A general cell–fluid Navier–Stokes model with inclusion of chemotaxis

Abstract

The main purpose of this work is to explore a general cell–fluid model which is based on a mixture theory formulation that accounts for the interplay between oxytactically (chemotaxis toward gradient in oxygen) moving bacteria cells in water and the buoyance forces caused by the difference in density between cells and fluid. The model involves two mass balance and two general momentum balance equations, respectively, for the cell and fluid phase, combined with a convection–diffusion–reaction equation for oxygen. In particular, the momentum balance equations include interaction terms which describe the cell–fluid drag force effect. Hence, the model is an extension of the classical Navier–Stokes equation in two different ways: (i) inclusion of two phases (cell and fluid) instead of one; (ii) inclusion of a chemotactic transport mechanism. The model can be understood as a natural generalization of the much studied chemotaxis-Stokes model explored by [I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA 102 (2005) 2277–2282]. First, we explore the model for parameters in a range which ensures that it lies close to the previously studied chemotaxis-Stokes model (essentially very low cell volume fraction). Main observations are (i) formation of sinking finger-shaped plumes and (ii) convergence to plumes that possibly can be stationary (i.e. persist over time). The general cell–fluid model provides new insight into the role played by the cell–fluid interaction term which controls the competition between gravity segregation and chemotaxis effect on the formation of cell plumes. Second, we explore cases with large cell volume fraction (far beyond the regime captured by the chemotaxis-Stokes model), which gives rise to rich pattern-formation behavior. The general cell–fluid model opens for exploring a hierarchy of different “submodels”. Hence, it seems to be an interesting model for further investigations of various, general cell–fluid spatio-temporal evolution dynamics, both from an experimental and mathematical point of view.