.5 | −5 | 5 | 0.1 | 1 | 2 | −25.6557 |

0.5 | −10 | 5 | 0.1 | 1 | 2 | −46.8270 |

0.5 | 5 | 3 | 0.1 | 1 | 2 | 11.2156 |

0.5 | 5 | 8 | 0.1 | 1 | 2 | 26.3903 |

0.5 | 5 | 5 | 0.01 | 1 | 2 | 35.8229 |

0.5 | 5 | 5 | 1 | 1 | 2 | 13.7920 |

0.5 | 5 | 5 | 0.1 | 0.5 | 2 | 8.7570 |

0.5 | 5 | 5 | 0.1 | 0.71 | 2 | 11.0189 |

0.5 | 5 | 5 | 0.1 | 1 | 4 | 28.0977 |

0.5 | 5 | 5 | 0.1 | 1 | 6 | 39.5676 |

S | Pr | R | Nu |
---|---|---|---|

0.0 | 1 | 2 | 0.0004 |

0.2 | 1 | 2 | 0.0757 |

0.5 | 1 | 2 | 0.2058 |

−0.2 | 1 | 2 | −0.0657 |

−0.5 | 1 | 2 | −0.1479 |

0.5 | 0.5 | 2 | 0.2174 |

0.5 | 0.71 | 2 | 0.2110 |

0.5 | 1 | 4 | 0.2028 |

0.5 | 1 | 6 | 0.2018 |

R is increased, velocity increases in the vicinity of the permeable plate while decreases in region close to non permeable wall. In

The values of Nusselt number is given in

・ Fluid flow slows down in the vicinity of the permeable plate while enhances in region close to nonpermeable

wall on increasing suction parameter correspond to cooling of the plate.

・ Fluid velocity profiles increase in the vicinity of the permeable plate while decrease in region close to non permeable wall with the increase in Grashof number correspond to cooling of the plate

・ Fluid velocity profiles increase in the vicinity of the permeable plate while decrease in region close to non permeable wall when Prandtl number and radiation parameter is increased.

・ Fluid velocity and temperature in the porous medium through the one period of oscillation oscillates up to a certain distance from the plate and this oscillation disappears far away from the plate.

・ The values of skin friction increase when magnetic parameter, Prandtl number and radiation parameter are increased while the values of skin friction decrease when Darcy number is increased.

・ Nusselt number decreases when Prandtl number and radiation parameter are increased.

Monika Miglani,Net Ram Garg,Mukesh Kumar Sharma, (2016) Radiative Effect on Flow and Heat Transfer over a Vertically Oscillating Porous Flat Plate Embedded in Porous Medium with Oscillating Surface Temperature. Open Journal of Fluid Dynamics,06,119-129. doi: 10.4236/ojfd.2016.62010