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.

1. Introduction

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  and the one by Juraj and Lubomir .

The inverse Laplace transform of a function is given by the following contour integral

Here and is greater than the real part of any singularity in the transformed function G(s). Weeks method  for numerical inversion of Laplce transform is based on the use of the Laguerre functions and is given as Whereas, the method of Juraj and Lumboir  is based on the use of infinite series for and along with the application of residue theorems to compute the contour integral given in (1). It gives us the following expression, which is also implemented here, for the numerical inversion of Laplace transform 2. Basic Equations

The momentum balance equation in the presence of a body force and the equation of conservation of mass are respectively given as

with being the density, the velocity, the stress tensor and represents the material derivative. The stress tensor in Equation (2) for a second grade fluid has form

where is the identity tensor, is the pressure, is the dynamic viscosity, is the elastic coefficient and is the transverse viscosity coefficient and and represent the Rivlin-Ericksen tensors. The tensors and are defined by the following expressions

The material constants, and also satisfy

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

3. Governing Equations

We consider an unsteady gravity driven thin film flow of a second grade fluid of uniform film thickness flowing over a semi-infinite plate, inclined at an angle with the horizontal, through a porous medium, with stationary ambient air and negligible surface tension. The velocity field thus becomes. Using such a velocity field in Equations (2)-(7) yields the following governing equation

Symbols and, respectively, represent the porosity and permeability of the porous space. The associated boundary conditions are

Here we would like to point out the presence of in Equation (8). For time independent velocity field, the presence of the fluid parameter is completely wiped out from the governing equation. The first boundary condition in Equation (9) is due to the no-slip assumption at the boundary. Now considering the following nondimensional variables

This gives us the following

where and. The boundary conditions take the form

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

with and. Also, the transformed boundary conditions are

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.

4. Results and Discussion

The dimensionless velocity field for an unsteady thin film flow of a second grade fluid is plotted in Figure 1. The values of different parameters used are and It is evident from these graphical results that the fluid velocity increases as the time progresses. Figure 2 provides us with the behaviour of the velocity profile due to changing. The values of different parameters involved are and. A drop in the value of velocity is observed due to an increase of.

Figure 3, which depicts the change in velocity profile, and, due to changing porosity also shows a decrease in velocity for an increase in.

5. Conclusion

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.

[<xref ref-type="bibr" rid="scirp.42658-ref1">1</xref>] REFERENCES

 W. T. Weeks, Journal of the ACM, Vol. 13, 1966, pp. 419-429. http://dx.doi.org/10.1145/321341.321351

 J. Valsa and L. Brancik, International Journal of Numerical Modelling, Vol. 11, 1998, pp. 153-166. http://dx.doi.org/10.1002/(SICI)1099-1204(199805/06)11:3<153::AID-JNM299>3.0.CO;2-C

 F. R. de Hoog, J. H. Knight and A. N. Stokes, SIAM Journal on Scientific Computing, Vol. 3, 1982, pp. 357-366.

 A. Talbot, Journal of Applied Mathematics, Vol. 23, 1979, pp. 97-120.

 B. Raftari, F. Parvaneh and K. Vajravelu, Energy, Vol. 59, 2013, pp. 625-632. http://dx.doi.org/10.1016/j.energy.2013.07.054

 Y. D. Yao and Y. H. Liu, Nonlinear Analysis: Real World Applications, Vol. 11, 2010, pp. 4442-4450. http://dx.doi.org/10.1016/j.nonrwa.2010.05.027

 M. E. Erdogan and C. E. Imrak, International Journal of Engineering Science, Vol. 48, 2010, pp. 1225-1232. http://dx.doi.org/10.1016/j.ijengsci.2010.06.007

 M. E. Erdogan and C. E. Imrak, International Journal of Non-Linear Mechanics, Vol. 46, 2011, pp. 986-989. http://dx.doi.org/10.1016/j.ijnonlinmec.2011.04.013

  M. E. Erdogan and C. E. Imrak, Applied Mathematical Modelling, Vol. 31, 2007, pp. 170-180. http://dx.doi.org/10.1016/j.apm.2005.08.019

  P. Huang and Z. Li, Journal of Tribology, Vol. 124, 2002, pp. 547-552. http://dx.doi.org/10.1115/1.1467636

  S. Miladinova, G. Lebon and E. Toshev, Journal of NonNewtonian Fluid Mechanics, Vol. 122, 2004, pp. 69-78. http://dx.doi.org/10.1016/j.jnnfm.2004.01.021

  R. Zhang and X. Li, Journal of Engineering Mathematics, Vol. 51, 2005, pp. 1-13. http://dx.doi.org/10.1007/s10665-004-1342-z

  A. M. Siddiqui, R. Mahmood and Q. K. Ghori, Physics Letters A, Vol. 352, 2006, pp. 404-410. http://dx.doi.org/10.1016/j.physleta.2005.12.033

  A. M. Siddiqui, M. Ahmed and Q. K. Ghori, Chaos Solitons Fractals, Vol. 33, 2007, pp. 1006-1016. http://dx.doi.org/10.1016/j.chaos.2006.01.101

  P. Huang and Z. Li, Journal of Tribology, Vol. 124, 2002, pp. 547-552. http://dx.doi.org/10.1115/1.1467636

  Q. Z. Gan and L. Z. Ji, Applied Mathematics and Mechanics, Vol. 8, 2006, pp. 655-665. http://dx.doi.org/10.1007/BF02458263

  C. I. Chen, Communications in Nonlinear Science and Numerical Simulation, Vol. 12, 2007, pp. 760-775. http://dx.doi.org/10.1016/j.cnsns.2005.04.010

  M. Sajid, M. Awais, S. Nadeem and T. Hayat, Computers and Mathematics with Applications, Vol. 56, 2008, pp. 2019-2026. http://dx.doi.org/10.1016/j.camwa.2008.04.022

  S. Nadeem and M. Awais, Physics Letters A, Vol. 372, 2008, pp. 4965-4972. http://dx.doi.org/10.1016/j.physleta.2008.05.048

  S. Nadeem and M. Awais, Journal of Porous Media, Vol. 13, 2010, pp. 973-980. http://dx.doi.org/10.1615/JPorMedia.v13.i11.30

  J. E. Dunn and R. L. Fosdick, Archive for Rational Mechanics and Analysis, Vol. 56, 1974, pp. 191-252. http://dx.doi.org/10.1007/BF00280970

NOTESReferencesW. T. Weeks, Journal of the ACM, Vol. 13, 1966, pp. 419-429. http://dx.doi.org/10.1145/321341.321351J. Valsa and L. Brancik, International Journal of Numerical Modelling, Vol. 11, 1998, pp. 153-166. http://dx.doi.org/10.1002/(SICI)1099-1204(199805/06)11:3<153::AID-JNM299>3.0.CO;2-CF. R. de Hoog, J. H. Knight and A. N. Stokes, SIAM Journal on Scientific Computing, Vol. 3, 1982, pp. 357-366.A. Talbot, Journal of Applied Mathematics, Vol. 23, 1979, pp. 97-120.B. Raftari, F. Parvaneh and K. Vajravelu, Energy, Vol. 59, 2013, pp. 625-632. http://dx.doi.org/10.1016/j.energy.2013.07.054Y. D. Yao and Y. H. Liu, Nonlinear Analysis: Real World Applications, Vol. 11, 2010, pp. 4442-4450. http://dx.doi.org/10.1016/j.nonrwa.2010.05.027M. E. Erdogan and C. E. Imrak, International Journal of Engineering Science, Vol. 48, 2010, pp. 1225-1232. http://dx.doi.org/10.1016/j.ijengsci.2010.06.007M. E. Erdogan and C. E. Imrak, International Journal of Non-Linear Mechanics, Vol. 46, 2011, pp. 986-989. http://dx.doi.org/10.1016/j.ijnonlinmec.2011.04.013M. E. Erdogan and C. E. Imrak, Applied Mathematical Modelling, Vol. 31, 2007, pp. 170-180. http://dx.doi.org/10.1016/j.apm.2005.08.019P. Huang and Z. Li, Journal of Tribology, Vol. 124, 2002, pp. 547-552. http://dx.doi.org/10.1115/1.1467636S. Miladinova, G. Lebon and E. Toshev, Journal of Non-Newtonian Fluid Mechanics, Vol. 122, 2004, pp. 69-78. http://dx.doi.org/10.1016/j.jnnfm.2004.01.021R. Zhang and X. Li, Journal of Engineering Mathematics, Vol. 51, 2005, pp. 1-13. http://dx.doi.org/10.1007/s10665-004-1342-zA. M. Siddiqui, R. Mahmood and Q. K. Ghori, Physics Letters A, Vol. 352, 2006, pp. 404-410. http://dx.doi.org/10.1016/j.physleta.2005.12.033A. M. Siddiqui, M. Ahmed and Q. K. Ghori, Chaos Solitons Fractals, Vol. 33, 2007, pp. 1006-1016. http://dx.doi.org/10.1016/j.chaos.2006.01.101P. Huang and Z. Li, Journal of Tribology, Vol. 124, 2002, pp. 547-552. http://dx.doi.org/10.1115/1.1467636Q. Z. Gan and L. Z. Ji, Applied Mathematics and Mechanics, Vol. 8, 2006, pp. 655-665. http://dx.doi.org/10.1007/BF02458263C. I. Chen, Communications in Nonlinear Science and Numerical Simulation, Vol. 12, 2007, pp. 760-775. http://dx.doi.org/10.1016/j.cnsns.2005.04.010M. Sajid, M. Awais, S. Nadeem and T. Hayat, Computers and Mathematics with Applications, Vol. 56, 2008, pp. 2019-2026. http://dx.doi.org/10.1016/j.camwa.2008.04.022S. Nadeem and M. Awais, Physics Letters A, Vol. 372, 2008, pp. 4965-4972. http://dx.doi.org/10.1016/j.physleta.2008.05.048S. Nadeem and M. Awais, Journal of Porous Media, Vol. 13, 2010, pp. 973-980. http://dx.doi.org/10.1615/JPorMedia.v13.i11.30J. E. Dunn and R. L. Fosdick, Archive for Rational Mechanics and Analysis, Vol. 56, 1974, pp. 191-252.http://dx.doi.org/10.1007/BF00280970