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Peristaltic transport of Herschel–Bulkley fluids in tubes of variable cross section induced by dilating peristaltic waves: application to sliding hiatus hernia

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Abstract

The presented formulation deals with the flow of Herschel–Bulkley fluid induced by progressive transverse dilating peristaltic waves in a circular cylindrical tube of non-uniform cross sectional area. This is an intended act to model swallowing of various types of chewed foods in oesophagus which suffers from hiatus hernia. Due to sliding hiatus hernia, the cross section of the lower oesophagus does not remain uniform. The impact of bulging, which is formed by various combinations of divergence and convergence, is examined. Effects of dilating amplitude, the flow behavior index etc. are also investigated. The non-linear governing equations which represent the model are linearized by low Reynolds number and long wavelength approximations and after that computer simulation is used to evaluate the numerical results. Pressure in the oesophagus along its axis and the time averaged volume flow rate over a period for a single wave propagation are also evaluated. It is inferred that pressure drops right from the beginning and lowers further in the lower part even if only the lower portion of the cylindrical tube diverges. Experimental observations confirm this when the oesophagus suffers from sliding hiatus hernia causing the lower oesophagus to bulge towards the end. If the lower part of the tube subsequently converges towards the end after divergence, pressure is found to increase. Pressure is found to increase with the flow behavior index even when the tube changes its cross sectional area. When the bolus is nearing the cardiac sphincter, pressure-rise and pressure-drop are both less in a diverging tube than that in a uniform tube. Therefore, the pressure requirement to deliver bolus in stomach is less if oesophagus diverges. The presented investigation is significant for the patients suffering from sliding hiatus hernia.

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Acknowledgements

Funding was provided by University Grants Commission (Grant No. F.2-2/2013(SA-I), April 04, 2014) to the second author in term of fellowship and contingency grant.

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  1. Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, 221005, India

    Sanjay Kumar Pandey & Amirlal Singh

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  1. Sanjay Kumar Pandey
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  2. Amirlal Singh
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Correspondence to Sanjay Kumar Pandey.

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Pandey, S.K., Singh, A. Peristaltic transport of Herschel–Bulkley fluids in tubes of variable cross section induced by dilating peristaltic waves: application to sliding hiatus hernia. Int. J. Dynam. Control 7, 407–418 (2019). https://doi.org/10.1007/s40435-018-0454-7

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  • DOI: https://doi.org/10.1007/s40435-018-0454-7

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