Abstract
The unsteady dispersion of a solute by an imposed pulsatile pressure gradient in a tube is studied by modeling the flowing fluid as a Casson fluid. The generalized dispersion model is applied to study the dispersion process, and according to this process, the entire dispersion process is expressed in terms of two coefficients, the convection and the dispersion coefficients. This model mainly brings out the effects of yield stress and flow pulsatility on the dispersion process. It is observed that the dispersion phenomenon in the pulsatile flow inherently differs from the steady flow, which is due to a change in the plug flow radius during a cycle of oscillation. Also, it was found that the dispersion coefficient fluctuates due to the oscillatory nature of the velocity. It is seen that the dispersion coefficient changes cyclically, and the amplitude and magnitude of the dispersion coefficient increases initially with time and reaches a non-transient state after a certain critical time. It is also seen that this critical time varies with Womersley frequency parameter and Schmidt number and is independent of yield stress and fluctuating pressure component. It is observed that the yield stress and Womersley frequency parameter inhibit the dispersion of a solute. It is also observed that the dispersion coefficient decreased approximately 4 times as the Womersley frequency parameter increases from 0.5 to 1. The study can be used in the understanding of the dispersion process in the cardiovascular system and blood oxygenators.
Similar content being viewed by others
References
Taylor G.I.: Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. A 219, 186–203 (1953)
Aris R.: On dispersion of solute in a fluid flowing through a tube. Proc. R. Soc. Lond. Ser. A 235, 67–77 (1956)
Gill W.N., Sankarasubramanian R.: Exact analysis of unsteady convective diffusion. Proc. Roy. Soc. Lond. A 316, 341–350 (1970)
Aris R.: On dispersion of solute in a pulsating flow through a tube. Proc. R. Soc. Lond. A 259, 370–376 (1960)
Chatwin P.C.: On the longitudinal dispersion of passive contaminant in oscillatory flow in tube. J. Fluid Mech. 7, 513–527 (1975)
Purtel L.P.: Molecular diffusion in oscillating laminar flow in a pipe. Phys. Fluids 24, 789–793 (1981)
Smith R.: Contaminant dispersion in oscillating flows. J. Fluid Mech. 114, 379–398 (1982)
Bandyopadhyay S., Mazumder B.S.: Unsteady convective diffusion in a pulsatile flow through a channel. Acta Mech. 134, 1–16 (1999)
Mazumder B.S, Mandal K.K.: On the solute transport in oscillatory flow through an annular pipe with a reactive wall and its application to a catheterized artery. Q. J. Mech. Appl. Math. 58, 349–365 (2005)
Sarkar A., Jayaraman G.: The effect of wall absorption on dispersion in oscillatory flow in an annulus. Acta Mech. 172, 151–167 (2004)
Ng C.-O.: Dispersion in steady and oscillatory flows through a tube with reversible and irreversible wall reactions. Proc. R. Soc. A 462, 481–515 (2006)
Kumar S., Jayaraman G.: Method of moments for laminar dispersion in an oscillatory flow through curved channels with absorbing walls. Heat Mass Transf. 44, 1323–1336 (2008)
Paul S.: Axial dispersion in pressure perturbed flow through an annular pipe oscillating around its axis. ZAMP 60, 899–920 (2009)
Paul S.: Effect of wall oscillation on dispersion in axis-symmetric flows between two coaxial cylinders. ZAMM 91, 23–37 (2011)
Agarwal S., Jayaraman G.: Numerical simulation of dispersion in the flow of power-law fluids in curved tubes. Appl. Math. Model. 18, 504–512 (1994)
Lighthill M.J.: Initial development of diffusion in Poiseuille flow. J. Inst. Math. Appl. 51, 97–108 (1966)
Hazra S.B., Gupta A.S., Niyogi P.: On the dispersion of solute in oscillatory flow of a non-Newtonian fluid in a channel. Heat Mass Transf. 32, 481–487 (1997)
Sharp M.K.: Shear augmented dispersion in non-Newtonian fluids. Ann. Biomed. Eng. 21, 407–415 (1993)
Dash R.K., Jayaraman G., Mehta K.N.: Shear augmented dispersion of a solute in a Casson fluid flowing in a conduit. Ann. Biomed. Eng. 28, 373–385 (2000)
Nagarani P., Sarojamma G., Jayaraman G.: Effect of boundary absorption in dispersion in Casson fluid flow in a tube. Ann. Biomed. Eng. 32, 706–719 (2004)
Aroesty J., Gross J.F.: The mathematics of pulsatile flow in small vessels I. Casson theory. Microvasc. Res. 4, 1–12 (1972)
Aroesty J., Gross J.F.: Pulsatile flow in small blood vessels, I. Casson theory. Biorheology 9, 33–42 (1972)
Fung Y.C.: Biomechanics: Mechanical Properties of Living Tissues. Springer, New York (1981)
Cokelet G.R., Merrill E.W., Gilliand E.R., Shin H., Britten A., Wells R.E.: The rheology of human blood measurement near and at zero shear rate. Trans. Soc. Rheol. 7, 303–317 (1963)
Merrill E.W., Benis A.M., Gilland E.R., Shearwood T.K., Salzman E.W.: Pressure flow relations of human blood in hollow fiber at low shear rates. J. Appl. Physiol. 20, 954–967 (1965)
Dash R.K., Jayaraman G., Mehta K.N.: Estimation of increased flow resistance in a narrow catheterized artery-a theoretical model. J. Biomech. 29, 917–930 (1996)
Nagarani P., Sarojamma G.: Effect of body acceleration on pulsatile flow of Casson fluid through a mild stenosed artery. Korea Aust. Rheol. J. 20, 189–196 (2008)
Boyce W.E., DiPrima R.C.: Elementary Differential Equations and Boundary Value Problems. Wiley, USA (2004)
Turskey G.A., Yuan F., Katz D.F.: Transport Phenomena in Biological Systems. Pearson Prentice Hall, New Jersey (2004)
Ananthkrishnan V., Gill W.N., Barduhan A.J.: Laminar dispersion in capillaries: part I. Math. Anal. AICHE J. 11, 1063–1072 (1965)
Reejhsinghani N.S., Gill W.N., Barduhan A.J.: Laminar dispersion in capillaries: part III. Experiments in horizontal tubes including observations on natural convection effects. AICHE J. 12, 916–923 (1966)
Lighthill M.J.: Physiological fluid dynamics: a survey. J. fluid. Mech. 52, 475–497 (1972)
Caro C.G., Pedley T.J., Schroter R.C., Seed W.A.: The Mechanics of Circulation. Oxford University Press, New York (1978)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nagarani, P., Sebastian, B.T. Dispersion of a solute in pulsatile non-Newtonian fluid flow through a tube. Acta Mech 224, 571–585 (2013). https://doi.org/10.1007/s00707-012-0753-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-012-0753-6