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The aim of this investigation is to analyze the effectiveness of Lorentz force, viscous dissipation and internal heating on the heat and flow characteristics of a non-Newtonian Casson fluid thin film resting on a stretching surface under the influence of a magnetic field. Employing suitable similarity variables and shooting technique and integrating scheme numerical solutions for velocity and temperature are obtained. The results of this analysis are compared with the published work and are found to be in good agreement. The thickness of the thin film is evaluated and is observed that Lorentz force and the non-Newtonian nature of the fluid have a thinning influence on the film. Velocity and temperature distributions in the thin film are discussed for various flow parameters.

During the recent years, study of liquid film flows has received significant attention from researchers due to its possible practical applications in several branches of science and technology. The characteristics of flow and heat transfer in a thin film are extremely useful to understand the coating process, design of different heat exchangers and chemical processing equipments. Similar situations prevail in wire and fiber coating, transpiration cooling, food stuff processing, reactor fluidization, manufacture of plastic and rubber sheets. Primary objective of all applications of extrusion is to maintain the surface quality of the final extrudate as best. It is reported that quality of the product depends on rate of heat transfer at the stretching surface. Hence, study of heat transfer under different thermal conditions shall be helpful to industry. Rassoulinejad-Mousavi and Abbasbandy [

The effect of viscous dissipation in heat transfer problems is of importance as it plays significant role as an energy source. The presence of viscous dissipation affects both the temperature and heat transfer rates. Rassoulinejad-Mousavi and Hessameddin [

In industry free surface flows of non-Newtonian fluids in thin films are frequently encountered in industry especially in polymer and plastic fabrication and in coating equipment. It is well known that the process of protective coatings on an extrudate use paints or fluids that are generally non-Newtonian in nature. Studies on film flow of power-law fluids have been examined by several researchers [

However, studies on flows in Casson fluid film are very limited in literature. Casson fluid is a non-Newtonian fluid that can be described as a shear thinning fluid having an infinite viscosity at zero shear rates and a yield stress with no flow below this critical yield value. The empirical model was suggested by Casson [

In this paper, influence of magnetic field on the thin film flow and heat transfer of a Casson fluid over an unsteady stretching sheet is investigated. The effects of viscous dissipation and internal heating are taken into consideration. Application of suitable similarity variables reduced the governing PDEs into a set of coupled non-linear ODEs for which numerical solutions have been obtained. To the best of knowledge of the authors, influence of transverse magnetic field, viscous dissipation and internal heating on the flow and heat transfer of a Casson fluid film over an unsteady stretching has not yet been studied in literature.

Consider a conducting incompressible non-Newtonian Casson thin film over a heated elastic sheet that emerges from a narrow slit at the origin of the Cartesian coordinate system for investigations as shown schematically in

The continuous sheet is parallel to x-axis and moves in its own plane with a velocity

where

where

stant reference temperature such that

The constitutive of the Casson fluid can be written as [

where

on non-Newtonian model.

Under these assumptions, the motion of liquid film due to stretching is governed by

where

The term Q is heat generation (>0) or absorption (<0) per unit volume which is modeled as [

where

The boundary conditions on the stretching sheet are no slip, no penetration and imposed sheet temperature distribution and they are represented respectively as

The following similarity transformation are introduced

Also,

where prime denotes differentiation with respect to

Subject to the boundary conditions

where

Since

Thus, the kinematic constraint at

The parameters of engineering interest in heat transfer problems are Skin friction coefficient

where

The coupled ordinary differential Equations (13) and (14) are non linear and exact analytical solutions are not possible. Equations (13) and (14) with the pertinent boundary conditions (15) and (16) are solved numerically by the most efficient numerical shooting technique with fourth order Runge-Kutta algorithm. First we convert these equations into a set of first order equations as follows:

Corresponding boundary conditions take the form,

Here

value of ^{−6}. Accuracy of the present scheme is ensured by comparing the present results, viz., non dimensional thickness of the film

The coupled non-linear differential Equations (13) and (14) along with the appropriate boundary conditions (15) and (16) are numerically solved employing the BVP4C technique. In order to have a physical insight of the flow in the thin film, numerical computations of flow variables for various sets of governing parameters have been carried out and graphically presented.

Variation of film thickness versus magnetic field parameter for different values of the unsteadiness parameter is plotted in

Behavior of free surface velocity

S | Wang [ | Abel et al. [ | Megahed [ | Present study | ||||
---|---|---|---|---|---|---|---|---|

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 | 5.122490 3.131250 2.151990 2.543620 1.127780 0821032 0.576173 0.356389 | −1.307785 −1.195155 −1.245795 −1.277762 −1.279177 −1.233549 −1.491137 −0.867414 | 4.981455 3.131710 2.151990 1.543617 1.127780 0.821033 0.576176 0.356390 | −1.134098 −1.195128 −1.245805 −1.277769 −1.279171 −1.233545 −1.114937 −0.867416 | 4.98145 3.131710 2.151994 1.543616 1.127781 0.821032 0.576173 0.356389 | −1.134096 −1.195126 −1.245806 −1.277769 −1.279172 −1.233545 −1.114938 −0.867414 | 4.981455 3.131710 2.151990 1.543617 1.127780 0.821033 0.576176 0.356390 | −1.134098 −1.195128 −1.245805 −1.277769 −1.279171 −1.233545 −1.114937 −0.867416 |

unsteadiness parameter is highlighted in

variation of unsteadiness parameter S. It is observed that velocity distribution in the film decreases monotonically for small values of unsteadiness parameter. As the unsteadiness parameter takes higher values fluids gets accelerated and hence higher velocities occur. Films become thinner for increasing values of unsteadiness parameter. When unsteadiness parameter S = 1.4, film thickness is reduced by two and half times than that of the film corresponding to S = 0.8. Fluid in the film gets cooled as unsteadiness parameter increases. However, reduction in the temperature is small.

on the surface. For higher values of Pr, temperature falls rapidly near the boundary. When

indicate that increasing values of

in the presence of heat sink

Surface drag coefficient and Nusselt number on the surface for different variations of

the flow parameters are tabulated in

In this investigation, effect of viscous dissipation and velocity slip on the characteristics of flow and heat transfer in a thin Casson liquid film resting on an unsteady stretching sheet under the influence of a uniform transverse magnetic field and thermal radiation is studied. The partial differential equations governing the flow and energy are transformed into a set of ordinary differential equations by employing appropriate similarity transformations. These ODEs are later solved numerically using shooting technique along with the efficient Runge-Kutta method. Results of the present analysis are validated with available data in literature and are found to be in excellent agreement. The following are some of the salient features of the investigation:

Ø Film thickness is found to decrease as unsteady parameter increases. Qualitatively

S | M | |||
---|---|---|---|---|

0.8 1.0 1.2 1.4 | 0.5 | 0.5 | −2.473527 −2.504972 −2.478698 −2.365328 | 1.505621 1.592773 1.675480 1.743821 |

0.8 | 1.0 2.0 3.0 4.0 | 0.5 | −1.577477 −1.585218 −1.591939 −1.596655 | 1.449664 1.437879 1.429346 1.423580 |

0.8 | 0.5 | 0.0 1.0 2.0 3.0 | −2.157798 −2.752908 −3.239791 −3.662303 | 1.539927 1.474249 1.416704 1.363561 |

Pr | Ec | ||
---|---|---|---|

0.7 1.0 2.0 3.0 | 0.1 | 0.5 | 1.165108 1.505621 2.298035 2.882486 |

1.0 | 0.0 1.0 2.0 3.0 | 0.5 | 1.569157 0.933798 0.298439 −0.336921 |

1.0 | 0.1 | −1.0 −0.5 0.5 1.0 | 1.826436 1.675676 1.505621 1.301671 |

similar behavior is noticed for increasing values of magnetic field and Casson parameters.

Ø Free surface velocity is found to be a decreasing function of the magnetic field parameter.

Ø Temperature distribution in the thin film is enhanced with increase in magnetic field strength and non-Newtonian nature of fluid.

Ø Viscous heating enhances rate of heat transfer.

The authors express their sincere thanks to the editor and referee for their valuable comment and suggestions, which lead to a significant improvement of the paper.

Vijaya, N., Sreelakshmi, K. and Sarojamma, G. (2016) Effect of Magnetic Field on the Flow and Heat Transfer in a Casson Thin Film on an Unsteady Stretching Surface in the Presence of Viscous and Internal Heating. Open Journal of Fluid Dynamics, 6, 303-320. http://dx.doi.org/10.4236/ojfd.2016.64023