DIFFERENTIAL TRANSFORMATION METHOD FOR ANALYSIS OF NONLINEAR FLOW AND MASS TRANSFER THROUGH A CHANNEL FILLED WITH A POROUS MEDIUM

Abstract

In this paper, mass transfer and chemical reaction effects on laminar viscous flow through a porous channel with moving or stationary walls are studied. The governing partial differential equations of the physical problem are transformed into a set of coupled nonlinear ordinary differential equations using similarity transformation. The coupled nonlinear ordinary differential equations are solved using differential transform method (DTM). The results obtained through the approximate analytical method are compared with the results of numerical method and high accuracy of the present approximate analytical solution is observed. The valuable achievement of the present study is imbedding a precise and efficient analytical method for the flow of viscous fluid in a porous channel with a chemical reaction. Also, the effects of some pertinent parameters such as Reynolds number, Darcy number, Schmidt number and suction/injection parameter on velocity components, heat transfer, concentration, and Sherwood distribution are presented in this work.

Keywords

• Heat Transfer,Chemical Reaction,Boundary Value Problem; Differential Transformation Method (DTM)

References

1. [1] Cox SM. Two-dimensional flow of a viscous fluid in a channel with porous walls. Journal of Fluid Mechanics. 1991 Jun;227:1-33. doi:10.1017/S0022112091000010.
2. [2] Debruge LL, Han LS. Heat transfer in a channel with a porous wall for turbine cooling application. Journal of Heat Transfer, 1972, 94: 385-390. doi: 10.1115/1.3449956.
3. [3] Durst F, Haas R, Interthal W. The nature of flows through porous media. Journal of Non-Newtonian Fluid Mechanics. 1987 Jan 1;22(2):169-89. doi: 10.1016/0377-0257(87)80034-4.
4. [4] Rajagopal KR, Saccomandi G, Vergori L. Linear stability of Hagen–Poiseuille flow in a chemically reacting fluid. Computers & Mathematics with Applications. 2011 Jan 1;61(2):460-9. doi: 10.1016/j.camwa.2010.11.026.
5. [5] Bég OA, Makinde OD. Viscoelastic flow and species transfer in a Darcian high-permeability channel. Journal of Petroleum Science and Engineering. 2011 Mar 1;76(3-4):93-9. doi: 10.1016/j.petrol.2011.01.008.
6. [6] Hayat T, Abbas Z. Channel flow of a Maxwell fluid with chemical reaction. Zeitschrift für angewandte Mathematik und Physik. 2008 Jan 1;59(1):124-44. doi: 10.1007/s00033-007-6067-1.
7. [7] Rundora L, Makinde OD. Effects of suction/injection on unsteady reactive variable viscosity non-Newtonian fluid flow in a channel filled with porous medium and convective boundary conditions. Journal of Petroleum Science and Engineering. 2013 Aug 1;108:328-35. doi: 10.1016/j.petrol.2013.05.010.
8. [8] Chinyoka T, Makinde OD. Computational dynamics of unsteady flow of a variable viscosity reactive fluid in a porous pipe. Mechanics Research Communications. 2010 Apr 1;37(3):347-53. doi: 10.1016/j.mechrescom.2010.02.007.

9. [9] Srinivas S, Muthuraj R. Effects of chemical reaction and space porosity on MHD mixed convective flow in a vertical asymmetric channel with peristalsis. Mathematical and Computer Modelling. 2011 Sep 1;54(5-6):1213-27. doi: 10.1016/j.mcm.2011.03.032.
10. [10] Makinde OD. Computer extension and bifurcation study by analytic continuation of porous tube flow. Jour. Math. Phys. Sci. 1996;30:1-24.
11. [11] Makinde OD. Extending the utility of perturbation series in problems of laminar flow in a porous pipe and a diverging channel. The ANZIAM Journal. 1999 Jul;41(1):118-28. doi: 10.1017/S0334270000011073.
12. [12] Makinde OD and Tay G. Numerical computation of bifurcation for steady flow in a converging channel with accelerating surface. A. M. S. E. Modelling, Measurement & Control B. 1999: 68 (1): 33-43.
13. [13] Makinde OD, Moitsheki RJ. On nonperturbative techniques for thermal radiation effect on natural convection past a vertical plate embedded in a saturated porous medium. Mathematical Problems in Engineering. 2008;2008.
14. [14] Chinyoka T, Makinde OD. Viscoelastic modeling of the diffusion of polymeric pollutants injected into a pipe flow. Acta Mechanica Sinica. 2013 Apr 1;29(2):166-78. doi: 10.1007/s10409-013-0016-3.
15. [15] Makinde OD, Chinyoka T. Numerical investigation of buoyancy effects on hydromagnetic unsteady flow through a porous channel with suction/injection. Journal of Mechanical Science and Technology. 2013 May 1;27(5):1557-68. doi: 10.1007/s12206-013-0221-9.
16. [16] Patil PM, Kulkarni PS. Effects of chemical reaction on free convective flow of a polar fluid through a porous medium in the presence of internal heat generation. International Journal of Thermal Sciences. 2008 Aug 1;47(8):1043-54. doi: 10.1016/j.ijthermalsci.2007.07.013.
17. [17] Subramanyam Reddy A, Srinivas S, Ramamohan TR. Analysis of heat and chemical reaction on an asymmetric laminar flow between slowly expanding or contracting walls. Heat Transfer—Asian Research. 2013 Jul;42(5):422-43. doi: 10.1002/htj.21036.
18. [18] Matin MH, Pop I. Forced convection heat and mass transfer flow of a nanofluid through a porous channel with a first order chemical reaction on the wall. International Communications in Heat and Mass Transfer. 2013 Aug 1;46:134-41. doi: 10.1016/j.icheatmasstransfer.2013.05.001.
19. [19] Sivaraj R, Kumar BR. Chemically reacting dusty viscoelastic fluid flow in an irregular channel with convective boundary. Ain Shams Engineering Journal. 2013 Mar 1;4(1):93-101. doi: 10.1016/j.asej.2012.06.005.
20. [20] Srinivas S, Gupta A, Gulati S, Reddy AS. Flow and mass transfer effects on viscous fluid in a porous channel with moving/stationary walls in presence of chemical reaction. International communications in heat and mass transfer. 2013 Nov 1;48:34-9. doi: 10.1016/j.icheatmasstransfer.2013.09.002.
21. [21] Mahdy A. Effect of chemical reaction and heat generation or absorption on double-diffusive convection from a vertical truncated cone in porous media with variable viscosity. International Communications in Heat and Mass Transfer. 2010 May 1;37(5):548-54. doi: 10.1016/j.icheatmasstransfer.2010.01.007.
22. [22] Rashidi MM, Shahmohamadi H, Dinarvand S. Analytic Approximate Solutions for Unsteady Two-Dimensional and Axisymmetric Squeezing Flows between Parallel Plates. Mathematical Problems in Engineering. 2008: Article ID 935095. doi: 10.1155/2008/935095.
23. [23] Dinarvand S, Rashidi M, Doosthoseini A. Analytical approximate solutions for two-dimensional viscous flow through expanding or contracting gaps with permeable walls. Open Physics. 2009 Dec 1;7(4):791-9. doi: 10.2478/s11534-009-0024-x.
24. [24] Rashidi MM, Domairry G, Dinarvand S. Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method. Communications in Nonlinear Science and Numerical Simulation. 2009 Mar 1;14(3):708-17. doi: 10.1016/j.cnsns.2007.09.015.
25. [25] Rashidi MM, Ganji DD, Dinarvand S. Approximate traveling wave solutions of coupled Whitham-Broer-Kaup shallow water equations by homotopy analysis method. International Journal of Differential Equations. 2008 May 15;2008: Article ID 243459. doi: 10.1155/2008/243459.
26. [26] Raftari B, Yildirim A. The application of homotopy perturbation method for MHD flows of UCM fluids above porous stretching sheets. Computers & Mathematics with Applications. 2010 May 1;59(10):3328-37. doi: 10.1016/j.camwa.2010.03.018.
27. [27] Esmaeilpour M, Ganji DD. Application of He's homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate. Physics Letters A. 2007 Dec 10;372(1):33-8. doi: 10.1016/j.physleta.2007.07.002.
28. [28] Rashidi MM, Ganji DD, Dinarvand S. Explicit analytical solutions of the generalized Burger and Burger–Fisher equations by homotopy perturbation method. Numerical Methods for Partial Differential Equations: An International Journal. 2009 Mar;25(2):409-17. doi: 10.1002/num.20350.
29. [29] Mohyud-Din ST, Yildirim A, Anıl Sezer S, Usman M. Modified variational iteration method for free-convective boundary-layer equation using padé approximation. Mathematical Problems in Engineering. 2010;2010: Article ID 318298. doi: 10.1155/2010/318298.
30. [30] Rashidi MM, Shahmohamadi H. Analytical solution of three-dimensional Navier–Stokes equations for the flow near an infinite rotating disk. Communications in Nonlinear Science and Numerical Simulation. 2009 Jul 1;14(7):2999-3006. doi: 10.1016/j.cnsns.2008.10.030.
31. [31] Wazwaz AM. The variational iteration method for solving two forms of Blasius equation on a half-infinite domain. Applied Mathematics and Computation. 2007 May 1;188(1):485-91. doi: 10.1016/j.amc.2006.10.009.
32. [32] Wazwaz AM. The modified decomposition method and Padé approximants for a boundary layer equation in unbounded domain. Applied Mathematics and Computation. 2006 Jun 15;177(2):737-44. doi: 10.1016/j.amc.2005.09.102.
33. [33] Alizadeh E, Sedighi K, Farhadi M, Ebrahimi-Kebria HR. Analytical approximate solution of the cooling problem by Adomian decomposition method. Communications in Nonlinear Science and Numerical Simulation. 2009 Feb 1;14(2):462-72. doi: 10.1016/j.cnsns.2007.09.008.
34. [34] Kechil SA, Hashim I. Non-perturbative solution of free-convective boundary-layer equation by Adomian decomposition method. Physics Letters A. 2007 Mar 19;363(1-2):110-4. doi: 10.1016/j.physleta.2006.11.054
35. [35] Zhou, J. Differential transformation and its applications for electrical circuits. ed: Huazhong University Press. 1986. Wuhan. China.
36. [36] Ayaz F. Applications of differential transform method to differential-algebraic equations. Applied Mathematics and Computation. 2004 May 13;152(3):649-57. doi: 10.1016/S0096-3003(03)00581-2.
37. [37] Liu H, Song Y. Differential transform method applied to high index differential–algebraic equations. Applied Mathematics and Computation. 2007 Jan 15;184(2):748-53. doi: 10.1016/j.amc.2006.05.173.
38. [38] Mosayebidorcheh S, Sheikholeslami M, Hatami M, Ganji DD. Analysis of turbulent MHD Couette nanofluid flow and heat transfer using hybrid DTM–FDM. Particuology. 2016 Jun 1;26:95-101. doi: 10.1016/j.partic.2016.01.002.
39. [39] Mosayebidorcheh S, Hatami M, Ganji DD, Mosayebidorcheh T, Mirmohammadsadeghi SM. Investigation of Transient MHD Couette flow and Heat Transfer of Dusty Fluid with Temperature-Dependent Oroperties. Journal of Applied Fluid Mechanics. 2015 Sep 1;8(4).
40. [40] Hatami M, Ghasemi SE, Sahebi SA, Mosayebidorcheh S, Ganji DD, Hatami J. Investigation of third-grade non-Newtonian blood flow in arteries under periodic body acceleration using multi-step differential transformation method. Applied Mathematics and Mechanics. 2015 Nov 1;36(11):1449-58. doi: 10.1007/s10483-015-1995-7.
41. [41] Mosayebidorcheh S, Rahimi-Gorji M, Ganji DD, Moayebidorcheh T, Pourmehran O, Biglarian M. Transient thermal behavior of radial fins of rectangular, triangular and hyperbolic profiles with temperature-dependent properties using DTM-FDM. Journal of Central South University. 2017 Mar 1;24(3):675-82. doi: 10.1007/s11771-017-3468-y.
42. [42] Mosayebidorcheh S, Rahimi-Gorji M, Ganji DD, Moayebidorcheh T, Pourmehran O, Biglarian M. Transient thermal behavior of radial fins of rectangular, triangular and hyperbolic profiles with temperature-dependent properties using DTM-FDM. Journal of Central South University. 2017 Mar 1;24(3):675-82. doi: 10.1007/s11771-017-3468-y.
43. [43] Mosayebidorcheh S, Farzinpoor M, Ganji DD. Transient thermal analysis of longitudinal fins with internal heat generation considering temperature-dependent properties and different fin profiles. Energy conversion and management. 2014 Oct 1;86:365-70. doi: 10.1016/j.enconman.2014.05.033.
44. [44] Odibat ZM. Differential transform method for solving Volterra integral equation with separable kernels. Mathematical and Computer Modelling. 2008 Oct 1;48(7-8):1144-9. doi: 10.1016/j.mcm.2007.12.022.
45. [45] Arikoglu A, Ozkol I. Solutions of integral and integro-differential equation systems by using differential transform method. Computers & Mathematics with Applications. 2008 Nov 1;56(9):2411-7. doi: 10.1016/j.camwa.2008.05.017.
46. [46] Arikoglu A, Ozkol I. Solution of boundary value problems for integro-differential equations by using differential transform method. Applied Mathematics and Computation. 2005 Sep 15;168(2):1145-58. doi: 10.1016/j.amc.2004.10.009.
47. [47] Mosayebidorcheh S. Solution of the boundary layer equation of the power-law pseudoplastic fluid using differential transform method. Mathematical Problems in Engineering. 2013: Article ID 685454. doi: 10.1155/2013/685454.
48. [48] Mosayebidorcheh S, Mosayebidorcheh T. Series solution of convective radiative conduction equation of the nonlinear fin with temperature dependent thermal conductivity. International Journal of Heat and Mass Transfer. 2012 Nov 1;55(23-24):6589-94. doi: 10.1016/j.ijheatmasstransfer.2012.06.066.
49. [49] Mosayebidorcheh S. Taylor series solution of the electrohydrodynamic flow equation. Journal of Mechanical Engineering and Technology. 2013 Sep;1(2):40-5.
50. [50] Mosayebidorcheh S. Analytical investigation of the micropolar flow through a porous channel with changing walls. Journal of Molecular Liquids. 2014 Aug 1;196:113-9. doi: 10.1016/j.molliq.2014.03.022.
51. [51] Mosayebidorcheh S. Comments on “Numerical solutions of the space-and time-fractional coupled Burgers equations by generalized differential transform method, Applied Mathematics and Computation 217 (2011) 7001–7008”. Applied Mathematics and Computation. 2014 Sep 15;243:960-2. doi: 10.1016/j.amc.2014.06.055.
52. [52] Odibat Z, Momani S, Erturk VS. Generalized differential transform method: application to differential equations of fractional order. Applied Mathematics and Computation. 2008 Apr 1;197(2):467-77. doi: 10.1016/j.amc.2007.07.068.
53. [53] Erturk VS, Momani S, Odibat Z. Application of generalized differential transform method to multi-order fractional differential equations. Communications in Nonlinear Science and Numerical Simulation. 2008 Oct 1;13(8):1642-54. doi: 10.1016/j.cnsns.2007.02.006.
54. [54] Arikoglu A, Ozkol I. Solution of fractional differential equations by using differential transform method. Chaos, Solitons & Fractals. 2007 Dec 1;34(5):1473-81. doi: 10.1016/j.chaos.2006.09.004.
55. [55] Reza Seyf H, Moein Rassoulinejad-Mousavi S. An analytical study for fluid flow in porous media imbedded inside a channel with moving or stationary walls subjected to injection/suction. Journal of fluids engineering. 2011 Sep 1;133(9). doi: 10.1115/1.4004822.