Mathematical Modeling of Diffusion Phenomenon in a Moderately Constricted Geometry

ABSTRACT

The incorporation of fluid flow through modelled normal and stenosed capillary-tissue exchange system has highlighted issues that may have major applications for the study of diffusion phenomenon. Results clearly demonstrate the important roles played by various physiological characteristics and diffusion variables involved in the analysis on blood flow. Assessment of the severity of the disease could be made possible through the variation of a parameter named as retention parameter. An attempt has been made to study the effects of local variation of viscosity on flow, wall-shearing stress and distribution of dissolved material in diseased artery as compared to the normal.

The incorporation of fluid flow through modelled normal and stenosed capillary-tissue exchange system has highlighted issues that may have major applications for the study of diffusion phenomenon. Results clearly demonstrate the important roles played by various physiological characteristics and diffusion variables involved in the analysis on blood flow. Assessment of the severity of the disease could be made possible through the variation of a parameter named as retention parameter. An attempt has been made to study the effects of local variation of viscosity on flow, wall-shearing stress and distribution of dissolved material in diseased artery as compared to the normal.

Cite this paper

nullS. Mishra, N. Verma and S. Siddiqui, "Mathematical Modeling of Diffusion Phenomenon in a Moderately Constricted Geometry,"*Applied Mathematics*, Vol. 2 No. 2, 2011, pp. 241-246. doi: 10.4236/am.2011.22027.

nullS. Mishra, N. Verma and S. Siddiqui, "Mathematical Modeling of Diffusion Phenomenon in a Moderately Constricted Geometry,"

References

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[9] P. Neofytou and D. Drikakis, “Non-Newtonian Flow Instability in a Channel with a Sudden Expansion,” Journal of Non-Newtonian Fluid Mechanics, Vol. 111, No. 2-3, 2003, pp. 127-150. doi:10.1016/S0377-0257(03)00041-7

[10] K. K. Yeleswarapu, “Evaluation of Continuum Models for Characterizing the Constitutive Behavior of Blood,” Ph.D. Thesis, Department of Mechanical Engineering, University of Pittsburgh, Pittsburgh, 1996.

[11] J. Perkkio and R. Keskinen, “Hematocrit Reduction in Bifurcation due to Plasma Skimming,” Bulletin of Mathematical Biology, Vol. 45, No. 1, 1983, pp. 41-50.

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[13] P. N. Tandon and T. S. Pal, “On Transmural Fluid Exchange and Variation of Viscosity of Blood Flowing through Permeable Capillaries,” Medical and Life Science Engineering, Vol. 5, No. 1, 1979, pp. 18-29.

[14] P. N. Tandon and R. Agarwal, “A Study on Nutritional Transport in Synovial Joints,” International Journal of Computers and Mathematics with Application, Vol. 17, No. 7, 1989, pp. 1101-1141.

[15] J. B. Shukla, R. S. Parihar and B. R. P. Rao, “Effects of Stenosis on Non-Newtonian Flow of the Blood in an Artery,” Bulletin of Mathematical Biology, Vol. 42, No. 3, 1980, pp. 283-294.

[16] J. C. Mishra and S. Chakravarty, “Flow in Arteries in the Presence of Stenosis,” Journal of Biomechanics, Vol. 19, No. 11, 1986, pp. 907-918. doi:10.1016/0021-9290(86)90186-7

[1] V. W. Bowry and K. U. Ingold, “The Unexpected Role of Vitamin E (α-Tocopherol) in the Peroxidation of Human Low-Density Lipoprotein,” Accounts of Chemical Research, Vol. 32, No. 1, 1999, pp. 27-34. doi:10.1021/ar950059o

[2] M. J. Davies and N. Woolf, “Atherosclerosis in Ischaemic Heart Disease: The Mechanisms,” Science Press, London, 1990.

[3] H. Esterbauer, J. Gebicki, H. Puhl and G. Jurgens, “The Role of Lipid Peroxidation and Antioxidants in Oxidative Modification of LDL,” Free Radical Biology and Medicine, Vol. 13, No. 4, 1992, pp. 341-390. doi:10.1016/0891-5849(92)90181-F

[4] B. V. R. Kumar and K. B. Naidu, “A Pulsatile Suspension Flow Simulation in a Stenosed Vessel,” Mathematical and Computer Modeling, Vol. 23, No. 5, 1996, pp. 75-86. doi:10.1016/0895-7177(96)00013-1

[5] P. N. Tandon and U. V. S. Rana, “A New Model for Blood Flow through an Artery with Axisymmetric Stenosis,” International Journal of Biomedical Computing, Vol. 38, No. 3, 1995, pp. 257-267. doi:10.1016/S0020-7101(05)80008-X

[6] F. J. H. Gijsen, F. N. van de Vosse and J. D. Janssen, “The Influence of the Non-Newtonian Properties of Blood on the Flow in Large Arteries: Steady Flow in a Carotid Bifurcation Model,” Journal of Biomechanics, Vol. 32, No. 6, 1999, pp. 601-608. doi:10.1016/S0021-9290(99)00015-9

[7] B. Johnston, P. R. Johnston, S. Corney and D. Kilpatrick, “Non-Newtonian Blood Flow in Human Right Coronary Arteries: Steady State Simulations,” Journal of Biomechanics, Vol. 37, No. 5, 2004, pp. 709-720. doi:10.1016/j.jbiomech.2003.09.016

[8] A. Leuprecht and K. Perktold, “Computer Simulation of Non-Newtonian Effects on Blood Flows in Large Arteries,” Computer Methods in Biomechanics & Biomedical Engineering, Vol. 4, No. 2, 2001, pp. 149-163. doi:10.1080/10255840008908002

[9] P. Neofytou and D. Drikakis, “Non-Newtonian Flow Instability in a Channel with a Sudden Expansion,” Journal of Non-Newtonian Fluid Mechanics, Vol. 111, No. 2-3, 2003, pp. 127-150. doi:10.1016/S0377-0257(03)00041-7

[10] K. K. Yeleswarapu, “Evaluation of Continuum Models for Characterizing the Constitutive Behavior of Blood,” Ph.D. Thesis, Department of Mechanical Engineering, University of Pittsburgh, Pittsburgh, 1996.

[11] J. Perkkio and R. Keskinen, “Hematocrit Reduction in Bifurcation due to Plasma Skimming,” Bulletin of Mathematical Biology, Vol. 45, No. 1, 1983, pp. 41-50.

[12] C. K. Kang and A. C. Eringen, “The Effect of Microstructure on the Rheological Properties of Blood,” Bulletin of Mathematical Biology, Vol. 38, No. 2, 1976, pp. 135-159.

[13] P. N. Tandon and T. S. Pal, “On Transmural Fluid Exchange and Variation of Viscosity of Blood Flowing through Permeable Capillaries,” Medical and Life Science Engineering, Vol. 5, No. 1, 1979, pp. 18-29.

[14] P. N. Tandon and R. Agarwal, “A Study on Nutritional Transport in Synovial Joints,” International Journal of Computers and Mathematics with Application, Vol. 17, No. 7, 1989, pp. 1101-1141.

[15] J. B. Shukla, R. S. Parihar and B. R. P. Rao, “Effects of Stenosis on Non-Newtonian Flow of the Blood in an Artery,” Bulletin of Mathematical Biology, Vol. 42, No. 3, 1980, pp. 283-294.

[16] J. C. Mishra and S. Chakravarty, “Flow in Arteries in the Presence of Stenosis,” Journal of Biomechanics, Vol. 19, No. 11, 1986, pp. 907-918. doi:10.1016/0021-9290(86)90186-7