On the Motion Past a Slip-Stick Sphere and a Shear Free Sphere in a Viscous Fluid

Md Kamran Hussain Chowdhury; Sujit Kumar Sen; Anjan Kumar Chowdhury; Mohammad Jalal Ahammad

doi:doi:10.11648/j.eas.20240902.11

Research Article |

| Peer-Reviewed

On the Motion Past a Slip-Stick Sphere and a Shear Free Sphere in a Viscous Fluid

Md Kamran Hussain Chowdhury, Sujit Kumar Sen, Anjan Kumar Chowdhury, Mohammad Jalal Ahammad^*

Published in Engineering and Applied Sciences (Volume 9, Issue 2)

Received: 14 January 2024 Accepted: 18 March 2024 Published: 2 April 2024

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

The theoretical development of the complex fluid flows such as more than one obstacle with different shapes have great interest for scientists to understand flow phenomena and verify the model or approximate solution. The complex physical properties due to a uniform streaming motion past two fixed spheres is investigated having one with shear stress and another being shear stress-free. This study concerns analytical technique of a steady incompressible viscous fluid past to two fixed spheres. The Gegenbaur function and associated Legendre polynomials is used to derive the solution that simplify the process of the theoretical calculations. The mathematical expression for the flow fields are obtained in terms of stream functions by Gegenbaur function and associated Legendre polynomials. The physical properties of interest such as the Stokes stream function, the stress and its drag are calculated. It is understandable that for the uniform streaming motion around a sphere with the stress and its drag are affected owing to the presence of another stress-free are analyzed. The present result can be considered as a generalized by making other established results as a corollary of this solution. This theoretical study helps to the numerical or computational work for the verification of their approximated results of interest.

Published in	Engineering and Applied Sciences (Volume 9, Issue 2)
DOI	10.11648/j.eas.20240902.11
Page(s)	34-42
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), 2024. Published by Science Publishing Group

Keywords

Uniform Stream, Strokes’ Flow, Incompressible Viscous Fluid, Gegenbaur Function, Legendre Polynomials

References

[1]	G. G. Stokes, (1851) “On the Effect of the Internal Friction of Fluids on the Motion of Pendulums.” Transactions of the Cambridge Philosophical Society, Part II, 9, 8-106. https://doi.org/10.1017/CBO9780511702266.002
[2]	Sen, S. K., M. Kamran Chowdhury, and M. Jalal Ahammad. (2015) "The Axisymmetric Slow Viscous Flow About A Shear Stress Free Sphere." Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 15 Issue 1 Version 1.0, 1-11.
[3]	Ingham D. B. 1983, Steady flow past a rotating cylinder. Computers & Fluids, 11(4), pp 351-366. https://doi.org/10.1016/0045-7930(83)90020-8
[4]	Stimson, M. A. and Jeffery, G. B. (1926). The motion of two spheres in a viscous fluid, Proc. Roy. Soc. A, 111, 110. https://doi.org/10.1098/rspa.1926.0053
[5]	Payne L. E and Pell. W. H. (1960). The Stokes flow problem for a class of axially symmetric bodies, J. Fluid. Mech. https://doi.org/10.1017/S002211206000027X
[6]	Kawaguti. M. (1964). The flow of a perfect fluid around two moving bodies, J. Phys. Soc. Japan, 19, 14001415. https://doi.org/10.1143/JPSJ.19.1409
[7]	Weihs. D and Small. R. D. A. An exact solution of the motion of two adjacent spheres in axisymmetrie potential flow, israel J. of Tech. 13, 16.
[8]	A. B. Basset, 1961, A treatise on Hydrodynamics, Vol. 2. New York: Dover.
[9]	B. U. Felderhof, Force density on a sphere in linear hydrodynamics. I. Fixed sphere, stick boundary conditions, Physica A 84(1976a) 557–568. https://doi.org/10.1016/0378-4371(76)90104-7
[10]	B. U. Felderhof, Force density on a sphere in linear hydrodynamics. I. Moving sphere, mixed boundary conditions, Physica A 84(1976b) 569–576. https://doi.org/10.1016/0378-4371(76)90105-9
[11]	Schmitz, R., and B. U. Felderhof. "Creeping flow about a spherical particle." Physica A: Statistical Mechanics and its Applications 113, no. 1-2(1982): 90-102. https://doi.org/10.1016/0378-4371(82)90007-3
[12]	S. A. Wymer, A. Lakhtakia, R. S. Engel, Extinction cross-section of an arbitrary body in a viscous incompressible fluid, Phys. Rev. E 52(1995) 1857–1865. https://doi.org/10.1103/PhysRevE.52.1857
[13]	S. A. Wymer, A. Lakhtakia, R. S. Engel, The Huygens principle for flow around an arbitrary body in a viscous incompressible fluid, Fluid Dyn. Res. 17(1996) 212–223. https://doi.org/10.1016/0169-5983(95)00029-1
[14]	Palaniappan, D. and Daripa, P., 2001. Interior Stokes flows with stick-slip boundary conditions. Physica A: Statistical Mechanics and its Applications, 297(1-2), pp. 37-63. https://doi.org/10.1016/S0378-4371(01)00226-6
[15]	Ahammad, M. Jalal. "Non-existence of an Inviscid Fluid Motion between Two Fixed Cylinders." Journal of Bangladesh Academy of Sciences 34, no. 1(2010): 83-87.
[16]	Ahammad, M. Jalal, and Sujit K. Sen. "A Problem on the Inviscid Fluid Motion Exterior to Two Cylinders." Chiang Mai Journal of Science 38, no. 3(2011): 360-369.
[17]	Sen, S. K., and M. J. Ahammad. "A problem on viscous fluid motion between two fixed cylinders." Journal of Bangladesh Academy of Sciences 33, no. 1(2009): 107-115.
[18]	James W. Swan & Aditya S. Khair. On the hydrodynamics of ‘slip–stick’ spheres. Journal of Fluid Mechanics. 2008, 606, 115-132. https://doi.org/10.1017/S0022112008001614
[19]	Lamb. H. (1932). Hydrodynamics, Cambridge at the University Press.
[20]	Harper, J. F. "Axisymmertic Stokes flow images in spherical free surfaces with applications to rising bubbles." The ANZIAM Journal 25, no. 2(1983): 217-231. https://doi.org/10.1017/S0334270000004021
[21]	Happel, John, and Howard Brenner. Low Reynolds number hydrodynamics: with special applications to particulate media. Vol. 1. Springer Science & Business Media, 2012.
[22]	Rybczynski, W. "On the translatory motion of a fluid sphere in a viscous medium." Bull. Acad. Sci., Cracow, Series A 40, no. 3(1911): 073605-18.
[23]	Padrino, Juan C., Daniel D. Joseph, Toshio Funada, Jing Wang, and William A. Sirignano. "Stress-induced cavitation for the streaming motion of a viscous liquid past a sphere." Journal of Fluid Mechanics 578(2007): 381-411. https://doi.org/10.1017/S002211200700506X
[24]	Collins, W. D. "A Note on Stokes’ Stream Function for the Slow Steady Motion of Viscous Fluid Before a Plane and Spherical Boundary." Mathematika, 1954, 1, 125-130.
[25]	Palaniappan, D., Nigam, S. D., Amaranath, T., & Usha, R. (1990). A theorem for a shear-free sphere in Stokes’ flow. Mechanics Research Communications, 17(6), 429–435. https://doi.org/10.1016/0093-6413(90)90062-H
[26]	R. Usha., Hemalatha, K.. A note on plane Stokes flow past a shear-free impermeable cylinder. Z. angew. Math. Phys., 1993 44, 73–84(1943). https://doi.org/10.1007/BF00914354
[27]	Arunachalam, P. V., and S. N. Majhi. "Secondary flow due to slow rotation of two spheres." The Quarterly Journal of Mechanics and Applied Mathematics 40, no. 1 1987): 47-55. https://doi.org/10.1093/qjmam/40.1.47
[28]	Padmavathi, B. S., T. Amaranath, and S. D. Nigam. "Stokes flow past a sphere with mixed slip-stick boundary conditions." Fluid dynamics research 11, no. 5(1993): 229-234. https://doi.org/10.1016/0169-5983(93)90113-O
[29]	Chowdhury M. K. (2010), Fluid motion past rigid bodies, M. Phil, thesis, RCMPS, University of Chittagong.

Cite This Article

Plain Text BibTeX RIS

APA Style

Chowdhury, M. K. H., Sen, S. K., Chowdhury, A. K., Ahammad, M. J. (2024). On the Motion Past a Slip-Stick Sphere and a Shear Free Sphere in a Viscous Fluid. Engineering and Applied Sciences, 9(2), 34-42. https://doi.org/10.11648/j.eas.20240902.11

Copy | Download

ACS Style

Chowdhury, M. K. H.; Sen, S. K.; Chowdhury, A. K.; Ahammad, M. J. On the Motion Past a Slip-Stick Sphere and a Shear Free Sphere in a Viscous Fluid. Eng. Appl. Sci. 2024, 9(2), 34-42. doi: 10.11648/j.eas.20240902.11

Copy | Download

AMA Style

Chowdhury MKH, Sen SK, Chowdhury AK, Ahammad MJ. On the Motion Past a Slip-Stick Sphere and a Shear Free Sphere in a Viscous Fluid. Eng Appl Sci. 2024;9(2):34-42. doi: 10.11648/j.eas.20240902.11

Copy | Download

@article{10.11648/j.eas.20240902.11,
  author = {Md Kamran Hussain Chowdhury and Sujit Kumar Sen and Anjan Kumar Chowdhury and Mohammad Jalal Ahammad},
  title = {On the Motion Past a Slip-Stick Sphere and a Shear Free Sphere in a Viscous Fluid
},
  journal = {Engineering and Applied Sciences},
  volume = {9},
  number = {2},
  pages = {34-42},
  doi = {10.11648/j.eas.20240902.11},
  url = {https://doi.org/10.11648/j.eas.20240902.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.eas.20240902.11},
  abstract = {The theoretical development of the complex fluid flows such as more than one obstacle with different shapes have great interest for scientists to understand flow phenomena and verify the model or approximate solution. The complex physical properties due to a uniform streaming motion past two fixed spheres is investigated having one with shear stress and another being shear stress-free. This study concerns analytical technique of a steady incompressible viscous fluid past to two fixed spheres. The Gegenbaur function and associated Legendre polynomials is used to derive the solution that simplify the process of the theoretical calculations. The mathematical expression for the flow fields are obtained in terms of stream functions by Gegenbaur function and associated Legendre polynomials. The physical properties of interest such as the Stokes stream function, the stress and its drag are calculated. It is understandable that for the uniform streaming motion around a sphere with the stress and its drag are affected owing to the presence of another stress-free are analyzed. The present result can be considered as a generalized by making other established results as a corollary of this solution. This theoretical study helps to the numerical or computational work for the verification of their approximated results of interest.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - On the Motion Past a Slip-Stick Sphere and a Shear Free Sphere in a Viscous Fluid

AU  - Md Kamran Hussain Chowdhury
AU  - Sujit Kumar Sen
AU  - Anjan Kumar Chowdhury
AU  - Mohammad Jalal Ahammad
Y1  - 2024/04/02
PY  - 2024
N1  - https://doi.org/10.11648/j.eas.20240902.11
DO  - 10.11648/j.eas.20240902.11
T2  - Engineering and Applied Sciences
JF  - Engineering and Applied Sciences
JO  - Engineering and Applied Sciences
SP  - 34
EP  - 42
PB  - Science Publishing Group
SN  - 2575-1468
UR  - https://doi.org/10.11648/j.eas.20240902.11
AB  - The theoretical development of the complex fluid flows such as more than one obstacle with different shapes have great interest for scientists to understand flow phenomena and verify the model or approximate solution. The complex physical properties due to a uniform streaming motion past two fixed spheres is investigated having one with shear stress and another being shear stress-free. This study concerns analytical technique of a steady incompressible viscous fluid past to two fixed spheres. The Gegenbaur function and associated Legendre polynomials is used to derive the solution that simplify the process of the theoretical calculations. The mathematical expression for the flow fields are obtained in terms of stream functions by Gegenbaur function and associated Legendre polynomials. The physical properties of interest such as the Stokes stream function, the stress and its drag are calculated. It is understandable that for the uniform streaming motion around a sphere with the stress and its drag are affected owing to the presence of another stress-free are analyzed. The present result can be considered as a generalized by making other established results as a corollary of this solution. This theoretical study helps to the numerical or computational work for the verification of their approximated results of interest.

VL  - 9
IS  - 2
ER  -

Copy | Download

Author Information

Md Kamran Hussain Chowdhury

Department of Mathematics, Directorate of Secondary and Higher Education, Dhaka, Bangladesh

Contact Email
Sujit Kumar Sen

Jamal Nazrul Islam Research Centre for Mathematical and Physical Sciences, University of Chittagong, Chattogram, Bangladesh
Anjan Kumar Chowdhury

Jamal Nazrul Islam Research Centre for Mathematical and Physical Sciences, University of Chittagong, Chattogram, Bangladesh

Contact Email
Mohammad Jalal Ahammad

Department of Mathematics, University of Chittagong, Chattogram, Bangladesh

Contact Email

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Chowdhury, M. K. H., Sen, S. K., Chowdhury, A. K., Ahammad, M. J. (2024). On the Motion Past a Slip-Stick Sphere and a Shear Free Sphere in a Viscous Fluid. Engineering and Applied Sciences, 9(2), 34-42. https://doi.org/10.11648/j.eas.20240902.11

Copy | Download

ACS Style

Chowdhury, M. K. H.; Sen, S. K.; Chowdhury, A. K.; Ahammad, M. J. On the Motion Past a Slip-Stick Sphere and a Shear Free Sphere in a Viscous Fluid. Eng. Appl. Sci. 2024, 9(2), 34-42. doi: 10.11648/j.eas.20240902.11

Copy | Download

AMA Style

Chowdhury MKH, Sen SK, Chowdhury AK, Ahammad MJ. On the Motion Past a Slip-Stick Sphere and a Shear Free Sphere in a Viscous Fluid. Eng Appl Sci. 2024;9(2):34-42. doi: 10.11648/j.eas.20240902.11

Copy | Download

@article{10.11648/j.eas.20240902.11,
  author = {Md Kamran Hussain Chowdhury and Sujit Kumar Sen and Anjan Kumar Chowdhury and Mohammad Jalal Ahammad},
  title = {On the Motion Past a Slip-Stick Sphere and a Shear Free Sphere in a Viscous Fluid
},
  journal = {Engineering and Applied Sciences},
  volume = {9},
  number = {2},
  pages = {34-42},
  doi = {10.11648/j.eas.20240902.11},
  url = {https://doi.org/10.11648/j.eas.20240902.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.eas.20240902.11},
  abstract = {The theoretical development of the complex fluid flows such as more than one obstacle with different shapes have great interest for scientists to understand flow phenomena and verify the model or approximate solution. The complex physical properties due to a uniform streaming motion past two fixed spheres is investigated having one with shear stress and another being shear stress-free. This study concerns analytical technique of a steady incompressible viscous fluid past to two fixed spheres. The Gegenbaur function and associated Legendre polynomials is used to derive the solution that simplify the process of the theoretical calculations. The mathematical expression for the flow fields are obtained in terms of stream functions by Gegenbaur function and associated Legendre polynomials. The physical properties of interest such as the Stokes stream function, the stress and its drag are calculated. It is understandable that for the uniform streaming motion around a sphere with the stress and its drag are affected owing to the presence of another stress-free are analyzed. The present result can be considered as a generalized by making other established results as a corollary of this solution. This theoretical study helps to the numerical or computational work for the verification of their approximated results of interest.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - On the Motion Past a Slip-Stick Sphere and a Shear Free Sphere in a Viscous Fluid

AU  - Md Kamran Hussain Chowdhury
AU  - Sujit Kumar Sen
AU  - Anjan Kumar Chowdhury
AU  - Mohammad Jalal Ahammad
Y1  - 2024/04/02
PY  - 2024
N1  - https://doi.org/10.11648/j.eas.20240902.11
DO  - 10.11648/j.eas.20240902.11
T2  - Engineering and Applied Sciences
JF  - Engineering and Applied Sciences
JO  - Engineering and Applied Sciences
SP  - 34
EP  - 42
PB  - Science Publishing Group
SN  - 2575-1468
UR  - https://doi.org/10.11648/j.eas.20240902.11
AB  - The theoretical development of the complex fluid flows such as more than one obstacle with different shapes have great interest for scientists to understand flow phenomena and verify the model or approximate solution. The complex physical properties due to a uniform streaming motion past two fixed spheres is investigated having one with shear stress and another being shear stress-free. This study concerns analytical technique of a steady incompressible viscous fluid past to two fixed spheres. The Gegenbaur function and associated Legendre polynomials is used to derive the solution that simplify the process of the theoretical calculations. The mathematical expression for the flow fields are obtained in terms of stream functions by Gegenbaur function and associated Legendre polynomials. The physical properties of interest such as the Stokes stream function, the stress and its drag are calculated. It is understandable that for the uniform streaming motion around a sphere with the stress and its drag are affected owing to the presence of another stress-free are analyzed. The present result can be considered as a generalized by making other established results as a corollary of this solution. This theoretical study helps to the numerical or computational work for the verification of their approximated results of interest.

VL  - 9
IS  - 2
ER  -

Copy | Download