Surface tension coupled non-uniformly imposed flows modulate the activity of reproducing chemotactic bacteria in porous media

Abstract

This paper investigates a non-homogeneous two-dimensional model for reproducing chemotactic bacteria, immersed in a porous medium that experiences non-uniformly imposed flows. It is shown that independently of the form of the fluid velocity field, the compressible/incompressible nature of the fluid significantly shifts the Turing stability-instability transition line. In dry media, Gaussian perturbations travel faster than the hyperbolic secant ones, yet the latter exhibit better stability properties. The system becomes highly unstable under strong flows and high surface tension. Approximated solutions recovered by injecting Gaussian perturbations overgrow, in addition to triggering concentric breathing features that split the medium into high and low-density domains. Secant perturbations on the other hand scatter slowly and form patterns of non-uniformly distributed peaks for strong flows and high surface tension. These results emphasize that Gaussian perturbations strongly modulate the activity of bacteria, hence can be exploited to perform fast spreading in environments with changing properties. In this sense, Gaussian profiles are better candidates to explain quick bacterial responses to external factors. Secant-type approximated solutions slowly modulate the bacterial activity, hence are better alternatives to dive into weak bacterial progressions in heterogeneous media.

Figure 1

Growth rate, phase and envelope speeds of the localized perturbations in absence of a background velocity field ($u=v=0$) $D_2=1.62,{\sigma }_x={\sigma }_y=2,k_x=k_y=0.05,x_0=y_0=2,\mathrm{\Phi }=2·{10}^{-3}$. Top panels are obtained with the secant function, and bottom panels are related to the Gaussian distribution.

Figure 2

Reduction of instability domains in presence of a strong incompressible velocity field. Top panels are obtained by injecting two-dimensional hyperbolic secant distribution, and bottom panels are recovered by using a two-dimensional Gaussian profile. Left panels ${\tau }_x={\tau }_y={10}^2$, middle ${\tau }_x={\tau }_y=1$ and right panels ${\tau }_x={\tau }_y=0.01$. Other parameters as in Fig. 1.

Figure 3

Spreading of unstable domains with surface tension, and for weak incompressible flows ${\tau }_x={\tau }_y={10}^2$. Top panels are obtained by injecting the hyperbolic secant distribution, and bottom panels are derived using a Gaussian profile. Left panels $D_2=1.62·{10}^{-3}$, middle $D_2=1.62$ and right panels $D_2=162$. Other parameters as in Fig. 1.

Figure 4

Propagation of bacterial waves at different times. The top panels depict approximate solutions calculated by injecting the hyperbolic secant distribution, and the bottom panels represent solutions derived using a Gaussian profile. $k_x=k_y=0.05,{\tau }_x={\tau }_y={10}^2,D_2=1.62$. Left panels $t=5·{10}^{-3}$, middle $t=0.5$ and right panels $t=1$. Other parameters as in Fig. 1.

Figure 5

The amplitude of approximated bacterial density constructed using hyperbolic secant increases with surface tension at t = 0.1. ${\tau }_x={\tau }_y={10}^2,k_x=k_y=0.05$. (a) $D_2=1.62·{10}^{-3}$, (b) $D_2=16.2$, and (c) $D_2=162$. Other parameters as in Fig. 1.

Figure 6

Strong incompressible flows significantly increase the amplitude approximated localized bacterial density determined using hyperbolic secant function at t = 0.1, and for $D_2=1.62$. (a): ${\tau }_x={\tau }_y={10}^2$, (b) ${\tau }_x={\tau }_y=1$, and (c) ${\tau }_x={\tau }_y={10}^{-2}$.

Figure 7

Diffusion initiates a fast scattering of initially localized bacterial density across the domain t = 0.1, and for $D_2=1.62$. (a): $\alpha =1$, (b) $\alpha =20$, and (c) $\alpha =50$.