^{1}

^{*}

^{1}

^{*}

The unsteady two-dimensional, laminar flow of a viscous, incompressible, electrically conducting fluid towards a shrinking surface in the presence of a uniform transverse magnetic field is studied. Taking suitable similarity variables, the governing boundary layer equations are transformed to ordinary differential equations and solved numerically by a perturbation technique for a small magnetic parameter. The effects of various parameters such as unsteadiness parameter, velocity parameter, magnetic parameter, Prandtl number and Eckert number for velocity and temperature distributions along with local Skin friction coefficient and local Nusselt number have been discussed in detail through numerical and graphical representations.

Stagnation flow of an incompressible viscous fluid over a shrinking sheet has many important practical applications in engineering and industrial processes, such as the extrusion of a polymer in a melt-spinning process, continuous casting of metals, the aerodynamic extrusion of plastic sheets, the cooling of metallic sheets or electronic chips and many others. In all these cases, a study of fluid flow and heat transfer has important significance because the quality of the final product depends on the rate of cooling and the process of stretching.

In recent years, the boundary layer flow due to a shrinking sheet has attraction of many researchers because of its useful applications. A very interesting example in which the shrinking sheet situation occurs is of a rising shrinking balloon. Shrinking film is also a common application of shrinking sheet problems in engineering and industries. Shrinking film is very useful in packaging of bulk products because it can be unwrapped easily with adequate heat.

From the stretching case, the flow of shrinking sheet is different and the fluid is attracted towards a slot. Physically, the generated velocity at shrinking sheet has an unsteady flow due to the application of inadequate suction and is not confined within the boundary layer.

In view of all these applications, Sakiadis [

Realizing the increasing technical applications of the magnetohydrodynamic effects, the aim of the present work is concerned with a steady, two-dimensional unsteady stagnation flow of an electrically conducting fluid over a shrinking surface in the presence of a uniform transverse magnetic field.

Consider an unsteady two-dimensional steady flow

highly elastic and is shrinking in the x-direction with a velocity is

where

bility,

The boundary conditions are

where c is a constant,

The continuity Equation (1) is identically satisfied by stream function

For the solution of the momentum and the energy Equations (2) and (3), the following dimensionless variables are defined:

Equations (5) to (8), transform Equations (2) and (3) into

where a prime (') denotes differentiation with respect to

The corresponding boundary conditions are

For numerical solution of the Equations (9) and (10), through a perturbation technique, by assuming the following power series in a small magnetic parameter

Substituting Equations (12) and (13) and its derivatives in Equations (9) and (10) and then equating the coefficients of like powers of

with the boundary conditions

The Equation (14) is obtained by Mahapatra and Nandy [

The physical quantities of interest, the local skin friction coefficient

and

which, in the present case can be expressed in the following forms

and

where

Numerical values of the functions

Re_{m} = 0.0 | Re_{m} = 0.5 | Re_{m} = 0.7 | Re_{m} = 0.0 | Re_{m} = 0.5 | Re_{m} = 0.7 | Re_{m} = 0.0 | Re_{m} = 0.5 | Re_{m} = 0.7 | |

−0.1 | 1.3288 | 1.3164 | 1.3104 | 1.2041 | 1.1955 | 1.1937 | 1.0623 | 1.0586 | 1.0512 |

−0.8 | 1.2614 | 1.2406 | 1.2251 | 1.1985 | 1.1805 | 1.1801 | 1.0210 | 1.0190 | 1.0070 |

−0.9 | 1.2516 | 1.2303 | 1.2207 | 1.1803 | 1.1795 | 1.1780 | 1.0108 | 1.0099 | 1.0047 |

−1.0 | 1.2305 | 1.2285 | 1.2178 | 1.1776 | 1.1706 | 1.1701 | 1.0098 | 1.0058 | 1.0015 |

α | Pr | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Re_{m} = 0.0 | Re_{m} = 0.5 | Re_{m} = 0.7 | Re_{m} = 0.0 | Re_{m} = 0.5 | Re_{m} = 0.7 | Re_{m} = 0.0 | Re_{m} = 0.5 | Re_{m} = 0.7 | ||

−1.0 | 0.05 | 0.1493 | 0.1530 | 0.1590 | 0.1493 | 0.1533 | 0.1595 | 0.1493 | 0.1549 | 0.1610 |

0.50 | 0.2259 | 0.2440 | 0.2646 | 0.2259 | 0.2443 | 0.2653 | 0.2259 | 0.2448 | 0.2663 | |

1.00 | 0.2227 | 0.2450 | 0.2710 | 0.2227 | 0.2455 | 0.2720 | 0.2227 | 0.2462 | 0.2736 | |

2.00 | 0.1815 | 0.2016 | 0.2203 | 0.1815 | 0.2106 | 0.2227 | 0.1815 | 0.2157 | 0.2270 | |

−0.9 | 0.50 | 0.2571 | 0.2705 | 0.2937 | 0.2571 | 0.2712 | 0.2948 | 0.2571 | 0.2719 | 0.2805 |

−0.8 | 0.50 | 0.2813 | 0.2993 | 0.3017 | 0.2813 | 0.2999 | 0.3024 | 0.2813 | 0.3011 | 0.3021 |

α | Pr | |||||||||

Re_{m} = 0.0 | Re_{m} = 0.5 | Re_{m} = 0.7 | Re_{m} = 0.0 | Re_{m} = 0.5 | Re_{m} = 0.7 | Re_{m} = 0.0 | Re_{m} = 0.5 | Re_{m} = 0.7 | ||

−1.0 | 0.05 | 0.1511 | 0.1562 | 0.1620 | 0.1511 | 0.1563 | 0.1622 | 0.1511 | 0.1563 | 0.1623 |

0.50 | 0.2330 | 0.2520 | 0.2741 | 0.2330 | 0.2523 | 0.2748 | 0.2330 | 0.2528 | 0.2758 | |

1.00 | 0.2322 | 0.2559 | 0.2842 | 0.2322 | 0.2563 | 0.2852 | 0.2322 | 0.2570 | 0.2866 | |

2.00 | 0.1919 | 0.2194 | 0.2536 | 0.1919 | 0.2201 | 0.2553 | 0.1919 | 0.2213 | 0.2578 | |

−0.9 | 0.50 | 0.2661 | 0.2842 | 0.3040 | 0.2661 | 0.2845 | 0.3046 | 0.2661 | 0.2849 | 0.3056 |

−0.8 | 0.50 | 0.2939 | 0.3102 | 0.3273 | 0.2939 | 0.3105 | 0.3279 | 0.2939 | 0.3109 | 0.3288 |

α | Pr | |||||||||

Re_{m} = 0.0 | Re_{m} = 0.5 | Re_{m} = 0.7 | Re_{m} = 0.0 | Re_{m} = 0.5 | Re_{m} = 0.7 | Re_{m} = 0.0 | Re_{m} = 0.5 | Re_{m} = 0.7 | ||

−1.0 | 0.05 | 0.1517 | 0.1576 | 0.1623 | 0.1517 | 0.1585 | 0.1637 | 0.1517 | 0.1595 | 0.1643 |

0.50 | 0.2413 | 0.2623 | 0.2895 | 0.2413 | 0.2653 | 0.2705 | 0.2413 | 0.2693 | 0.2917 | |

1.00 | 0.2385 | 0.2639 | 0.2957 | 0.2385 | 0.2644 | 0.2966 | 0.2385 | 0.2650 | 0.2980 | |

2.00 | 0.2217 | 0.2415 | 0.2713 | 0.2217 | 0.2503 | 0.2802 | 0.2217 | 0.2505 | 0.2817 | |

−0.9 | 0.50 | 0.2951 | 0.3105 | 0.3317 | 0.2951 | 0.3101 | 0.3320 | 0.2951 | 0.3140 | 0.3303 |

−0.8 | 0.50 | 0.3216 | 0.3401 | 0.3625 | 0.3216 | 0.3501 | 0.3703 | 0.3216 | 0.3501 | 0.3701 |

_{m}. It may be observed that, for the fixed value of the velocity parameter α velocity distribution increases with the decreasing value of the unsteadiness parameter β, and opposite phenomenon occur for the magnetic parameter Re_{m}, for a fixed η.

_{m}, the Prandtl number Pr and the Eckert number Ec. From these figures it may be observed that the temperature distribution decreases with increasing values of the unsteadiness parameter β, the velocity parameter α, the magnetic parameter Re_{m}, the Prandtl number Pr and the Eckert number Ec.

In _{m} are given. It may be observed from the table that the boundary values _{m} respectively taking other parameters constant and reverse phenomenon occurs for the velocity parameter α.

In _{m}, the Prandtl number Pr and the Eckert number Ec are given. It may be observed from the table that the boundary values _{m}, the Prandtl number

The present work extends the two-dimensional unsteady stagnation flow of an electrically conducting fluid, over shrinking surface in the presence of magnetic field. Under some special conditions, the problem will reduce the results obtained by previous researchers. The effects of different parameters such as the unsteadiness parameter, the velocity parameter, the magnetic parameter, the Prandtl number and the Eckert number are studied in detail. The velocity as well as thermal boundary layer thickness decreases with the increasing values of the unsteadiness parameter, the velocity parameter, the magnetic parameter, the Prandtl number and the Eckert number whereas in the velocity reverse phenomenon occurs for the magnetic parameter. From the results it can be concluded that skin friction and Nusselt number vary according to the velocity and thermal boundary layers thickness respectively with different parameters.

SantoshChaudhary,PradeepKumar, (2015) Unsteady Magnetohydrodynamic Boundary Layer Flow near the Stagnation Point towards a Shrinking Surface. Journal of Applied Mathematics and Physics,03,921-930. doi: 10.4236/jamp.2015.37112