Peristaltic-Ciliary Flow of A Casson Fluid through An Inclined Tube

Saravana Ramachandran; Kuppalapalle Vajravelu; K.V. Prasad; S. Sreenadh

doi:10.5614/cbms.2021.4.1.3

Authors

Saravana Ramachandran Department of Mathematics, Madanapalle Institute of Technology & Science, Madanapalle 517 325, India https://orcid.org/0000-0002-7360-1179
Kuppalapalle Vajravelu Department of Mathematics, University of Central Florida, Orlando, Florida 32816, USA
K.V. Prasad Department of Mathematics, VSK University, Vinayaka Nagar, Ballari-583 105, Karnataka, India
S. Sreenadh Department of Mathematics, Sri Venkateswara University, Tirupati 517 502, India

DOI:

https://doi.org/10.5614/cbms.2021.4.1.3

Keywords:

peristalsis-cilia flow, trapping phenomena, Casson fluid, inclined tube, pressure gradient

Abstract

The paper is concerned with the peristaltic-ciliary transport of a viscoplastic fluid (Casson fluid) through an inclined cylindrical tube. The peristalsis-cilia induced motion is analysed in the moving frame of reference under the lubrication approximations. Solutions to the flow characteristics petering to yielded and unyielded regions are obtained. The effects of various physical parameters on the axial velocity, the pumping characteristics, the pressure rise, and the frictional force over one wavelength, along with the trapping phenomenon are presented through graphs. Further, the peristaltic flow and peristaltic-ciliary flow results are compared. It is noticed that the axial velocity and the size of trapping bolus in the unplug flow region decrease with an increase in the yield stress. In addition, the axial velocity and the axial pressure gradient in the peristaltic-ciliary pumping are higher than those in the peristaltic pumping.

References

Shapiro, A.H., Jaffrin, M.Y. and Weinberg, S.L., Peristaltic pumping with long wave lengths at low Reynolds number, Journal of Fluid Mechanics, 37(4), pp. 799-825, July 1969.

Ramachandra Rao, A. and Mishra, M., Peristaltic transport of a power-law fluid in a porous tube, Journal of Non-Newtonian Fluid Mechanics, 121(2-3), pp. 163-174, 2004. https://doi.org/10.1016/j.jnnfm.2004.06.006.

Hayat, T., and Ali, N., A mathematical description of peristaltic hydromagnetic flow in a tube, Applied Mathematics and Computation, 188(2), pp. 1491-1502, 2007. https://doi.org/10.1016/j.amc.2006.11.035.

Saravana, R., Sreenadh, S., Kumar, P.R. and Babu, V.R., Peristaltic pumping of Ellis fluid through a flexible tube with complete slip effects, Journal of Naval Architecture and Marine Engineering, 17(2), pp. 79-88. 2020. https://doi.org/10.3329/jname.v17i2.49559.

Vajravelu, K., Sreenadh, S. and Saravana, R., Combined influence of velocity slip, temperature and concentration jump conditions on MHD peristaltic transport of a Carreau fluid in a non-uniform channel with wall properties, Applied Mathematics and Computation, 225, pp. 656-676, 2013. https://doi.org/10.1016/j.amc.2013.10.014.

Saravana, R., Hemadri Reddy, R., Suresh Goud, J., and Sreenadh, S., MHD peristaltic flow of a Hyperbolic tangent fluid in a non-uniform channel with heat and mass transfer, IOP Conf. Series: Materials Science and Engineering 263, pp. 1-15, 2017. https://doi.org/10.1088/1757-899x/263/6/062006.

Lardner, T.J. and Shack, W.J., Cilia transport, The bulletin of mathematical biophysics, 34(3), pp. 325-335. 1972. https://doi.org/10.1007/BF02476445.

Nadeem, S., Munim, A., Shaheen, A. and Hussain, S., Physiological flow of Carreau fluid due to ciliary motion, AIP Advances, 6(3), Article ID: 035125, 2016. https://doi.org/10.1063/1.4945270.

Nadeem, S. and Sadaf, H., Metachronal wave of cilia transport in a curved channel, Zeitschrift fur Naturforschung A, 70(1), pp. 33-38, 2015. https://doi.org/10.1515/zna-2014-0117

Maiti, S. and Pandey, S.K., Rheological fluid motion in tube by metachronal waves of cilia. Applied Mathematics and Mechanics, 38(3), pp.393-410, 2017. https://doi.org/10.1007/s10483-017-2179-8.

Leese, H.J., The formation and function of oviduct fluid. Reproduction, 82(2), pp. 843-856, 1988. https://doi.org/10.1530/jrf.0.0820843.

Croxatto, H.B., Physiology of gamete and embryo transport through the fallopian tube. Reproductive biomedicine online, 4(2), pp. 160-169, 2002. https://doi.org/10.1016/S1472-6483(10)61935-9.

Lyons, R.A., Saridogan, E. and Djahanbakhch, O., The reproductive significance of human Fallopian tube cilia. Human reproduction update, 12(4), pp. 363-372. 2006. https://doi.org/10.1093/humupd/dml012.

Ashraf, H., Siddiqui, A.M. and Rana, M.A., Fallopian tube assessment of the peristaltic-ciliary flow of a linearly viscous fluid in a finite narrow tube. Applied Mathematics and Mechanics, 39(3), pp. 437-454. 2018. https://doi.org/10.1007/s10483-018-2305-9.

Bird, R.B., Dai, G.C. and Yarusso, B.J., The rheology and flow of viscoplastic materials. Reviews in Chemical Engineering, 1(1), pp. 1-70, 1983. https://doi.org/10.1515/revce-1983-0102.

Srivastava, L.M. and Srivastava, V.P., Peristaltic transport of blood: Casson model - II. Journal of Biomechanics, 17(11), pp. 821-829, 1984. https://doi.org/10.1016/0021-9290(84)90140-4.

Misra, J.C. and Pandey, S.K., Peristaltic transport of blood in small vessels: study of a mathematical model. Computers & Mathematics with Applications, 43(8-9), pp. 1183-1193, 2002. https://doi.org/10.1016/S0898-1221(02)80022-0.

Mernone, A.V., Mazumdar, J.N. and Lucas, S.K., A mathematical study of peristaltic transport of a Casson fluid. Mathematical and Computer Modelling, 35(7-8), pp. 895-912, 2002. https://doi.org/10.1016/S0895-7177(02)00058-4.

Nagarani, P. and Sarojamma, G., 2004. Peristaltic transport of a Casson fluid in an asymmetric channel. Australasian Physics & Engineering Sciences in Medicine, 27(2), pp. 49-59, 2004. https://doi.org/10.1007/BF03178376.

Hayat, T., Yasmin, H. and Al-Yami, M., Soret and Dufour effects in peristaltic transport of physiological fluids with chemical reaction: A mathematical analysis. Computers & Fluids, 89, pp. 242-253, 2014. https://doi.org/10.1016/j.compfluid.2013.10.038.

Siddiqui, A.M., Farooq, A.A. and Rana, M.A., A mathematical model for the flow of a Casson fluid due to metachronal beating of cilia in a tube. The Scientific World Journal, 2015. Article ID:487819, pp. 1-12, https://doi.org/10.1155/2015/487819.

Saravana, R., Vajravelu, K. and Sreenadh, S., Influence of compliant walls and heat transfer on the peristaltic transport of a Rabinowitsch fluid in an inclined channel. Zeitschrift fur Naturforschung A, 73(9), pp. 833-843, 2018. https://doi.org/10.1515/zna-2018-0181.

Vajravelu, K., Sreenadh, S. and Ramesh Babu, V., Peristaltic transport of a Herschel?Bulkley fluid in an inclined tube. International Journal of Non-Linear Mechanics, 40(1), pp. 83-90, 2005. https://doi.org/10.1016/j.ijnonlinmec.2004.07.001.

Wakeley, P.W., Optimisation and Properties of Gamete Transport, Ph. D. dissertation, University of Birmingham, pp. 139-166, 2008.