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 JAMP  Vol.5 No.12 , December 2017
The Oscillatory Motion of Oldroyd-B Fluid by Incorporating Some of the Mechanical Factors
Abstract: The aim of this paper is to present a numerical study of oscillatory motion of Oldroyd-B fluid in a uniform magnetic field through a small circular pipe. First, we derive the orientation stress tensor by considering the Brownian force. Then, the orientation stress tensor is incorporated by taking Hookean dumbbells on Brownian configuration fields in the Oldroyd-B model. The Oldroyd-B model is then reformulated coupled with the momentum equation and the total stress tensor. Finally, we analyze the orientation stress tensor in the pipe by the numerical simulations of the model and showed that the effect of orientation stress tensor is considerable although the Brownian force is sufficiently small.
Cite this paper: Khan, A. , Zaman, G. , Li, Y. , Ahmad, S. and Hussain, A. (2017) The Oscillatory Motion of Oldroyd-B Fluid by Incorporating Some of the Mechanical Factors. Journal of Applied Mathematics and Physics, 5, 2402-2410. doi: 10.4236/jamp.2017.512196.
References

[1]   Maxwell, J.C. (1866) On the Dynamical Theory of Gases. Philosophical Transactions of the Royal Society, 157, 49-88.

[2]   Rajagopal, K.R. and Srinivasa, A. (2000) A Thermodynamic Framework for Rate Type Fluid Models. Journal of Non-Newtonian Fluid Mechanics, 88, 207-227.
https://doi.org/10.1016/S0377-0257(99)00023-3

[3]   Zaman, G., Islam, S., Kang, Y.H. and Jung, I.H. (2012) Blood Flow of an Oldroyd-B Fluid in a Blood Vessel Incorporating a Brownian Stress. Science China Physics, Mechanics and Astronomy, 55, 125-131.
https://doi.org/10.1007/s11433-011-4571-y

[4]   Khan, A. and Zaman, G. (2017) Hydromagnetic Flow near an Accelerating Plate in the Presence of Magnetic Field through Porous Medium. Georgian Mathematical Journal.
https://doi.org/10.1515/gmj-2017-0017

[5]   Khan, A. and Zaman, G. (2015) The Motion of a Generalized Oldroyd-B Fluid Between Two Side Walls of a Plate. South Asian Journal of Mathematics, 5, 42-52.

[6]   Khan, A. Zaman, G. and Rahman, G. (2015) Hydromagnetic Flow near a Non-Uniform Accelerating Plate in the Presence of Magnetic Field through Porous Medium. Journal of Porous Media, 18, 801-809.
https://doi.org/10.1615/JPorMedia.v18.i8.50

[7]   Khan, A. and Zaman, G. (2016) Unsteady Magneto-Hydrodynamic Flow of Second Grade Fluid Due To Uniform Accelerating Plate. Journal of Applied Fluid Mechanics, 9, 3127-3133.

[8]   Pontrelli, G. (2000) Blood Flow through a Circular Pipe with an Impulsive Pressure Gradient. Mathematical Models and Methods in Applied Sciences, 10, 187-202.
https://doi.org/10.1142/S0218202500000124

[9]   Pontrelli, G. (2001) Blood Flow through an Axisymmetric Stenosis. Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine, 215, 1-10.
https://doi.org/10.1177/095441190121500101

[10]   Yeleswarapu, K.K., Kameneva, M.V., Rajagopal, K.R. and Antaki, J.F. (1998) The Flow of Blood in Tubes: Theory and Experiment. Mechanics Research Communications, 25, 257-262.
https://doi.org/10.1016/S0093-6413(98)00036-6

[11]   Lee, S.W., Sohn, S.M., Ryu, S.H., Kim, C. and Song, K.W. (2001) Experimental Studies on the Axisymmetric Sphere-Wall Interaction in Newtonian and Non-Newtonian Fluids. Korea-Australia Rheology Journal, 13, 141-148.

[12]   Mekheimer, Kh.S. (2008) Effect of the Induced Magnetic Field on Peristaltic Flow of a Couple Stress Fluid. Physics Letters A, 372, 4271-4278.
https://doi.org/10.1016/j.physleta.2008.03.059

[13]   Quarteroni, A. (2006) What Mathematics Can Do for Simulation of Blood Circulation.MOX Report, Jan 16.

[14]   Bellert, S.L. (2001) Computational Fluid Dynamics of the Human Carotid Bifuraction. B. E. Thesis, University of Queensland, Queensland.

[15]   Pries, A.R., Neuhaus, D. and Gaehtgens Freie, P. (1992) Blood Viscosity in Tube Flow: Dependence on Diameter and Hematocrit. American Journal of Physiology, 263, H1770-H1778.
https://doi.org/10.1152/ajpheart.1992.263.6.H1770

[16]   Fung, Y.C. (1993) Biomechanics: Mechanical Propreties of Living Tissues. Springer-Verlag, New York.
https://doi.org/10.1007/978-1-4757-2257-4

[17]   Song, Y.S. and Youn, J.R. (2004) Modeling of Rheological Behavior of Nanocomposites by Brownian Dynamics Simulation. Korea-Australia Rheology Journal, 16, 201-212.

[18]   Wilson, H.J. (2006) Polymeric Fluid Lecture, GM05 part 1. Jan 9.

[19]   Underhilla, P.T. and Doyle, P.S. (2007) Accuracy of Bead-Spring Chains in Strong Flows. Journal of Non-Newtonian Fluid Mechanics, 145, 109-123.
https://doi.org/10.1016/j.jnnfm.2007.05.011

[20]   Everitt, S.L. Harlen, O.G., Wilson, H.J. and Read, D.J. (2003) Bubble Dynamics in Viscoelastic Fluids with Application to Reacting and Non-Reacting Polymer Foams, Journal of Non-Newtonian Fluid Mechanics, 114, 83-107.
https://doi.org/10.1016/S0377-0257(03)00108-3

 
 
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