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Numerical Study of Slip Effect of Unsteady MHD Pulsatile Flow through Porous Medium in an Artery Using Generalized Differential Quadrature Method (Comparative Study)

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DOI: 10.4236/wjet.2014.22015    3,670 Downloads   5,800 Views   Citations

ABSTRACT

The unsteady pulsatile flow of blood through porous medium in an artery has been studied under the influence of periodic body acceleration and slip condition by considering blood as incompressible Newtonian electrically conducting fluid in the presence of magnetic field. In this paper, a new technique of differential quadrature method is introduced to find numerical solution of non-linear partial differential equations such as the equation of motion of this problem “Navier-Stokes equation”. The presence of the nonlinearity in the problem leads to severe difficulties in the solution approximation. In construction of the numerical scheme “a new algorithm” a generalized differential quadrature method (GDQM) is to use for derivatives with respect to space variables of differential equations and for the time derivative applying fourth order RungeKutta Method (RKM). The GDQM changed the nonlinear partial differential equations into a system of nonlinear ordinary differential equations (ODEs). The obtained system of ODEs is solved by 4th order RKM. This combination of DQM and 4th order RKM gives a very good numerical technique for solving time dependent problems. The algorithm is coded in Matlab 7.14.0.739 and the simulations are run on a Pentium 4 CPU 900 MHz with 1 GB memory capacity. The effects of slip condition, magnetic field, porous medium, and body acceleration have been discussed. The numerical results show that the proposed method is more accurate and convergent than other numerical methods in literature. The method is illustrated and compared with the exact and analytical solutions and it is found that the proposed method gives a better accuracy and is quite easy to implement.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Eldesoky, I. , Kamel, M. and Abumandour, R. (2014) Numerical Study of Slip Effect of Unsteady MHD Pulsatile Flow through Porous Medium in an Artery Using Generalized Differential Quadrature Method (Comparative Study). World Journal of Engineering and Technology, 2, 131-148. doi: 10.4236/wjet.2014.22015.

References

[1] Bellman, R.E. and Casti, J. (1971) Differential Quadrature and Long-Term Integration. Journal of Mathematical Analysis and Applications, 34, 235-238.
http://dx.doi.org/10.1016/0022-247X(71)90110-7
[2] Bellman, R.E., Kashef, B.G. and Casti, J. (1972) Differential Quadrature: A Technique for the Rapid Solution of Non-Linear Partial Differential Equations. Journal of computational Physics, 10, 40-52.
http://dx.doi.org/10.1016/0021-9991(72)90089-7
[3] Shu, C. (2000) Differential Quadrature and Its Application in Engineering. Springer-Verlag, London.
http://dx.doi.org/10.1007/978-1-4471-0407-0
[4] Quan, J.R. and Chang, C.T. (1989) New Insights in Solving Distributed System Equations by the Quadrature method-I, Analysis. Journal of Computers and Chemical Engineering, 13, 779-788.
http://dx.doi.org/10.1016/0098-1354(89)85051-3
[5] Shu, C. and Richards, B. (1992) Application of Generalized Differential Quadrature to Solve Two-Dimensional Incompressible Navier-Stokes Equations. International Journal for Numerical Methods in Fluids, 15, 791-798.
http://dx.doi.org/10.1002/fld.1650150704
[6] Wen, D. and Yu, Y. (1993) Calculation and Analysis of Weighting Coefficient Matrices in Differential Quadrature Method. Computational Engineering, Elsevier Oxford, 157-162.
[7] Shu, C. and Richards, B.E. (1990) High Resolution of Natural Convection in a Square Cavity by Generalized Differential Quadrature. Proceedings of 3rd International Conference on Advanced in numerical Methods in Engineering: Theory and Applications, Swansea, Vol. 2, 978-985.
[8] Shu, C. (1991) Generalized Differential-Integral Quadrature and Application to the Simulation of Incompressible Viscous Flows Including Parallel Computation. Ph.D. Thesis, University of Glasgow, Glasgow.
[9] Wen, D. and Yu, Y. (1993) Differential Quadrature Method for High Order Boundary Value Problems. Computational Engineering, 163-168.
[10] Tanaka, M. and Chen, W. (2001) Coupling Dual Reciprocity Boundary Element Method and Differential Quadrature Method for Time Dependent Diffusion Problems. Applied Mathematical Modelling, 25, 257-268.
http://dx.doi.org/10.1016/S0307-904X(00)00052-4
[11] Pu, J.-P. and Zheng, J.-J. (2006) Structural Dynamic Responses Analysis Applying Differential Quadrature Method. Journal of Zhejiang University Science, 7, 1831-1838.
http://dx.doi.org/10.1631/jzus.2006.A1831
[12] Zong, Z. and Zhang, Y.Y. (2009) Advanced Differential Quadrature Methods. Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series.
[13] Korkmaz, A., Aksoy, A.M. and Dag, I. (2011) Quartic B-Spline Differential Quadrature Method. International Journal of Nonlinear Science, 11, 403-411.
[14] Krowiak, A. (2008) Methods Based on the Differential Quadrature in Vibration Analysis of Plates. Journal of Theoretical and Applied Mechanics, 46, 123-139.
[15] Meral, G. (2013) Differential Quadrature Solution of Heat-and Mass-Transfer Equations. Applied Mathematical Modelling, 37, 4350-3459.
http://dx.doi.org/10.1016/j.apm.2012.09.012
[16] Ramezani, M., Shahrezaee, M., Kharazi, H. and Kashany, L.H. (2012) Numerical Solutions of Differential Algebraic Equation by Differential Quadrature Method. Journal of Basic and Applied Scientific Research, 2, 11821-11828.
[17] Al-Saif, A.S.J. and Zhu, Z.Y. (2003) Application of Mixed Differential Quadrature Method for Solving the Coupled Two-Dimensional Incompressible Navier-Stokes Equation and Heat Equation. Journal of Shanghai University, 7, 343-351.
http://dx.doi.org/10.1007/s11741-003-0007-x
[18] Al-Saif, A.S.J. and Zhu, Z.Y. (2006) Differential Quadrature Method for Steady Flow of an Incompressible Second-Order Viscoelastic Fluid and Heat Transfer Model. Journal of Shanghai University, 9, 298-306.
[19] Elshehawey, E.F., Elbarbary, E.M.E., Elsoud, A.S.N. and Elshahed, M. (1999) Blood Flow through a Porous Medium under Periodic Body Acceleration. Mathematical Sciences Research Hot-Line, 11, 27-31.
[20] Elshehawey, E.F., Elbarbary, E.M.E., Elsoud, A.S.N. and Elshahed, M. (1999) Womersley Problem for Pulsatile Flow of Blood through a Porous Medium. Proceedings of the 6th Conference on Theoretical and Applied Mechanics, Cairo, March 1999, 115-122.
[21] Elshehawey, E.F., Elbarbary, E.M.E., Afifi, N.A.S. and El-Shahed, M. (2001) MHD Flow of Blood under Body Acceleration. Integral Transforms and Special Functions, 12, 1-6.
http://dx.doi.org/10.1080/10652460108819329
[22] El-Shehawey, E.F., Elbarbary, E.M.E., Afifi, N.A.S. and Elshahed, M. (2000) MHD Flow of an Elastico-Viscous Fluid under Periodic Body Acceleration. International Journal of Mathematics and Mathematical Sciences, 23, 795-799.
http://dx.doi.org/10.1155/S0161171200002817
[23] Elshehawey, E.F., Elbarbary, E.M.E., Afifi, N.A.S. and El-Shahed, M. (2000) Pulsatile Flow of Blood through a Porous Mediumunder Periodic Body Acceleration. International Journal of Theoretical Physics, 39, 183-188.
http://dx.doi.org/10.1023/A:1003611604207
[24] El-Shahed, M. (2003) Pulsatile Flow of Blood through a Stenosed Porous Medium under Periodic Body Acceleration. Applied Mathematics and Computation, 138, 479-488.
http://dx.doi.org/10.1016/S0096-3003(02)00164-9
[25] El-Shehawy, E.F., El-Dabe, N.T. andEl-Desoky, I.M. (2006) Slip Effects on the Peristaltic Flow of a Non-Newtonian Maxwellian Fluid. Acta Mechanica, 186, 141-159.
http://dx.doi.org/10.1007/s00707-006-0343-6
[26] Sanyal, D.C., Das, K. and Debnath, S. (2007) Effect of Magnetic Field on Pulsatile Blood Flow through an Inclined Circular Tube with Periodic Body Acceleration. Journal of Physical Science, 11, 43-56.
[27] Das, K. and Saha, G.C. (2009) Arterial MHD Pulsatile Flow of Blood under Periodic Body Acceleration. Bulletin of Society of Mathematicians Banja Luka, 16, 21-42.
[28] Sanyal, D.C. and Biswas, A. (2010) Pulsatile Motion through an Axi-Symmetric Artery in Presence of Magnetic Field. Assam University Journal of Science and Technology: Physical Science and Technology, 5, 12-20.
[29] Mathur, P. and Jain, S. (2011) Pulsatile Flow of Blood through a Stenosed Tube: Effect of Periodic Body Acceleration and a Magnetic Field. Journal of Biorheology, 25, 71-77.
http://dx.doi.org/10.1007/s12573-011-0040-5
[30] Misra, J.C., Sinha, A. and Shit, G.C. (2011) Mathematical Modeling of Blood Flow in a Porous Vessel Having Double Stenoses in the Presence of an External Magnetic Field. International Journal of Biomathematics, 4, 207-225.
http://dx.doi.org/10.1142/S1793524511001428
[31] Eldesoky, I.M. (2012) Mathematical Analysis of Unsteady MHD Blood Flow through Parallel Plate Channel with Heat Source. World Journal of Mechanics, 2, 131-137.
http://www.SciRP.org/journal/wjm.
[32] Eldesoky, I.M. (2010) Slip Effects on the Unsteady MHD Pulsatile Blood Flow through Porous Medium in an Artery under the Effect of Body Acceleration. International Journal of Mathematics and Mathematical Sciences, 2012, Article ID 860239.
http://dx.doi.org/10.1155/2012/860239
[33] Mohan, V., Prasad, V., Varshney, N.K. and Gupta, P.K. (2013) Effect of Magnetic Field on Blood Flow (Elastico-Viscous) Under Periodic Body Acceleration in Porous Medium. IOSR Journal of Mathematics, 6, 43-48.
http://dx.doi.org/10.9790/5728-0644348
[34] Tzirtzilakis, E.E. (2005) A Mathematical Model for Blood Flow in Magnetic Field. Physics of Fluids, 17, Article ID: 077103.
[35] Abdelnaby, M.A., Eldabe, N.T.M. and Abouzeid, M.Y. (2006) Numerical Study of Pulsatile MHD Non-Newtonian Fluid Flow with Heat and Mass Transfer through a Porous Medium between Two Permeable Parallel Plates. Jour. Mech. Cont. & Math. Sci., 1, 1-15.
[36] Malekzadeh, A., Heydarinasab, A. and Dabir, B. (2011) Magnetic Field Effect on Fluid Flow Characteristics in a Pipe for Laminar Flow. Journal of Mechanical Science and Technology, 25, 333-339.
[37] Sankar, D.S. and Lee, U. (2011) FDM Analysis for MHD Flow of a Non-Newtonian Fluid for Blood Flow in Stenosed Arteries. Journal of Mechanical Science and Technology, 25, 2573-2581.
http://dx.doi.org/10.1007/s12206-011-0728-x
[38] Amira, H.T., Ilyani, A. and Izzati, C.M.S.S.N. (2012) Effects of Body Acceleration on the Blood Flow through Irregular Stenosis. UMT 11th International Annual Symposium on Sustainability Science and Management, Terengganu, 9-11 July 2012, 735-744.
[39] Eldesoky, I.M., Kamel, M.H., Hussien, R.M. and Abumandour, R.M. (2013) Numerical Study of Unsteady MHD Pulsatile Flow through Porous Medium in an Artery Using Generalized Differential Quadrature Method (GDQM). International Journal of Materials, Mechanics and Manufacturing, 1, 200-206.
[40] Wang, C.Y. (2003) Stagnation Flows with Slip: Exact Solutions of the Navier-Stokes Equations. Zeitschrift fur Angewandte Mathematik und Physik, 54, 184-189.
http://dx.doi.org/10.1007/PL00012632
[41] Megahed, A.A. and Kamel, M.H. (1988) Unsteady Motion of a Conducting Fluid through a Porous Medium in a Circular Pipe with Heat Transfer. Proceedings of 5th International Conference of Multiphase Transfer Phenomena, Miami, 1988.
[42] Chaturani, P. and Palanisamy, V. (1991) Pulsatile Flow of Blood with Periodic Body Acceleration. International Journal of Engineering Science, 29, 113-121.
http://dx.doi.org/10.1016/0020-7225(91)90081-D
[43] Kamel, M.H. and El-Tawil, M.A. (2001) Stochastic Blood Flow through an Overlapping Arterial Stenosis. Journal of Engineering and Applied Science, 48, 623-635.
[44] El-Shahed, M. (2003) Pulsatile Flow of Blood through a Stenosed Porous Medium under Periodic Body Acceleration. Applied Mathematics and Computation, 138, 479-488.
http://dx.doi.org/10.1016/S0096-3003(02)00164-9
[45] Megahed, A.A., Maher, B.M. and Eldesoky, I.M. (2004) Unsteady MHD Pulsatile Flow through Porous Medium in an Infinite Circular Pipe under the Effect of the Body Acceleration. Scientific Bulletin of the Faculty of Engineering, Ain Shams University, 39, 715-735.

  
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