Mathematical Models for Flow of Chyme during Gastrointestinal Endoscopy

ABSTRACT

Intestinal infection has become a common disease in human and endoscopy can be a powerful means in diagnosis of intestinal illnesses. Mathematical models are developed for an inserted endoscope on the flow of chyme in the small intestine considering a Newtonian incompressible fluid flow, under an axisymmetric condition, in a cylindrical annulus between the small intestine and the endoscope. We obtain novel mathematical expressions for the pressure drop, forces exerted by the endoscope on the flow of chyme, and the force exerted by the chyme on the intestine for one wave length of the peristaltic rush wave. We also investigate and calculate the flow velocity and pressure for different flow rates and the wave lengths. The results are presented, and discussed for the cases and conditions under which pressure, pressure drop can be positive or negative and the forces can be acted either by the intestine or endoscope on the flow or vice-versa.

Intestinal infection has become a common disease in human and endoscopy can be a powerful means in diagnosis of intestinal illnesses. Mathematical models are developed for an inserted endoscope on the flow of chyme in the small intestine considering a Newtonian incompressible fluid flow, under an axisymmetric condition, in a cylindrical annulus between the small intestine and the endoscope. We obtain novel mathematical expressions for the pressure drop, forces exerted by the endoscope on the flow of chyme, and the force exerted by the chyme on the intestine for one wave length of the peristaltic rush wave. We also investigate and calculate the flow velocity and pressure for different flow rates and the wave lengths. The results are presented, and discussed for the cases and conditions under which pressure, pressure drop can be positive or negative and the forces can be acted either by the intestine or endoscope on the flow or vice-versa.

KEYWORDS

Mathematical Models, Gastrointestinal, Chyme Flow, Endoscopy, Flow Modeling, Newtonian Incompressible Fluid Flow

Mathematical Models, Gastrointestinal, Chyme Flow, Endoscopy, Flow Modeling, Newtonian Incompressible Fluid Flow

Cite this paper

nullR. Roy, F. Rios and D. Riahi, "Mathematical Models for Flow of Chyme during Gastrointestinal Endoscopy,"*Applied Mathematics*, Vol. 2 No. 5, 2011, pp. 600-607. doi: 10.4236/am.2011.25080.

nullR. Roy, F. Rios and D. Riahi, "Mathematical Models for Flow of Chyme during Gastrointestinal Endoscopy,"

References

[1] M. Y. Jaffrin and A. H. Shapiro, “Peristaltic Pumping,” Annual Review of Fluid Mechanics, Vol. 3, 1971, pp. 13-36. doi:10.1146/annurev.fl.03.010171.000305

[2] T. W. Latham, “Fluid Motion in a Peristaltic Pump,” Master’s Thesis, Massachusetts Institute of Technology, Cambridge, 1966.

[3] A. H. Shapiro, M. Y. Jaffrin and S. L. Weinberg, “Peristaltic Pumping with Long Wavelength at Low Reynolds Number,” Journal of Fluid Mechanics, Vol. 37, 1969, pp. 799-825. doi:10.1017/S0022112069000899

[4] L. M. Srivastava and V. P. Srivastava, “Peristaltic Transport of Blood. Casson Model II,” Journal Biomechanics, Vol. 17, No. 11, 1984, pp. 821-829. doi:10.1016/0021-9290(84)90140-4

[5] L. M. Srivastava and V. P. Srivastava, “Peristaltic Tran- sport of a non-Newtonian Fluid: Applications to Vas De- ferens and Small Intestine,” Annals Biomedical Engineering, Vol. 13, No. 2, 1985, pp. 137-153. doi:10.1007/BF02584235

[6] V. P. Srivastava, “Particle-Fluid Suspension Flow Induced by Peristaltic Waves in a Circular Cylindrical Tube,” Bulletin of the Calcutta Mathematical Society, Vol. 5, No. 1, 2002, pp. 167-184.

[7] V. P. Srivastava, “Effects of an Inserted Endoscope on Chyme Movement in Small Intestine—A Theoretical Model,” Applications and Applied Mathematics, Vol. 2, No. 2, 2007, pp. 79-91.

[8] F. M. White, “Viscous Fluid Flow,” McGraw-Hill, Inc., New York, 1991.

[9] H. S. Lew, Y. C. Fung and C. B. Lowenstein, “Peristaltic Carrying and Mixing of Chyme in Small Intestine,” Journal Biomechanics, Vol. 4, No. 4, 1971, pp. 297-315. doi:10.1016/0021-9290(71)90036-4

[10] L. D. Landau and E. M. Lifshitz, “Fluid Mechanics,” 2nd Edition, Pergamum, London, 1987.

[11] V. P. Srivastava and M. Saxena, “A Two-Fluid Model of non-Newtonian Blood Flow Induced by Peristaltic Waves,” Rheologica Acta, Vol. 34, No. 4, 1995, pp. 406-414. doi:10.1007/BF00367155

[1] M. Y. Jaffrin and A. H. Shapiro, “Peristaltic Pumping,” Annual Review of Fluid Mechanics, Vol. 3, 1971, pp. 13-36. doi:10.1146/annurev.fl.03.010171.000305

[2] T. W. Latham, “Fluid Motion in a Peristaltic Pump,” Master’s Thesis, Massachusetts Institute of Technology, Cambridge, 1966.

[3] A. H. Shapiro, M. Y. Jaffrin and S. L. Weinberg, “Peristaltic Pumping with Long Wavelength at Low Reynolds Number,” Journal of Fluid Mechanics, Vol. 37, 1969, pp. 799-825. doi:10.1017/S0022112069000899

[4] L. M. Srivastava and V. P. Srivastava, “Peristaltic Transport of Blood. Casson Model II,” Journal Biomechanics, Vol. 17, No. 11, 1984, pp. 821-829. doi:10.1016/0021-9290(84)90140-4

[5] L. M. Srivastava and V. P. Srivastava, “Peristaltic Tran- sport of a non-Newtonian Fluid: Applications to Vas De- ferens and Small Intestine,” Annals Biomedical Engineering, Vol. 13, No. 2, 1985, pp. 137-153. doi:10.1007/BF02584235

[6] V. P. Srivastava, “Particle-Fluid Suspension Flow Induced by Peristaltic Waves in a Circular Cylindrical Tube,” Bulletin of the Calcutta Mathematical Society, Vol. 5, No. 1, 2002, pp. 167-184.

[7] V. P. Srivastava, “Effects of an Inserted Endoscope on Chyme Movement in Small Intestine—A Theoretical Model,” Applications and Applied Mathematics, Vol. 2, No. 2, 2007, pp. 79-91.

[8] F. M. White, “Viscous Fluid Flow,” McGraw-Hill, Inc., New York, 1991.

[9] H. S. Lew, Y. C. Fung and C. B. Lowenstein, “Peristaltic Carrying and Mixing of Chyme in Small Intestine,” Journal Biomechanics, Vol. 4, No. 4, 1971, pp. 297-315. doi:10.1016/0021-9290(71)90036-4

[10] L. D. Landau and E. M. Lifshitz, “Fluid Mechanics,” 2nd Edition, Pergamum, London, 1987.

[11] V. P. Srivastava and M. Saxena, “A Two-Fluid Model of non-Newtonian Blood Flow Induced by Peristaltic Waves,” Rheologica Acta, Vol. 34, No. 4, 1995, pp. 406-414. doi:10.1007/BF00367155