Efficient monolithic projection-based method for chemotaxis-driven bioconvection problems

Abstract

We propose a non-iterative monolithic projection-based method to examine the nonlinear dynamics of time-dependent chemotaxis-driven bioconvection problems. In the proposed method, all the terms are advanced using the Crank–Nicolson scheme in time along with the second-order central difference in space. Linearizations, approximate block lower–upper decompositions, and an approximate factorization technique are adopted to improve the computational efficiency while preserving the second-order temporal accuracy. We perform numerical simulations of quasi-homogeneous bioconvection, two-dimensional forced chemotaxis bioconvection, and two-dimensional chemotaxis-driven bioconvection to test the numerical performance of the proposed method. The results show that the proposed method provides predictions that are in good agreement with those in previous works. Moreover, it preserves the second-order accuracy in time, significantly reduces the time-step limitation, and improves the computational efficiency. Finally, the proposed method is employed to investigate the nonlinear dynamics of chemotaxis-driven bioconvection problems with varying characteristic bacterial concentration and chamber depth. Four regimes were classified based on the fluid and bacterial motions: stable shallow-chamber, unstable shallow-chamber, unstable deep-chamber, and chaotic deep-chamber flows. We show the formation and merging of falling plumes and their surrounding fluid motion under random initial conditions as well as their convergence toward stationary or chaotic bacterial plumes. To track the dynamical regimes over the entire considered domain, we designed normalized variance and kurtosis, which reflect formation and merging of plumes and intermittency in chaotic cases, respectively. A posterior classification, which provides a rough outline of the characteristic features of the different regimes, was also carried out.

Introduction

Bioconvection, which was originally investigated by Wager over one hundred years ago [1], refers to the process of spontaneous pattern formation in suspensions of upswimming microorganisms, such as bacteria and algae, in still water [2]. In all such cases of pattern formation, the bacterial cells are denser than the water they swim in, and they tend to propel themselves upward on average. Although the cause of upswimming behavior may vary in each case (gravitaxis, phototaxis, chemotaxis, gyrotaxis, and so on), the patterns show significant similarities among various species. This paper is concerned with bioconvection due to the chemotaxis (corresponds to swimming up chemical gradients) of bacterial cells of the species Bacillus subtilis placed in a container filled with water such that the upper surface is exposed to the atmosphere; the bacteria are 10% denser than water, consume oxygen, and swim up an oxygen gradient. We refer to such bioconvection as chemotaxis-driven bioconvection.

Bioconvection driven by chemotaxis is of intrinsic interest because it plays an important role in medical sciences, agricultural sciences, and other fields [3]. Chemotaxis-driven bioconvection is based on dynamic synergism between physics and biology, which has been widely studied experimentally [4], [5], [6], [7], [8], [9], analytically [10], [11], [12], [13], [14], and numerically [8], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. This phenomenon involves not only the incompressibility and nonlinearity in a viscous fluid but also strong couplings among incompressible flows, bacterial concentration, and oxygen concentration. The oxygen concentration throughout the initially well-stirred suspension is assumed to be equal to the atmospheric value at the exposed surface. Consumption of oxygen by the bacteria leads to a decrease in oxygen concentration throughout, except near the exposed surface. Thus, oxygen gradients form, and the oxytactic bacteria swim up their gradients, generating an overturning and convective instability with falling plumes when the vertical oxygen gradient increases.

Many experimental studies have been conducted to visually investigate chemotaxis-driven bioconvection problems [4], [5], [6], [7], [8], [9]. Kessler et al. [4] experimentally investigated a suspension of bacterial cells, B. subtilis, placed in a chamber with its upper surface exposed to the atmosphere, and they found that a combination of physical and biological factors converts an originally static microbial habitat into a functional dynamic system, which significantly improves the transport and mixing of oxygen as well as the viability of the bacterial cells. An advanced analysis technique was proposed by Jánosi et al. [5], who quantitatively analyzed the onset of bioconvective instability through laboratory experiments on bacterial cultures based on the standard deviation of optical density. They discovered an inverse correlation between the delay time (the time interval between a homogeneous mixed initial condition and the appearance of nonuniformity in cell concentration) and the concentration of the organisms. Bioconvection has been shown to transport oxygen efficiently in bacterial suspensions, thereby increasing the entire microbial population [4], [5]. However, there remains a lack of agreement among empirical studies with regard to this conclusion. Through an experimental study, Jánosi et al. [7] showed that bioconvection does not produce a clear increase in the bacteria colony growth rate, while other numerical, analytical, and experimental results [8] have indicated that the occurrence of bioconvection should promote oxygen transport. Recently, Abe et al. [9] showed that bioconvection might maintain the viability of bacteria by preventing excessive concentration of bacteria.

To extensively examine the dynamics of chemotaxis-driven bioconvection, many researchers have conducted various analytical studies [10], [11], [12], [13], [14] based on published experimental observations [4], [5], [6], [7], [8], [9]. All these analytical studies involve a mathematical model constructed by Keller and Segel [26], [27], which describes bioconvection in a suspension of chemotactic cells. Another model of chemotaxis including stochastic fluctuations has been studied by Chavanis [28]. Hillesdon et al. [10] provided a quantitative description of a pattern-formation process observed in concentrated suspensions of swimming bacteria without considering the bulk fluid motion, and steady-state solutions were observed for both shallow- and deep-layer chambers. Subsequently, Hillesdon and Pedley [11] carried out a linear instability analysis of a chemotaxis-driven bioconvection model, which provided useful qualitative insights into the behavior of an unstable cell distribution at the onset of instability. Further, Metcalfe and Pedley [12] conducted a weakly nonlinear analysis to predict the initial planform of bioconvection patterns. Cherniha and Didovych [14] adopted Lie method to obtain the Lie symmetries of simplified one-dimensional Keller–Segel model, and constructed exact solutions of the simplified model. Although the above-mentioned analytical studies have provided useful insights into bacterial bioconvection dynamics or exact solutions of simplified models, these models do not consider the resuscitating of bacterial cells, which cannot be ignored, especially in deep chamber. Thus, numerical simulations are required to capture the dynamics of the entire system.

In recent decades, various numerical methods have been developed to investigate the long-time nonlinear dynamics of chemotaxis-driven bioconvection problems [8], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. Ghorai and Hill [15] numerically investigated the structure and stability of a two-dimensional (2D) bioconvection in tall and narrow chambers by applying the implicit Euler backward scheme in time and second-order central difference in space with a finite difference method (FDM), where an iterative line-by-line tri-diagonal matrix algorithm with a relaxation scheme was used to solve the nonlinear discretized equations. They found that the plume was always unstable to both varicose and meandering modes in a sufficiently deep chamber [15]. To avoid the time-consuming iterative procedure in solving the nonlinear bioconvection problems, many researchers pursued various semi-implicit or explicit approaches. Hopkins and Fauci [17] conducted numerical simulations by applying the semi-implicit pressure linked equation (SIMPLE) finite volume method to both Navier–Stokes and chemical transport equations, where the micro-organisms are represented as a suspension of discrete particles within a fluid domain. They studied the plume formation and stability for purely geotactic as well as gyrotactic microbes in rectangular chambers of varying depth, and they showed that the plume wavelength increases slightly whereas the plume stability decreases with the depth. Chertock et al. [19] developed a high-resolution semi-discrete hybrid finite volume–finite difference method based on the vorticity formulation of the incompressible Navier–Stokes equations, in which the second-order finite volume upwind method is used for the chemotaxis equations and the second-order central difference scheme is used for the vorticity equation along with the explicit third-order Runge–Kutta method. Chertock et al. [19] examined the chemotactic effects in much deeper chambers than those investigated in [17]. The formation and merging of plumes under random and deterministic initial conditions and their convergence toward stationary plumes have been described in detail in [19]. Prior to the analytical work conducted in [12], a fully nonlinear stationary state was achieved in [19]. Later, Deleuze et al. [22] adopted an upwind finite element method (FEM) with an inconsistent Petrov–Galerkin weighted residual scheme to solve coupled chemotaxis-driven bioconvection problems. They affirmed that chemotaxis has a stabilizing effect on the overall system by analyzing the time scales for each of the three competing physical mechanisms, namely chemotaxis, diffusion, and convection of bacteria. Very recently, Lenarda et al. [23] proposed a partitioned FEM for solving the advection–diffusion–reaction systems coupled with incompressible viscous flow using a nested Newton–Raphson iterative scheme, where an implicit backward Euler scheme with an operator splitting was used in temporal discretization with the first-order approximation.

Recently, we developed efficient projection schemes with a second-order implicit time integration for solving time-dependent natural convection problems [29], [30]. Using the basic rationale of the projection methods in [29], [30], we aim to develop a stable and efficient monolithic projection method in FDM framework for examining the long-time nonlinear dynamics of time-dependent chemotaxis-driven bioconvection problems. An implicit Crank–Nicolson scheme is used to discretize the buoyancy, nonlinear convection, linear diffusion, nonlinear chemotaxis, and nonlinear oxygen consumption terms in both incompressible Navier–Stokes and chemotaxis equations. This enables us to mitigate the excessively time-step restriction due to the coupled multiphysics. To avoid time-consuming iterative procedures, we apply a linearization technique to the nonlinear convection, chemotaxis, and oxygen consumption terms. Thus, we obtain a global linearized coupled system. We employ approximate block lower–upper (LU) decomposition along with the approximate factorization technique for the linearized coupled system, which not only improves the computational efficiency but also preserves the second-order accuracy in time. A non-iterative monolithic projection-based method for solving chemotaxis-driven bioconvection ($\text{MPBM-CDB}$) is then established. To verify the accuracy and stability of the proposed $\text{MPBM-CDB}$, we numerically simulate three different 2D chemotaxis-driven bioconvection examples: quasi-homogeneous bioconvection, 2D forced chemotaxis bioconvection, and 2D chemotaxis-driven bioconvection. Finally, we implement the developed method to examine different scenarios occurring in 2D chemotaxis-driven bioconvection with various bacterial concentrations and chamber depths in the absence of cut-off oxygen. Four flow regimes were initially classified according to the various features of the fluid and bacterial motions: stable shallow-chamber, unstable shallow-chamber, unstable deep-chamber, and chaotic deep-chamber flows, in which formation and merging of falling plumes may occur under random initial conditions. Normalized variance is introduced to track the formation and merging of plumes in these four regimes, while normalized kurtosis is designed for tracking the intermittency in the chaotic-chamber case. As a summary, a posterior regime classification was constructed based on the time-averaged ratio of the active region and integral length scale of the bacterial concentration fluctuation in the longitudinal direction.

The remainder of this paper is organized as follows. Section 2 presents the mathematical formulation and boundary conditions of chemotaxis-driven bioconvection. Section 3 describes the detailed procedures of the proposed $\text{MPBM-CDB}$. Section 4 discusses the numerical verification, temporal accuracy, and stability of the proposed $\text{MPBM-CDB}$. Section 5 describes the implementation of the $\text{MPBM-CDB}$ for examining different scenarios in fully nonlinear 2D chemotaxis-driven bioconvection while considering the cut-off oxygen. Finally, Section 6 concludes the paper.