Abstract
An asymptotic approach is considered for the transport and deposition of nanofibres in pipe flow. Convection and Brownian diffusion are included, and Brownian diffusion is assumed to be the dominant mechanism. The fibre position and orientation are modelled with a probability density function for which the governing equation is a Fokker–Planck equation. The focus is set on dilute fibres concentrations implying that interaction between individual fibres is neglected. At the entrance of the pipe, a fully developed velocity profile is set and it is assumed that the fibres enter the pipe with a completely random orientation and position. A small parameter \({\varepsilon =l/a}\) is introduced, where l is the fibre half-length and a is the pipe radius. The probability density function is expanded for small \({\varepsilon}\) and the solution turns out to be multi-structured with three areas, consisting of one outer solution and two boundary layers. For the deposition of fibres on the wall, it is found that for parabolic flow, and for the lowest order, the deposition can be obtained with a simplified angle averaged convective-diffusion equation. It is suggested that this simplification is valid also for more complex flows like when the inflow boundary condition yields a developing velocity profile and flows within more intricate geometries than here studied. With the model fibre, deposition rates in human respiratory airways are derived. The results obtained compare relatively well with those obtained with a previously published model.
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References
Poland C.A., Duffin R., Kinloch I., Maynard A., Wallace W.A.H., Seaton A., Stone V., Brown S., MacNee W., Donaldson K.: Carbon nanotubes introduced into the abdominal cavity of mice show asbestos-like pathogenicity in a pilot study. Nat. Nanotechnol. 3(7), 423–428 (2008)
Högberg S.M., Åkerstedt H.O., Lundström T.S., Freund J.F.: Respiratory deposition of fibres in the non-inertial regime: development and application of a semi-analytical model. Aerosol Sci. Technol. 44(10), 847–860 (2010)
Risken H.: The Fokker–Planck Equation. Springer, Berlin (1977)
Åkerstedt H.O., Högberg S.M., Lundström T.S., Sandström T.: The effect of cartilaginous rings on particle deposition by convection and Brownian diffusion. Nat. Sci. 2(7), 769–779 (2010)
Asgharian B., Yu C.P., Gradon L.: Diffusion of fibres in tubular flow. Aerosol Sci. Technol. 9(3), 213–219 (1988)
Sobey, I.: Introduction to Interactive Boundary Layer Theory. Oxford Applied Engineering and Mathematics (2000)
Jeffery G.B.: The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. 102(175), 161–179 (1922)
Högberg, S.M., Åkerstedt, H.O., Lundström, T.S., Freund, J.F.: Numerical model for fibre transport in the respiratory airways. In: The 19th International Symposium on Transport Phenomena. Reykjavik (2008)
Nitsche J.M., Brenner H.: On the formulation of boundary conditions for rigid nonspherical Brownian particles near solid walls: application to orientation-specific reactions with immobilized enzymes. J. Colloid. Interface Sci. 38(1), 21–41 (1990)
Duff G.F.D., Naylor D.: Differential Equations of Applied Mathematics. Wiley, New York (1966)
Dyke M.: Perturbation Methods in Fluid Mechanics. Academic press, New York (1975)
Weibel E.R.: Morphometry of the Human Lung. Academic Press, New York (1963)
Dahneke B.E.: Slip correction factors for nonspherical bodies III. The form of the general law. J. Aerosol Sci. 4(2), 163–170 (1973)
Högberg S.M., Lundström T.S.: Motion of dispersed carbon nanotubes during impregnation of fabrics. Plast. Rubber Compos. Macromol. Eng. 40(2), 70–79 (2011). doi:10.1179/1743289811Y.0000000005
Nordlund M., Fernberg S.P., Lundström T.S.: Particle deposition mechanisms during processing of advanced composite materials. Compos. Part A 38(10), 2182–2193 (2007)
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Åkerstedt, H.O., Högberg, S.M. & Lundström, T.S. An asymptotic approach of Brownian deposition of nanofibres in pipe flow. Theor. Comput. Fluid Dyn. 27, 561–575 (2013). https://doi.org/10.1007/s00162-012-0262-1
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DOI: https://doi.org/10.1007/s00162-012-0262-1