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The effect of weak inertia in rotating high-aspect-ratio vessel bioreactors

Published online by Cambridge University Press:  27 November 2017

Mohit P. Dalwadi Open the ORCID record for Mohit P. Dalwadi [Opens in a new window] ,
S. Jonathan Chapman ,
James M. Oliver  and
Sarah L. Waters
Mohit P. Dalwadi*
Affiliation:
Synthetic Biology Research Centre, University of Nottingham, Nottingham NG7 2RD, UK
S. Jonathan Chapman
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
James M. Oliver
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
Sarah L. Waters
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
*
†Email address for correspondence: mohit.dalwadi@nottingham.ac.uk
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Abstract

One method to grow artificial body tissue is to place a porous scaffold seeded with cells, known as a tissue construct, into a rotating bioreactor filled with a nutrient-rich fluid. The flow within the bioreactor is affected by the movement of the construct relative to the bioreactor which, in turn, is affected by the hydrodynamical and gravitational forces the construct experiences. The construct motion is thus coupled to the flow within the bioreactor. Over the time scale of a few hours, the construct appears to move in a periodic orbit but, over tens of hours, the construct drifts from periodicity. In the biological literature, this effect is often attributed to the change in density of the construct that occurs via tissue growth. In this paper, we show that weak inertia can cause the construct to drift from its periodic orbit over the same time scale as tissue growth. We consider the coupled flow and construct motion problem within a rotating high-aspect-ratio vessel bioreactor. Using an asymptotic analysis, we investigate the case where the Reynolds number is large but the geometry of the bioreactor yields a small reduced Reynolds number, resulting in a weak inertial effect. In particular, to accurately couple the bioreactor and porous flow regions, we extend the nested boundary layer analysis of Dalwadi et al. (J. Fluid Mech., vol. 798, 2016, pp. 88–139) to include moving walls and the thin region between the porous construct and the bioreactor wall. This allows us to derive a closed system of nonlinear ordinary differential equations for the construct trajectory, from which we show that neglecting inertia results in periodic orbits; we solve the inertia-free problem analytically, calculating the periodic orbits in terms of the system parameters. Using a multiple-scale analysis, we then systematically derive a simpler system of nonlinear ordinary differential equations that describe the long-time drift of the construct due to the effect of weak inertia. We investigate the bifurcations of the construct trajectory behaviour, and the limit cycles that appear when the construct is less dense than the surrounding fluid and the rotation rate is large enough. Thus, we are able to predict when the tissue construct will drift towards a stable limit cycle within the bioreactor and when it will drift out until it hits the bioreactor edge.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Biological Fluid Dynamics: Biological Fluid Dynamics Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 835 , 25 January 2018 , pp. 674 - 720
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Copyright
© 2017 Cambridge University Press 

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