Peristaltic transport of a Herschel-Bulkley fluid in contact with a Newtonian fluid

Authors:
K. Vajravelu, S. Sreenadh and V. Ramesh Babu

Journal:
Quart. Appl. Math. **64** (2006), 593-604

MSC (2000):
Primary 34B15, 92C10

DOI:
https://doi.org/10.1090/S0033-569X-06-01020-9

Published electronically:
September 11, 2006

MathSciNet review:
2284461

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Abstract | References | Similar Articles | Additional Information

Abstract: Peristaltic transport of Herschel-Bulkley fluid in contact with a Newtonian fluid in a channel is investigated for its various applications to flows with physiological fluids (blood, chyme, intrauterine fluid, etc.). The primary application is when blood flows through small vessels; blood has a peripheral layer of plasma and a core region of suspension of all the erythrocytes. That is, in the modeling of blood flow, one needs to consider the core region consisting of a yield stress fluid and the peripheral region consisting of a Newtonian fluid. Peristaltic pumping of a yield stress fluid in contact with a Newtonian fluid has not previously been studied in detail. Our goal is to initiate such a study. The Herschel-Bulkley fluid model considered here reduces to the power law model in the absence of yield stress.

The stream function, the velocity field, and the equation of the interface are obtained and discussed. When the yield stress and when the index , our results agree with those of Brasseur et al. (*J. Fluid Mech.* **174** (1987), 495) for peristaltic transport of the Newtonian fluid. It is observed that for a given flux the pressure rise increases with an increase in the amplitude ratio . Furthermore, the results obtained for the flow characteristics reveal many interesting behaviors that warrant further study of the peristaltic transport models with two immiscible physiological fluids.

**1.**T. W. Latham, Fluid motions in a peristaltic pump, M.S. Thesis, M.I.T., Cambridge, Massachusetts, 1966.**2.**A. H. Shapiro, Pumping and retrograde diffusion in peristaltic waves,*Proc. Workshop in Ureteral Reflux in Children*(1967), 109-126.**3.**J. C. Burns and T. Parkes, Peristaltic motion,*J. Fluid Mech.***29**(1967), 731-743.**4.**A. H. Shapiro, M. Y. Jaffrin and S. L. Weinberg, Peristaltic pumping with log wavelength at low Reynolds number,*J. Fluid Mech.***37**(1969), 799-825.**5.**M. Y. Jaffrin, Inertia and streamline curvature on peristaltic pumping,*Int. J. Engrg. Sci.***11**(1973), 681-699.**6.**C. Barton and S. Raynor, Peristaltic flow in tubes,*Bull. Math. Biophysics***30**(1968), 663-680.**7.**C. C. Yin and Y. C. Fung, Peristaltic waves in circular cylindrical tubes,*Trans. ASME J. Appl. Mech.***36**(1969), 579-587.**8.**M. Y. Jaffrin and A. H. Shapiro, Peristaltic pumping,*Ann. Rev. Fluid Mech.***3**(1971), 13-36.**9.**J. N. Kapur,*Mathematical models in biology and medicine*, Affiliated East-West Press Pvt. Ltd., New Delhi, 1985. MR**806352****10.**G. W. S. Blair and D. C. Spanner,*An Introduction to Amsterdam*, Oxford and New York; Elsevier Biorheology Scient. Pub. Co., 1974.**11.**P. Chaturani and R. P. Samy, A study of non-Newtonian aspects of blood flow through stenosed arteries and its application in arterial diseases,*J. Biorheology***22**(1985), 521-531.**12.**G. B. Bohme and R. Friedrich, Peristaltic flow of viscoelastic liquids,*J. Fluid Mech.***138**(1983), 109-122.**13.**E. F. E. Shehawey and K. S. Mekheimer, Couple-stresses in peristaltic transport of fluids,*J. Physics D: Appl. Phys.***27**(1994), 1163-1170.**14.**O. Eytan, A. J. Jaffa and D. Elad, Peristaltic flow in a tapered channel; application to embryo transport within the uterine cavity,*J. Medical Engrg. Physics***23**(2001), 473-482.**15.**J. M. Krishnan and K. R. Rajagopal, Review of the uses and modeling of bitumen from ancient to modern times,*Appl. Mech. Reviews***56**(2003), 149-214.**16.**J. Málek, J. Nečas, M. Rokyta, and M. R\ocirc užička,*Weak and measure-valued solutions to evolutionary PDEs*, Applied Mathematics and Mathematical Computation, vol. 13, Chapman & Hall, London, 1996. MR**1409366****17.**J. Málek, J. Nečas, and K. R. Rajagopal,*Global existence of solutions for flows of fluids with pressure and shear dependent viscosities*, Appl. Math. Lett.**15**(2002), no. 8, 961–967. MR**1925921**, https://doi.org/10.1016/S0893-9659(02)00070-8**18.**J. Málek, K. R. Rajagopal, and M. R\ocirc užička,*Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity*, Math. Models Methods Appl. Sci.**5**(1995), no. 6, 789–812. MR**1348587**, https://doi.org/10.1142/S0218202595000449**19.**A. N. Alexandrou, T. M. McGilvreay and G. Burgos, Steady Herschel-Bulkley fluid flow in three-dimensional expansions,*J. Non-Newtonian Fluid Mech.***100**(2001), 77-96.**20.**J. B. Shukla, R. S. Parihar, B. R. P. Rao, and S. P. Gupta,*Effects of peripheral-layer viscosity on peristaltic transport of a bio-fluid*, J. Fluid Mech.**97**(1980), no. 2, 225–237. MR**566650**, https://doi.org/10.1017/S0022112080002534**21.**J. G. Brasseur, S. Corrsin and N. W. Lu, The influence of a peripheral layer of different viscosity on peristaltic pumping with Newtonian fluid,*J. Fluid Mech.***174**(1987), 495-519.**22.**A. R. Rao and S. Usha, Peristaltic transport of two immiscible viscous fluids in a circular tube,*J. Fluid Mech.***298**(1995), 271-285.**23.**S. Usha and A. R. Rao, Peristaltic transport of two-layered power-law fluids,*J. Biomech. Engrg.***119**(1997), 483-488.**24.**J. M. Zahm, D. Pierrot, C. Fuchey, J. Levrier, D. Duval, K. G. Lloyd and E. Puchelle, Comparative rheological profile of rate gastric and duodenal gel mucus,*Biorheology***26**(1989), 813-822.

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Additional Information

**K. Vajravelu**

Affiliation:
Department of Mathematics, University of Central Florida, Orlando, Florida 32816

**S. Sreenadh**

Affiliation:
Department of Mathematics, Sri Venkateswara University, Tirupati 517502, India

**V. Ramesh Babu**

Affiliation:
Department of Mathematics, S. V. Arts College, Tirupati, India

DOI:
https://doi.org/10.1090/S0033-569X-06-01020-9

Keywords:
Peristaltic transport,
Herschel-Bulkley fluid,
Newtonian fluid,
volume flow rate

Received by editor(s):
June 21, 2004

Published electronically:
September 11, 2006

Article copyright:
© Copyright 2006
Brown University