Numerical Study of Creeping Flow Through Sinusoidally Periodic Tube

Asif Mahmud; Suhana Perveen; Md. Nazmul Hasan; Md. Samsuzzoha; Nazmul Islam

doi:doi:10.11648/j.ijamtp.20190503.14

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Numerical Study of Creeping Flow Through Sinusoidally Periodic Tube

Asif Mahmud, Suhana Perveen, Md. Nazmul Hasan, Md. Samsuzzoha, Nazmul Islam

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 5, Issue 3)

Received: 29 April 2019 Accepted: 4 September 2019 Published: 19 September 2019

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Abstract

There has been renewed interest in the flow behaviour within tubes with periodically varying cross-section with the recognition that they can be used as particle separation devices. In this paper, we present a numerical study of the effect of tube geometry on creeping flow of viscous incompressible fluid through sinusoidally constricted periodic tube which is axisymmetric but longitudinally asymmetric. The boundary element method is used to solve for the flow in the tube by specifying the pressure drop across the ends of the tube. The boundary element equations have been formulated for an infinite periodic tube by writing the velocity in terms of the integrals over the tube boundary and is used to calculate the force on the tube boundary, to obtain the detailed velocity distribution within the tube and to determine the effect of amplitude and wavelength of corrugation on the structure of the flow. We have found that the highest axial velocity is at throat region and lowest axial velocity is at expansion region. Also, we have discovered that the maximum radial velocity occurs at diverging cross-section and minimum radial velocity occurs at converging cross-section. The tangential force on the tube wall is examined for different amplitudes and wavelengths of corrugation and observed that the tangential force is greater in the constricted region than in the expansion region. The physical quantities (such as velocity and force) increase with increasing amplitude and decrease with increasing wavelength. Finally, we have compared our results with the work of Hemmat and Borhan [3] and have found good agreement with them.

Published in	International Journal of Applied Mathematics and Theoretical Physics (Volume 5, Issue 3)
DOI	10.11648/j.ijamtp.20190503.14
Page(s)	72-81
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Creeping Flow, Numerical Study, Periodic Tube, Sinusoidal Cross-Section, Boundary Element Method

References

[1]	Chow, J. C. F. and Soda, K. (1972) Laminar flow in tubes with constriction. Physics of Fluids. 15: 1700-1706.
[2]	Deiber, J. A. and Schowalter, W. R. (1979) Flow through tubes with sinusoidal axial variations in diameter. International Journal of American Institute of Chemical Engineers. 25: 638-645.
[3]	Hemmat, M. and Borhan, A. (1995) Creeping flow through sinusoidally constricted capillaries. Physics of Fluids. 7: 2111-2121.
[4]	Ralph, M. E. (1987) Steady flow structures and pressure drops in wavy-walled tubes. Journal of Fluids Engineering. 109: 255-261.
[5]	Sisavath, S., Jing, X. and Zimmerman, R. W. (2001) Creeping flow through a pipe of varying radius. Physics of Fluids. 13: 2762-2772.
[6]	Tilton, J. N. and Payatakes, A. C. (1984) Collocation solution of creeping Newtonian flow through sinusoidal tubes: a correction. International Journal of American Institute of Chemical Engineers 30: 1016-1021.
[7]	Islam, N. (2016) Fluid flow and particle transport through periodic capillaries. PhD dissertation, University of South Australia, Australia.
[8]	Sisavath, S., Jing, X. and Zimmerman, R. W. (2001) Creeping flow through a pipe of varying radius. Physics of Fluids. 13, 2762-2772.
[9]	Hagen, G. H. L. (1839) Uber die Bewegung des Wassers in engen cylindrischen Rohren. Poggendorf's Annalen der Physik und Chemie. 46: 423-42.
[10]	Poiseuille, J. L. M. (1830) Recherches sur les causes du mouvement du sang dans les veincs. Journal of Clinical and Experimental Pathology 10: 277-295.
[11]	Rothstein, J. P. and McKinley, G. H. (2001) The axisymmetric contraction expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop. Journal of Non-Newtonian Fluid Mechanics 98: 33-63.
[12]	Subramanya, K. (1982) Flow in Open Channels, 3^rd ed.; Tata McGraw-Hill Education, New Delhi, India.
[13]	Moffatt, H. K. (1964) Viscous and resistive eddies near a sharp corner. Journal of Fluid Mechanics. 18: 1-18.
[14]	Forrester, J. H. and Young, D. F. (1970) Flow through a converging-diverging tube and its implications in occlusive vascular disease. Journal of Biomechanics 3: 297-305.
[15]	Leneweit, G. and Auerbach, D. (1999) Detachment phenomena in low Reynolds number flows through sinusoidally constricted tubes. Journal of Fluid Mechanics 387: 129-150.
[16]	Nishimura, T., Bian, Y., Matsumoto, Y. and Kunitsugu, K. (2003) Fluid flow and mass transfer characteristics in a sinusoidal wavy-walled tube at moderate Reynolds numbers for steady flow. International Journal of Heat and Mass Transfer. 39 (3): 239-248.
[17]	Islam, N., Bradshaw-Hajek, B. H., Miklavcic, S. J. and White, L. R. (2015) The onset of recirculation flow in periodic capillaries: Geometric effects. European Journal of Mechanics B/Fluids. 53: 119-128.
[18]	Pozrikidis, C. (1992) Boundary Integral and singularity Methods for Linearised Viscous Flow, Cambridge University Press, Cambridge, U.K.%

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APA Style

Asif Mahmud, Suhana Perveen, Md. Nazmul Hasan, Md. Samsuzzoha, Nazmul Islam. (2019). Numerical Study of Creeping Flow Through Sinusoidally Periodic Tube. International Journal of Applied Mathematics and Theoretical Physics, 5(3), 72-81. https://doi.org/10.11648/j.ijamtp.20190503.14

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ACS Style

Asif Mahmud; Suhana Perveen; Md. Nazmul Hasan; Md. Samsuzzoha; Nazmul Islam. Numerical Study of Creeping Flow Through Sinusoidally Periodic Tube. Int. J. Appl. Math. Theor. Phys. 2019, 5(3), 72-81. doi: 10.11648/j.ijamtp.20190503.14

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AMA Style

Asif Mahmud, Suhana Perveen, Md. Nazmul Hasan, Md. Samsuzzoha, Nazmul Islam. Numerical Study of Creeping Flow Through Sinusoidally Periodic Tube. Int J Appl Math Theor Phys. 2019;5(3):72-81. doi: 10.11648/j.ijamtp.20190503.14

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@article{10.11648/j.ijamtp.20190503.14,
  author = {Asif Mahmud and Suhana Perveen and Md. Nazmul Hasan and Md. Samsuzzoha and Nazmul Islam},
  title = {Numerical Study of Creeping Flow Through Sinusoidally Periodic Tube},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {5},
  number = {3},
  pages = {72-81},
  doi = {10.11648/j.ijamtp.20190503.14},
  url = {https://doi.org/10.11648/j.ijamtp.20190503.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20190503.14},
  abstract = {There has been renewed interest in the flow behaviour within tubes with periodically varying cross-section with the recognition that they can be used as particle separation devices. In this paper, we present a numerical study of the effect of tube geometry on creeping flow of viscous incompressible fluid through sinusoidally constricted periodic tube which is axisymmetric but longitudinally asymmetric. The boundary element method is used to solve for the flow in the tube by specifying the pressure drop across the ends of the tube. The boundary element equations have been formulated for an infinite periodic tube by writing the velocity in terms of the integrals over the tube boundary and is used to calculate the force on the tube boundary, to obtain the detailed velocity distribution within the tube and to determine the effect of amplitude and wavelength of corrugation on the structure of the flow. We have found that the highest axial velocity is at throat region and lowest axial velocity is at expansion region. Also, we have discovered that the maximum radial velocity occurs at diverging cross-section and minimum radial velocity occurs at converging cross-section. The tangential force on the tube wall is examined for different amplitudes and wavelengths of corrugation and observed that the tangential force is greater in the constricted region than in the expansion region. The physical quantities (such as velocity and force) increase with increasing amplitude and decrease with increasing wavelength. Finally, we have compared our results with the work of Hemmat and Borhan [3] and have found good agreement with them.},
 year = {2019}
}

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TY  - JOUR
T1  - Numerical Study of Creeping Flow Through Sinusoidally Periodic Tube
AU  - Asif Mahmud
AU  - Suhana Perveen
AU  - Md. Nazmul Hasan
AU  - Md. Samsuzzoha
AU  - Nazmul Islam
Y1  - 2019/09/19
PY  - 2019
N1  - https://doi.org/10.11648/j.ijamtp.20190503.14
DO  - 10.11648/j.ijamtp.20190503.14
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 72
EP  - 81
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20190503.14
AB  - There has been renewed interest in the flow behaviour within tubes with periodically varying cross-section with the recognition that they can be used as particle separation devices. In this paper, we present a numerical study of the effect of tube geometry on creeping flow of viscous incompressible fluid through sinusoidally constricted periodic tube which is axisymmetric but longitudinally asymmetric. The boundary element method is used to solve for the flow in the tube by specifying the pressure drop across the ends of the tube. The boundary element equations have been formulated for an infinite periodic tube by writing the velocity in terms of the integrals over the tube boundary and is used to calculate the force on the tube boundary, to obtain the detailed velocity distribution within the tube and to determine the effect of amplitude and wavelength of corrugation on the structure of the flow. We have found that the highest axial velocity is at throat region and lowest axial velocity is at expansion region. Also, we have discovered that the maximum radial velocity occurs at diverging cross-section and minimum radial velocity occurs at converging cross-section. The tangential force on the tube wall is examined for different amplitudes and wavelengths of corrugation and observed that the tangential force is greater in the constricted region than in the expansion region. The physical quantities (such as velocity and force) increase with increasing amplitude and decrease with increasing wavelength. Finally, we have compared our results with the work of Hemmat and Borhan [3] and have found good agreement with them.
VL  - 5
IS  - 3
ER  -

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Author Information

Asif Mahmud

Mathematics Discipline, Khulna University, Khulna, Bangladesh
Suhana Perveen

Mathematics Discipline, Khulna University, Khulna, Bangladesh
Md. Nazmul Hasan

Mathematics Discipline, Khulna University, Khulna, Bangladesh
Md. Samsuzzoha

Department of Mathematics, Swinburne University of Technology, Victoria, Australia
Nazmul Islam

Mathematics Discipline, Khulna University, Khulna, Bangladesh

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APA Style

Asif Mahmud, Suhana Perveen, Md. Nazmul Hasan, Md. Samsuzzoha, Nazmul Islam. (2019). Numerical Study of Creeping Flow Through Sinusoidally Periodic Tube. International Journal of Applied Mathematics and Theoretical Physics, 5(3), 72-81. https://doi.org/10.11648/j.ijamtp.20190503.14

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ACS Style

Asif Mahmud; Suhana Perveen; Md. Nazmul Hasan; Md. Samsuzzoha; Nazmul Islam. Numerical Study of Creeping Flow Through Sinusoidally Periodic Tube. Int. J. Appl. Math. Theor. Phys. 2019, 5(3), 72-81. doi: 10.11648/j.ijamtp.20190503.14

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AMA Style

Asif Mahmud, Suhana Perveen, Md. Nazmul Hasan, Md. Samsuzzoha, Nazmul Islam. Numerical Study of Creeping Flow Through Sinusoidally Periodic Tube. Int J Appl Math Theor Phys. 2019;5(3):72-81. doi: 10.11648/j.ijamtp.20190503.14

Copy | Download

@article{10.11648/j.ijamtp.20190503.14,
  author = {Asif Mahmud and Suhana Perveen and Md. Nazmul Hasan and Md. Samsuzzoha and Nazmul Islam},
  title = {Numerical Study of Creeping Flow Through Sinusoidally Periodic Tube},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {5},
  number = {3},
  pages = {72-81},
  doi = {10.11648/j.ijamtp.20190503.14},
  url = {https://doi.org/10.11648/j.ijamtp.20190503.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20190503.14},
  abstract = {There has been renewed interest in the flow behaviour within tubes with periodically varying cross-section with the recognition that they can be used as particle separation devices. In this paper, we present a numerical study of the effect of tube geometry on creeping flow of viscous incompressible fluid through sinusoidally constricted periodic tube which is axisymmetric but longitudinally asymmetric. The boundary element method is used to solve for the flow in the tube by specifying the pressure drop across the ends of the tube. The boundary element equations have been formulated for an infinite periodic tube by writing the velocity in terms of the integrals over the tube boundary and is used to calculate the force on the tube boundary, to obtain the detailed velocity distribution within the tube and to determine the effect of amplitude and wavelength of corrugation on the structure of the flow. We have found that the highest axial velocity is at throat region and lowest axial velocity is at expansion region. Also, we have discovered that the maximum radial velocity occurs at diverging cross-section and minimum radial velocity occurs at converging cross-section. The tangential force on the tube wall is examined for different amplitudes and wavelengths of corrugation and observed that the tangential force is greater in the constricted region than in the expansion region. The physical quantities (such as velocity and force) increase with increasing amplitude and decrease with increasing wavelength. Finally, we have compared our results with the work of Hemmat and Borhan [3] and have found good agreement with them.},
 year = {2019}
}

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TY  - JOUR
T1  - Numerical Study of Creeping Flow Through Sinusoidally Periodic Tube
AU  - Asif Mahmud
AU  - Suhana Perveen
AU  - Md. Nazmul Hasan
AU  - Md. Samsuzzoha
AU  - Nazmul Islam
Y1  - 2019/09/19
PY  - 2019
N1  - https://doi.org/10.11648/j.ijamtp.20190503.14
DO  - 10.11648/j.ijamtp.20190503.14
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 72
EP  - 81
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20190503.14
AB  - There has been renewed interest in the flow behaviour within tubes with periodically varying cross-section with the recognition that they can be used as particle separation devices. In this paper, we present a numerical study of the effect of tube geometry on creeping flow of viscous incompressible fluid through sinusoidally constricted periodic tube which is axisymmetric but longitudinally asymmetric. The boundary element method is used to solve for the flow in the tube by specifying the pressure drop across the ends of the tube. The boundary element equations have been formulated for an infinite periodic tube by writing the velocity in terms of the integrals over the tube boundary and is used to calculate the force on the tube boundary, to obtain the detailed velocity distribution within the tube and to determine the effect of amplitude and wavelength of corrugation on the structure of the flow. We have found that the highest axial velocity is at throat region and lowest axial velocity is at expansion region. Also, we have discovered that the maximum radial velocity occurs at diverging cross-section and minimum radial velocity occurs at converging cross-section. The tangential force on the tube wall is examined for different amplitudes and wavelengths of corrugation and observed that the tangential force is greater in the constricted region than in the expansion region. The physical quantities (such as velocity and force) increase with increasing amplitude and decrease with increasing wavelength. Finally, we have compared our results with the work of Hemmat and Borhan [3] and have found good agreement with them.
VL  - 5
IS  - 3
ER  -

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