Vertical distribution and longitudinal dispersion of gyrotactic microorganisms in a horizontal plane Poiseuille flow

Dispersion of active Brownian particles is a fundamental issue in biological, environmental, and related applications. However, due to the restriction in former models, a detailed analysis of Taylor dispersion of gyrotactic microorganisms in a horizontal plane Poiseuille flow is still lacking. In the present paper, with a recently proposed method [Jiang and Chen, J. Fluid Mech. 877, 1 (2019)], we illustrate the influences of the swimming ability, gyrotaxis intensity, shape anisotropy of microorganisms, and velocity of the ambient fluid on the dispersion process. Compared with nongyrotactic ones, there is a double accumulation mechanism for gyrotactic microorganisms: gravitactic focusing and wall accumulation. By using different boundary conditions, we show the effects of gravitactic focusing alone and double accumulation together. The variations of vertical distribution, overall drift, and effective dispersivity are characterized by changing the characteristic parameters of the microorganisms and the flow. Consisting of a swimming-induced part and an advection-induced part, the overall drift and effective dispersivity are coupled with the shape factor, flow Péclet number, and swimming Péclet number, which leads to nonmonotonic variations as functions of these parameters.

Figure 1

Sketch of gyrotactic microorganisms in a horizontal plane Poiseuille flow.

Figure 2

Vertical distributions ${\langle P_0^{\infty }\rangle }_{\text{O}}$ of spherical microorganisms under the reflective boundary conditions with zero translational diffusivity ($D_t=0$). Swimming Péclet numbers ${\text{Pe}}_s$ at left, middle, and right are 0.1, 1, and 10, respectively. $\lambda$ denotes the bias parameter of microorganisms. “Asymptotic” denotes the exponential distribution (44).

Figure 3

Vertical distributions ${\langle P_0^{\infty }\rangle }_{\text{O}}$ of spherical microorganisms under the reflective boundary conditions. Swimming Péclet numbers ${\text{Pe}}_s$ at left, middle, and right are 0.1, 1, and 10, respectively. Flow Péclet numbers ${\text{Pe}}_f$ at top, middle, and bottom are 0.1, 1, and 10, respectively. $D_t=1/6$ in all cases. $\lambda$ denotes the bias parameter of microorganisms.

Figure 4

Vertical distributions ${\langle P_0^{\infty }\rangle }_{\text{O}}$ of microorganisms under the reflective boundary conditions with fixed bias parameter $\lambda =2$. Swimming Péclet numbers ${\text{Pe}}_s$ at left, middle, and right are 0.1, 1, and 10, respectively. Flow Péclet numbers ${\text{Pe}}_f$ at top, middle, and bottom are 0.1, 1, and 10, respectively. $D_t=1/6$ in all cases. ${\alpha }_0$ denotes the shape factor of microorganisms.

Figure 5

$P_0^{\infty }(z,\theta )$ of microorganisms in the local space $\mathit{q}$ under the reflective boundary conditions. First row, ${\text{Pe}}_f=0.1$; second row, ${\text{Pe}}_f=1$; third row, ${\text{Pe}}_f=10$. The left two columns are for nongyrotactic microorganisms and the right two columns are for gyrotactic microorganisms with $\lambda =2$. $\alpha =0$ in the first column and third column and $\alpha =1$ in the second column and fourth column. ${\text{Pe}}_s=1$, $D_t=1/6$ in all cases.

Figure 6

Overall drift $U_d$ and dispersivity $D_{\mathrm{T}}$ as functions of swimming Péclet number ${\text{Pe}}_s$ under the reflective boundary conditions with fixed shape factor ${\alpha }_0=0$. Top, ${\text{Pe}}_f$=0.1; middle, ${\text{Pe}}_f=1$; bottom, ${\text{Pe}}_f$=10. In all cases, $D_t=1/6$.

Figure 7

Overall drift $U_d$ and dispersivity $D_{\mathrm{T}}$ as functions of swimming Péclet number ${\text{Pe}}_s$ under the reflective boundary conditions with fixed bias parameter $\lambda =2$. Top, ${\text{Pe}}_f=0.1$; middle, ${\text{Pe}}_f=1$; bottom, ${\text{Pe}}_f=10$. In all cases, $D_t=1/6$.

Figure 8

Vertical distributions ${\langle P_0^{\infty }\rangle }_{\text{O}}$ of microorganisms under the Robin type boundary conditions. ${\text{Pe}}_s$ at left and right are 0.1 and 1, respectively. ${\text{Pe}}_f$ at top and bottom are 0.1 and 10, respectively. $D_t=1/6$ in all cases.

Figure 9

$P_0^{\infty }(z,\theta )$ of microorganisms in the local space $\mathit{q}$ under the Robin type boundary conditions. First row, ${\text{Pe}}_f=0.1$; second row, ${\text{Pe}}_f=1$; third row, ${\text{Pe}}_f=10$. The left two columns are for nongyrotactic microorganisms and the right two columns are for gyrotactic microorganisms with $\lambda =2$. $\alpha =0$ in the first column and third column and $\alpha =1$ in the second column and fourth column. ${\text{Pe}}_s=1$, $D_t=1/6$ in all cases.

Figure 10

Overall drift $U_d$ and dispersivity $D_{\mathrm{T}}$ as functions of swimming Péclet number ${\text{Pe}}_f$ under the Robin type boundary conditions. Top, ${\text{Pe}}_s=0.1$; bottom, ${\text{Pe}}_s=1$. $D_t=1/6$ in all cases.

Figure 11

$P_0^{\infty }(z,\theta )$ of microorganisms in the local space $\mathit{q}$. First row: Reflective boundary conditions. Second row: Robin type boundary conditions. Left column: Results of GTD. Right column: Results of IBM. In all cases, ${\text{Pe}}_s=1$, ${\text{Pe}}_f=10$, ${\alpha }_0=1$, $\lambda =2$, $D_t=1/6$.

Figure 12

Vertical distribution ${\langle P_0^{\infty }\rangle }_{\text{O}}$ and orientational distribution ${\langle P_0^{\infty }\rangle }_{\text{V}}$ of microorganisms under different boundary conditions. In all cases, ${\text{Pe}}_s=1$, ${\text{Pe}}_f=10$, ${\alpha }_0=1$, $\lambda =2$, $D_t=1/6$.

Figure 13

Evolution and steady state of drift $U_d$ and dispersivity $D_{\mathrm{T}}$.