Unsteady solute dispersion in Herschel-Bulkley fluid in a tube with wall absorption

Abstract

The axial dispersion of solute in a pulsatile flow of Herschel-Bulkley fluid through a straight circular tube is investigated considering absorption/reaction at the tube wall. The solute dispersion process is described by adopting the generalized dispersion model suggested by Sankarasubramanian and Gill [“Unsteady convective diffusion with interphase mass transfer,” Proc. R. Soc. A 333, 115–132 (1973)]. Firstly the exchange, convection, and dispersion coefficients are determined for small and large time, and then the axial mean concentration of a solute in the tube is determined. The effect of power-law index l, yield stress of fluid τy, wall absorption parameter β, amplitude of fluctuating pressure component e, and Womersley frequency parameter α on the convection, dispersion, and mean concentration of solute is discussed for a Herschel-Bulkley fluid in the tube. The single frequency period in the oscillation of dispersion coefficient K2 is observed for small values of α while the double frequency period is noticed for large values of α at small time. Only positive dispersion occurs for small values of α. Both positive and negative dispersion is seen for large values of α. Also, the occurrence of negative dispersion is influenced by the parameters l, τy, β, and e for large values of α. A comparative study of the convection, dispersion, and mean concentration of solute among the Newtonian and non-Newtonian Herschel-Bulkley, power-law, Bingham, and Casson [J. Rana and P. V. S. N. Murthy, “Solute dispersion in pulsatile casson fluid flow in a tube with wall absorption,” J. Fluid Mech. 793, 877–914 (2016)] fluid models is presented at small and large time. Also, large time behaviour of non-Newtonian Carreau and Carreau-Yasuda fluid models [J. Rana and P. V. S. N. Murthy, “Unsteady solute dispersion in non-Newtonian fluid flow in a tube with wall absorption,” Proc. R. Soc. A 472, 20160294 (2016)] is considered for comparison with other discussed fluid models. It is noticed that these fluid models exhibit significant differences during the solute dispersion in the presence of wall absorption. These models are applied to study the dispersion process of a solute in blood flow. For a Herschel-Bulkley fluid, the critical value of α at which fluctuations of K2 attain negative magnitude increases as l increases. The critical value of α for a Herschel-Bulkley fluid (l = 0.9 with τy = 0.05) is 2.9 but it is equal to 3 for a Casson fluid (τy = 0.05) [J. Rana and P. V. S. N. Murthy, “Solute dispersion in pulsatile casson fluid flow in a tube with wall absorption,” J. Fluid Mech. 793, 877–914 (2016)] with non-zero β. It is noticed that the amplitude of fluctuations of both negative convection coefficient −K1 and dispersion coefficient K2 for a Casson fluid is lying below that of Herschel-Bulkley fluid at all times. Therefore, the peak of mean concentration Cm for the Casson model is higher than that of the Herschel-Bulkley model. The present study may be useful to know the transportation process of drugs in blood flow through the blood vessels.