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Dispersion of solute released from a sphere flowing in a microchannel

  • Stephan Gekle (a1)
Abstract

A solute is released from the surface of a sphere flowing freely in a cylindrical channel mimicking a modern drug delivery agent in a blood vessel. The solute then disperses by the combined action of advection and diffusion. We consider reflecting boundary conditions on the sphere and absorbing boundary conditions on the channel surface mimicking a biochemical reaction between the drug and endothelial cells on the vessel surface. The drug is released either instantaneously or continuously in time. The two key observables are the mean residence time in the flow before the drug is absorbed and the width over which it is spread on the vessel surface upon reaction. We numerically solve the Fokker–Planck equation for the time-dependent substance concentration combined with an analytical solution of the flow field. As expected, we find that the presence of the sphere leads to a substantial reduction in mean residence time and reaction width. Surprisingly, however, even in the limit of very large Péclet numbers (high velocities) the sphere-free case is not generally recovered. This observation can be attributed mainly to the small, but non-negligible radial flow component induced by the moving sphere. We further identify a strong influence of the release position which sharply separates two qualitatively different regimes. If the release position is between $\unicode[STIX]{x1D703}_{0}=0$ (front) and a critical $\unicode[STIX]{x1D703}_{c}$ the substance is quickly advected away from the sphere and its overall behaviour is similar to free diffusion in an empty channel. For release between $\unicode[STIX]{x1D703}_{c}$ and $\unicode[STIX]{x1D703}_{0}=\unicode[STIX]{x03C0}$ (tail), on the other hand, the substance is pushed towards the sphere leading to behaviour reminiscent of confined diffusion between two infinitely long cylinders. The critical position $\unicode[STIX]{x1D703}_{c}$ is generally smaller than $\unicode[STIX]{x03C0}/2$ which would correspond to an equatorial release position.

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Email address for correspondence: stephan.gekle@uni-bayreuth.de
References
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Adrover, A. 2011 Convection-dominated dispersion in channels with fractal cross-section. Phys. Fluids 23 (1), 013603.
Adrover, A. 2013 Effect of secondary flows on dispersion in finite-length channels at high Peclet numbers. Phys. Fluids 25 (9), 093601.
Adrover, A. & Pedacchia, A. 2013 Mass transfer through laminar boundary layer in microchannels with nonuniform cross section: the effect of wall shape and curvature. Intl J. Heat Mass Transfer 60 (C), 624631.
Adrover, A. & Pedacchia, A. 2014 Mass/heat transfer through laminar boundary layer in axisymmetric microchannels with nonuniform cross section and fixed wall concentration/temperature. Intl J. Heat Mass Transfer 68 (C), 2128.
Aris, R. 1956 On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. A 235 (1200), 6777.
Balakotaiah, V. 2008 Comment on ‘Taylor dispersion with absorbing boundaries: a stochastic approach’. Phys. Rev. Lett. 100 (2), 029402.
Balasubramanian, V., Jayaraman, G. & Iyengar, S. R. F. 1997 Effect of secondary flows on contaminant dispersion with weak boundary absorption. Appl. Math. Model. 21 (5), 275285.
Barton, N. G. 1984 An asymptotic theory for dispersion of reactive contaminants in parallel flow. J. Austral. Math. Soc. B 25, 287310.
Berezhkovskii, A. M. 2012 Note: Aris–Taylor dispersion from single-particle point of view. J. Chem. Phys. 137 (6), 066101.
Berezhkovskii, A. M. & Skvortsov, A. T. 2013 Aris–Taylor dispersion with drift and diffusion of particles on the tube wall. J. Chem. Phys. 139 (8), 084101.
Bhaumik, S. K., Kannan, A. & Dasgupta, S. 2015 Taylor–Aris dispersion induced by axial variation in velocity profile in patterned microchannels. Chem. Engng Sci. 134 (C), 251259.
Biswas, R. R. & Sen, P. N. 2007 Taylor dispersion with absorbing boundaries: a stochastic approach. Phys. Rev. Lett. 98 (16), 164501.
Biswas, R. R. & Sen, P. N. 2008 Biswas and Sen reply. Phys. Rev. Lett. 100 (2), 029403.
Camassa, R., Lin, Z. & McLaughlin, R. M. 2010 The exact evolution of the scalar variance in pipe and channel flow. Commun. Math. Sci. 8 (2), 601626.
Chatwin, P. C. 1970 The approach to normality of the concentration distribution of a solute in a solvent flowing along a straight pipe. J. Fluid Mech. 43, 321352.
Daddi-Moussa-Ider, A. & Gekle, S. 2016 Hydrodynamic interaction between particles near elastic interfaces. J. Chem. Phys. 145 (1), 014905.
Daddi-Moussa-Ider, A., Guckenberger, A. & Gekle, S. 2016 Long-lived anomalous thermal diffusion induced by elastic cell membranes on nearby particles. Phys. Rev. E 93 (1), 012612.
Daddi-Moussa-Ider, A., Lisicki, M. & Gekle, S. 2017 Mobility of an axisymmetric particle near an elastic interface. J. Fluid Mech. 811, 210233.
Dagdug, L., Berezhkovskii, A. M. & Skvortsov, A. T. 2014 Aris–Taylor dispersion in tubes with dead ends. J. Chem. Phys. 141 (2), 024705.
Dagdug, L., Berezhkovskii, A. M. & Skvortsov, A. T. 2015 Trapping of diffusing particles by striped cylindrical surfaces. Boundary homogenization approach. J. Chem. Phys. 142 (23), 234902.
Davidson, M. R. & Schroter, R. C. 1983 A theoretical model of absorption of gases by the bronchial wall. J. Fluid Mech. 129, 313335.
Deutch, J. M. 1980 A simple method for determining the mean passage time for diffusion. J. Chem. Phys. 73, 4700.
Dorfman, K. D. 2009 Taylor–Aris dispersion during lubrication flow in a periodic channel. Chem. Engng Commun. 197 (1), 3950.
Dorfman, K. D. & Brenner, H. 2008 Comment on ‘Taylor dispersion with absorbing boundaries: a stochastic approach’. Phys. Rev. Lett. 100 (2), 029401.
Giona, M., Adrover, A., Cerbelli, S. & Garofalo, F. 2009 Laminar dispersion at high Péclet numbers in finite-length channels: effects of the near-wall velocity profile and connection with the generalized Leveque problem. Phys. Fluids 21 (12), 123601.
Giona, M. & Cerbelli, S. 2010 Perturbation analysis of mixing and dispersion regimes in the low and intermediate Péclet number region. Phys. Rev. E 81 (4), 046309.
Giona, M. & Garofalo, F. 2015 Dispersion of overdamped diffusing particles in channel flows coupled to transverse acoustophoretic potentials: transport regimes and scaling anomalies. Phys. Rev. E 92 (3), 032104.
Hänggi, P., Talkner, P. & Borkovec, M. 1990 Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys. 62, 251341.
Hansen, R., Bruus, H., Callisen, T. H. & Hassager, O. 2012 Transient convection, diffusion, and adsorption in surface-based biosensors. Langmuir 28 (19), 75577563.
Hlushkou, D., Gritti, F., Guiochon, G., Seidel-Morgenstern, A. & Tallarek, U. 2014 Effect of adsorption on solute dispersion: a microscopic stochastic approach. Anal. Chem. 86 (9), 44634470.
Howard, M. P., Gautam, A., Panagiotopoulos, A. Z. & Nikoubashman, A. 2016 Axial dispersion of Brownian colloids in microfluidic channels. Phys. Rev. Fluids 1 (4), 044203.
Jayaraman, G., Pedley, T. J. & Goyal, A. 1998 Dispersion of solute in a fluid flowing through a curved tube with absorbing walls. Q. J. Mech. Appl. Maths 51 (4), 577598.
Kumar, J. P., Umavathi, J. C. & Basavaraj, A. 2012 Effects of homogeneous and heterogeneous reactions on the dispersion of a solute for immiscible viscous fluids between two plates. J.  Appl. Fluid Mech. 5 (4), 1322.
Latini, M. & Bernoff, A. J. 2001 Transient anomalous diffusion in Poiseuille flow. J. Fluid Mech. 441, 399411.
Leichtberg, S., Pfeffer, R. & Weinbaum, S. 1976 Stokes flow past finite coaxial clusters of spheres in a circular cylinder. Intl J. Multiphase Flow 3, 147.
Lighthill, M. J. 1966 Initial development of diffusion in Poiseuille flow. IMA J. Appl. Maths 2 (1), 97108.
Mazumder, B. S. & Das, S. K. 1992 Effect of boundary reaction on solute-dispersion in pulsatile flow through a tube. J. Fluid Mech. 239, 523549.
Mazumder, B. S. & Paul, S. 2011 Dispersion of reactive species with reversible and irreversible wall reactions. Heat Mass Transfer 48 (6), 933944.
Mondal, K. K. & Mazumder, B. S. 2005 On the solute dispersion in a pipe of annular cross-section with absorption boundary. Z. Angew. Math. Mech. 85 (6), 422430.
Muradoglu, M. 2010 Axial dispersion in segmented gas–liquid flow: effects of alternating channel curvature. Phys. Fluids 22 (12), 122106.
Muradoglu, M., Günther, A. & Stone, H. A. 2007 A computational study of axial dispersion in segmented gas–liquid flow. Phys. Fluids 19 (7), 072109.
Nacev, A., Beni, C., Bruno, O. & Shapiro, B. 2011 The behaviors of ferromagnetic nano-particles in and around blood vessels under applied magnetic fields. J. Magn. Magn. Mater. 323 (6), 651668.
Ng, C.-O. 2006 Dispersion in steady and oscillatory flows through a tube with reversible and irreversible wall reactions. Proc. R. Soc. Lond. A 462, 481515.
Ng, C.-O. 2010 How does wall slippage affect hydrodynamic dispersion? Microfluid Nanofluid 10 (1), 4757.
Ng, C.-O. & Rudraiah, N. 2008 Convective diffusion in steady flow through a tube with a retentive and absorptive wall. Phys. Fluids 20 (7), 073604.
Phillips, C. G. & Kaye, S. R. 1998 Approximate solutions for developing shear dispersion with exchange between phases. J. Fluid Mech. 374, 195219.
Phillips, C. G., Kaye, S. R. & Robinson, C. D. 1995 Time-dependent transport by convection and diffusion with exchange between two phases. J. Fluid Mech. 297, 373401.
Popel, A. S. & Johnson, P. C. 2005 Microcirculation and hemorheology. Annu. Rev. Fluid Mech. 37 (1), 4369.
Purnama, A. 1988 Boundary retention effects upon contaminant dispersion in parallel flows. J. Fluid Mech. 195, 393412.
Purnama, A. 1995 The dispersion of chemically active solutes in parallel-flow. J. Fluid Mech. 290, 263277.
Ramachandra Rao, A. & Deshikachar, K. S. 1987 An exact analysis of unsteady convective diffusion in an annular pipe. Z. Angew. Math. Mech. 67 (3), 189195.
Sankarasubramanian, R. & Gill, W. N. 1971 Taylor diffusion in laminar flow in an eccentric annulus. Intl J. Heat Mass Transfer 14 (7), 905919.
Sankarasubramanian, R. & Gill, W. N. 1973 Unsteady convective diffusion with interphase mass transfer. Proc. R. Soc. Lond. A 333, 115.
Sarkar, A. & Jayaraman, G. 2002 The effect of wall absorption on dispersion in annular flows. Acta Mechanica 158 (1–2), 105119.
Skvortsov, A. T., Berezhkovskii, A. M. & Dagdug, L. 2015 Note: boundary homogenization for a circle with periodic absorbing arcs. Exact expression for the effective trapping rate. J. Chem. Phys. 143 (22), 226101.
Smith, R. 1983 Effect of boundary absorption upon longitudinal dispersion. J. Fluid Mech. 134, 161177.
Szabo, A., Schulten, K. & Schulten, Z. 1980 First passage time approach to diffusion controlled reactions. J. Chem. Phys. 72, 4350.
Taylor, G. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219 (1137), 186203.
Vedel, S. & Bruus, H. 2012 Transient Taylor–Aris dispersion for time-dependent flows in straight channels. J. Fluid Mech. 691, 95122.
Vedel, S., Hovad, E. & Bruus, H. 2014 Time-dependent Taylor–Aris dispersion of an initial point concentration. J. Fluid Mech. 752, 107122.
Yeh, H. Y. & Keh, H. J. 2013 Axisymmetric creeping motion of a prolate particle in a cylindrical pore. Eur. J. Mech. (B/Fluids) 39, 5258.
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