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In this article, we have effectively used the Numerical Inversion of Laplace transform to study the time-dependent thin film flow of a second grade fluid flowing down an inclined plane through a porous medium. The solution to the governing equation is obtained by using the standard Laplace transform. However, to transform the obtained solutions from Laplace space back to the original space, we have used the Numerical Inversion of Laplace transform. Graphical results have been presented to show the effects of different parameters involved and to show how the fluid flow evolves with time.

Laplace transform is a very useful tool for solving the differential equations. However, to analytically compute the inverse Laplace transform of the solutions obtained by the use of Laplace transform is a very important but complicated step. To overcome this issue, several algorithms for numerical inversion of Laplace transform have been proposed in literature [1-4]. Here we implement the idea of numerical inversion of Laplace transform presented by Weeks [

The inverse Laplace transform

Here

Whereas, the method of Juraj and Lumboir [

Non-Newtonian fluids have been a famous topic of research because of their diverse use in many industrial processes. Various complex fluids such as oils, polymer melts, different types of drilling muds and clay coatings and many emulsions are included in the category of non-Newtonian fluids. One of the very important models suggested for non-Newtonian fluids is called the secondgrade fluid. Flows of second grade fluid have been studied by many [5-9]. Also, the flow of thin films has vast applications in industry. Such flows have applications in microchip productions, in biology, in chemistry and many other fields. Very often, thin film flows are examined by using steady flows. Some investigators have recently obtained some results for thin film flows of nonNewtonian fluids [10-14]. An investigation of thin film flow of a second order fluid is made by Huang and Li [

The momentum balance equation in the presence of a body force

with

where

The material constants

due to Clausius-Duhem inequality and the condition that the Helmholtz free energy is minimum when the fluid is at rest [

We consider an unsteady gravity driven thin film flow of a second grade fluid of uniform film thickness

Symbols

Here we would like to point out the presence of

This gives us the following

where

Taking the Laplace transform of Equations (11)-(12), we get, after rearranging

with

Equations (13)-(14) have the following solution in the Laplace space

Now we use the method of numerical inversion of the Laplace transform [1,2] for the solution given in Equation (15) to give us the following plots.

The dimensionless velocity field for an unsteady thin film flow of a second grade fluid is plotted in

We have shown an efficient application of the numerical inverse Laplace algorithms [1,2] to study an unsteady thin film flow problem of a second grade fluid which is flowing through a porous medium along an inclined plane. This shows that the numerical inversion of the Laplace transform is a very effective and useful technique. Many unsteady problems of fluid flows, which are very hard to solve otherwise, can be dealt easily by the use of the Laplace transform. To accurately convert the solutions back to the original space, we can make use of

these efficient algorithms [1,2] available for the numerical inversion of the Laplace transform.

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