Proc R Soc A-Math Phys Eng Sci.2016 Sep;472(2193)

**Unsteady solute dispersion in non-Newtonian fluid flow in a tube with wall absorption.**

Jyotirmoy Rana, P. V. S. N. Murthy

**Abstract**

The theory of miscible dispersion in a straight circular pipe with interphase mass transfer that was investigated by Sankarasubramanian & Gill (1973 Proc. R. Soc. Lond. A 333, 115–132. (doi:10.1098/rspa.1973.0051); 1974 Proc. R. Soc. Lond. A 341, 407–408. (doi:10.1098/rspa.1974.0195)) in Newtonian fluid flow is extended by considering various non-Newtonian fluid models, such as the Casson (Rana & Murthy 2016 J. Fluid Mech. 793, 877–914. (doi:10.1017/jfm.2016.155)), Carreau and Carreau–Yasuda models. These models are useful to investigate the solute dispersion in blood flow. The three effective transport coefficients, i.e. exchange, convection and dispersion coefficients, are evaluated to analyse the dispersion process of solute. The convection and dispersion coefficients are determined asymptotically at large time which is sufficient to understand the nature of the solute dispersion process in a tube. The axial mean concentration is analysed, using the asymptotic expressions for these three coefficients. The effect of the wall absorption parameter, Weissenberg number, power-law index, Yasuda parameter and Peclet number on the dispersion process is discussed clearly in this study. A comparative study of the solute dispersion among the Newtonian and all other non-Newtonian models is presented. At low shear rate, it is observed that Carreau fluid behaves like Newtonian fluid, whereas the other fluids exhibit significant differences during the solute dispersion. This study may be applicable to understand the dispersion process of drugs in the blood stream.

**Supplement: **

In recent past, the study of solute dispersion becomes more popular to the researchers due to its outspread applications in different areas such as transportation of drugs or plasma proteins in cardiovascular systems, chromatographic separations in chemical processing and pollutant transport in the environmental systems. Several researchers [1, 2, 3] studied the solute dispersion process in Newtonian/non-Newtonian fluid flow in a straight tube.

This study finds interesting applications in the blood flow in the circulatory system. Owing to the non-Newtonian characteristics of blood and pulsatile nature of blood flow, the investigation of solute dispersion in pulsatile flow of non-Newtonian fluid through a circular tube is useful for understanding the transport of drugs or plasma proteins in the arterial blood flow. After the injection of drugs in the blood flow of cardiovascular system, it is important to know the rate of dispersion of drugs due to their therapeutic and toxic nature at low and high concentrations. Also, in the directed drug targeting to cure diseases like Cancer, nano-particle assisted multifunctional drug capsule (carrier particle) is injected into the blood and the dispersion of this drug capsule is analyzed for the better treatment of the patient. So, for the drug delivery process through the blood vessels, the concentration distribution of drugs in the blood is more important aspects. In this study, the distribution of mean concentration of solute is analyzed with the effect of absorption at the tube wall by considering blood as different non-Newtonian fluids, such as Carreau, Carreau-Yasuda and Casson fluids. These non-Newtonian fluid models are relevant to the blood flow analysis [3-6]. In this investigation, the generalized dispersion model is adopted to analyze the solute dispersion phenomenon at large time. The three transport coefficients i.e., exchange, convection and dispersion coefficients which describe the whole dispersion process in the system are determined. After obtaining these coefficients, mean concentration of solute is determined.

From figures 1 and 2 for mean concentration of solute at different time and different axial position in a tube, it is noted that at low shear rate, Carreau fluid behaves like Newtonian fluid, whereas the other non-Newtonian fluids exhibit the significant differences. It is also noted that maximum mean concentration is for the Casson model and minimum mean concentration is seen for the Carreau-Yasuda model. So, this investigation may be helpful for understanding the dispersion of drugs in the blood flow at different time after the injection and at different axial position of the blood vessels.

Figure 1: Variation of axial mean concentrationCwith time t at z=0.5 for different Carreau-Yasuda models with Carreau, Newtonian and Casson models when_{m}(t,z)We=0.04,β=0.01 andPe=10^3.

Figure 2: Variation of axial mean concentrationCwith axial distances z at t=0.5 for different Carreau-Yasuda models with Carreau, Newtonian and Casson models when_{m}(t,z)We=0.04,β=0.01 andPe=10^3.

**References**

[1] Sankarasubramanian R, Gill WN. 1973 Unsteady convective diffusion with interphase mass transfer. *In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences* **333** (1592), 115-132. The Royal Society. (doi:10.1098/rspa.1973.0051)

[2] Mazumder BS, Das SK. 1992 Effect of boundary reaction on solute dispersion in pulsatile flow through a tube. *Journal of Fluid Mechanics* **239**, 523-549. (doi:10.1017/S002211209200452X)

[3] Rana Jyotirmoy, Murthy PVSN. 2016 Solute dispersion in pulsatile Casson fluid flow in a tube with wall absorption. *Journal of Fluid Mechanics* **793**, 877-914. (doi:10.1017/jfm.2016.155)

[4] Boyd J, Buick JM, Green S. 2007 Analysis of the Casson and Carreau-Yasuda non-Newtonian blood models in steady and oscillatory flows using the lattice Boltzmann method. *Physics of Fluids* **19 **, 093103. (doi:10.1063/1.2772250)

[5] El Naby AEHA, El Misiery AEM. 2002 Effects of an endoscope and generalized Newtonian fluid on peristaltic motion. *Applied Mathematics and Computation* 128, 19-35. (doi:10.1016/S0096-3003(01)00153-9)

[6] Bernabeu, MO, Nash RW, Groen D, Carver HB, Hetherington J, Krüger T, Coveney PV. 2013 Impact of blood rheology on wall shear stress in a model of the middle cerebral artery. *Interface Focus* **3**, 20120094. The Royal Society. (doi:10.1098/rsfs.2012.0094)