The free surface effect on a chemotaxis–diffusion–convection coupling system

Abstract

Suspension of an oxytactic bacteria (e.g. the species Bacillus subtilis) placed in a container with its upper surface open to the atmosphere results in the formation of complex bioconvection patterns. The bacteria consume the oxygen diluted in the water, thereby causing the decrease of oxygen concentration everywhere except on the free surface. Through the free surface, which is in direct contact with the air, oxygen diffuses into the water. Slightly denser than water, the oxytactic bacteria are able to swim towards the higher concentration of oxygen (i.e. upwards) and they concentrate in a thin layer below the free surface. This causes the change of the suspension density and Rayleigh–Taylor type instabilities to occur. The chemotaxis phenomenon has been successfully modeled within continuum mechanics approach under certain simplifications. The set of (non-linearly) coupled equations describing the process involves the Boussinesq approximation of the Navier–Stokes equations governing the fluid motion and two convection–diffusion type equations governing the bacteria and oxygen concentrations. One of the simplifications that might significantly influence numerical simulations is the boundary condition for fluid equation on the free surface. This condition ensures that the vertical component of the velocity is zero, thus keeping the position of free surface fixed. This assumption significantly simplifies numerical procedure since the non-linearly coupled system can then be solved on stationary grid. However, allowing the motion of the free surface and completing the system with appropriate boundary conditions on contact line (liquid–solid–gas interface), a more realistic model is derived and new insights on nonlinear dynamics of the chemotaxis phenomenon are obtained. Our aims in this paper are to upgrade the currently available model into a more realistic one in both two and three dimensions, to propose a numerical procedure to deal with the new system (now posed on time-dependent domain) and, finally, to show the difference between this new model and the previous simplified one.