# American Institute of Mathematical Sciences

September  2020, 28(3): 1207-1225. doi: 10.3934/era.2020066

## A multiple-relaxation-time lattice Boltzmann method with Beam-Warming scheme for a coupled chemotaxis-fluid model

 1 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China 2 School of Mathematics and Computational Science, Hunan First Normal University, Changsha, China

* Corresponding author: Xuguang Yang

Received  April 2020 Revised  June 2020 Published  July 2020

Fund Project: Z. Qiao's work is partially supported by Hong Kong Research Council GRF grant 15300417. X. Yang's work is supported by the Natural Science Foundation of China (Grant No. 11802090) and was also supported by the Hong Kong Polytechnic University Postdoctoral Fellowships Scheme (No. 1-YW1D)

In this work, a multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) is proposed to solve a coupled chemotaxis-fluid model. In the evolution equation of the proposed LBM, Beam-Warming (B-W) scheme is used to enhance the numerical stability. In numerical experiments, at first, the comparison between the classical LBM and the present LBM with B-W scheme is carried out by simulating blow up phenomenon of the Keller-Segel (K-S) model. Numerical results show that the stability of the present LBM with B-W scheme is better than the classical one. Then, the second order convergence rate of the proposed B-W scheme is verified in the numerical study of the coupled Navier-Stokes (N-S) K-S model. Finally, through solving the coupled chemotaxis-fluid model, the formation of falling bacterial plumes is numerically investigated. Numerical results agree well with existing ones in the literature.

Citation: Zhonghua Qiao, Xuguang Yang. A multiple-relaxation-time lattice Boltzmann method with Beam-Warming scheme for a coupled chemotaxis-fluid model. Electronic Research Archive, 2020, 28 (3) : 1207-1225. doi: 10.3934/era.2020066
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##### References:
Numerical solutions of bacteria concentrations $n(\mathbf{x}, t)$ of (27)-(28) by the standard LBM (A = 0): (a) $t = 5.0\times10^{-6}$, (b) $t = 1.0\times10^{-5}$, (c) $t = 4.5\times10^{-5}$, (d) $t = 1.15\times10^{-4}$
Numerical solutions of bacteria concentrations $n(\mathbf{x}, t)$ of (27)-(28) by the present LBM with B-W scheme (A = 0.5), (a) $t = 5.0\times10^{-6}$, (b) $t = 1.0\times10^{-5}$, (c) $t = 4.5\times10^{-5}$, (d) $t = 1.15\times10^{-4}$
1D profiles of $n(\mathbf{x}, t)$ along y = 0 at $t = 1.15\times10^{-4}$: (a) by the standard LBM (A = 0); (b) by the present LBM with B-W scheme (A = 0.5)
(a) Vertical profiles of the oxygen $c$ and cell densities $n$ and (b) streamlines at t = 0.22
Vertical profiles of (a) oxygen $c$ and (b) cell densities $n$ at t = 0.22 with $\alpha = 5$, $\delta = 1$ and different $\beta$
Vertical profiles of (a) oxygen $c$ and (b) cell densities $n$ at t = 0.22 with $\beta = 10$, $\delta = 1$ and different $\alpha$
Bioconvection patterns of the cell density $n(x, y)$ at (a) $t = 0.32$, (b) $t = 0.37$, (c) $t = 0.42$, and (d) $t = 0.47$
For $\delta = 5$, cell concentration $n(x, y)$ at (a) $t = 0.27$, (b) $t = 0.34$, and (c) $t = 0.39$
For $\delta = 25$, cell concentration $n(x, y)$ at (a) $t = 2.9$, (b) $t = 3.4$, and (c) $t = 3.9$
For $\delta = 50$, cell concentration $n(x, y)$ at (a) $t = 1.9$, (b) $t = 2.4$, and (c) $t = 2.9$
Convergence test with different lattice spacings, $\delta t = 1.0\times10^{-5}$
 $Meshes$ $E_n$ order $E_c$ order $E_\mathbf{u}$ order $64\times 64$ $1.3934 \times 10^{-4}$ – $1.6327 \times 10^{-4}$ – $2.0029 \times 10^{-4}$ – $128\times 128$ $3.6892 \times 10^{-5}$ 1.9172 $4.3355 \times10^{-5}$ 1.9130 $5.3150 \times 10^{-5}$ 1.9139 $256\times 256$ $9.4955 \times 10^{-6}$ 1.9580 $1.1174\times 10^{-5}$ 1.9560 $1.3699 \times 10^{-5}$ 1.9560 $512\times 512$ $2.4090 \times 10^{-6}$ 1.9788 $2.8380\times 10^{-6}$ 1.9772 $3.4775 \times 10^{-6}$ 1.9779
 $Meshes$ $E_n$ order $E_c$ order $E_\mathbf{u}$ order $64\times 64$ $1.3934 \times 10^{-4}$ – $1.6327 \times 10^{-4}$ – $2.0029 \times 10^{-4}$ – $128\times 128$ $3.6892 \times 10^{-5}$ 1.9172 $4.3355 \times10^{-5}$ 1.9130 $5.3150 \times 10^{-5}$ 1.9139 $256\times 256$ $9.4955 \times 10^{-6}$ 1.9580 $1.1174\times 10^{-5}$ 1.9560 $1.3699 \times 10^{-5}$ 1.9560 $512\times 512$ $2.4090 \times 10^{-6}$ 1.9788 $2.8380\times 10^{-6}$ 1.9772 $3.4775 \times 10^{-6}$ 1.9779
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