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.

Unsteady solute dispersion in small blood vessels using a two-phase Casson model

“…Therefore, the following set of differential equations for f j are obtained as: 11) where j = 0, 1, 2 and f −1 = f −2 = 0. By using the method of eigenfunction expansion, we have solved the coupled system of partial differential equations (3.9) and (3.11) to obtain the unknown functions f 0 (t, r), K 0 (t), f 1 (t, r), K 1 (t), f 2 (t, r) and K 2 (t) in this specific order [7,9]. By following the analysis of Sankarasubramanian & Gill [6] and Rana & Murthy [7][8][9], the function f 0 (t, r) and exchange coefficient K 0 (t) are obtained as follows:…”

Hydrodynamic Dispersion of Solute under Homogeneous and Heterogeneous Reactions

“…The moment method no longer gives any immediate expression for mean concentration but it is conceivable to surmise the mean concentration which can also be achieved by means of Hermite polynomials for the representation of non-Gaussian curve [29,30] with the aid of first four central moments as follows, .5 e = Therefore, given the statistical parameters (26), the concentration distribution can be estimated from (30) at any given location in the axial direction and time. The variations of mean concentration distribution have been displayed in Figure 18 against the axial distance for different factors viz., yield stress, wall absorption, bulk flow reaction parameter etc.…”

“…The present work, attempts are made to examine dispersion process in Casson liquid through a tube in presence of both reactions. In short the present study probably the fresh attempt and a generalization of the work of Rana and Murthy [26]. The specific objective of the current study: (1) to cultivate a mathematical model of Taylor dispersion in oscillatory flow field; (2) to illustrate and characterize the behavior of dispersion coefficient w.r.t chemically active solute; and (3) to study the break through curve in above background.…”

“…The authors [24] likewise concentrated the dispersion phenomena in Casson liquid through a conduit by applying the generalized dispersion model and establishing the functional relationship of a dispersion coefficient with a Womersley frequency number, Schmidt number, yield stress, and fluctuating pressure component. As of late, Rana & Murthy [25,26] investigated the large time behavior of the axial dispersion process in an oscillatory flow in presence of wall reaction.…”

“…All three transport coefficient and mean concentration are function of central moment defined in Eq. (26). The interrelated relationship of the four elements with central moment, and the parametric effect on these four element are elaborated in the following subsections:…”