Imperial College London MS.c Individual Project *

Abstract

The term “Gyrotactic” represents the swimming behaviour of certain types of microorganisms in suspension subjected to gravity and shear torque. The interesting collective behaviour is shown in the vertical pipe flow, that is, the cells swim toward the centre and the wall in the downward flow and the upward flow, respectively. Instabilities arise from this effect and lead to bifurcations.

In this project, based on the self-developed semi-implicit unsteady FVM solver, the bifurcation was studied under two conditions of fixed flow and prescribed pressure gradient. In fixing the flow rate, bifurcations are found matched the previous results made by J Kessler’s 1 and L Fung .et.al 2.

Regarding the pressure gradient, one bifurcation related to the initial flow conditions is found. And the explanation is given from the perspective of combined influence on the velocity profile of cell distribution and pressure gradient.

The unsteady solver is also utilised to investigate the flow with the time-period pressure gradient. starting from a derived linearized state-space system, the effects of different parameters on the system are studied. And found “diverge gap” in certain Reynolds numbers, and proposed an explanation.

Key pictures and conclusions

Modelling

In this project, an unsteady solver with constant flow rate and pressure gradient based on Kessler’s 1986 model1 are developed.

  • For cell conservation law, Semi-implicit first order disicritisation scheme are taken with compensation coefficient $K_p$

  • For momentum equation, pure implicit first order discritisation scheme are taken

The code is validate with a series of typical parameters.

Bifurcations Under Different Fixed Flow Rate Conditions

Keep a steady flow rate $Q$ when run simulations

Change flow rate $Q$, and the Richardson number $Ri$

The converged cell concentrations for different $Q$ and $Ri$ are plotted together as below

Centerline_cell_concentration_with_Ri_and_q0

Two bifurcation points are validated:

  • Kessler’s analytical result is validated by setting a series of Ri values and determining that the bifurcation point occurs at $Ri = 8$.

  • Fung’s simulation result shows a similar qualitative match, with the bifurcation point at 0 point occurring near $Ri = 25$

Bifurcations Under Different Fixed Pressure Gradient Conditions

Keep a steady pressure gradient $Pz$ when run simulation

Change the pressure gradient $Pz$, initial flow rate $Q_0$, and the Richardson number $Ri$

Similarly, plot the converged cell concentrations with different $Q_0$, $P_z$, $Ri$

Centerline_cell_concentration_with_Ri_pz_and_q0

video 1, the concentration and velocity distribution with time near bifurcation initial condition

Another bifurcation is found, when $pz < 0$ and $Q0 < 0$, a performance pattern of divergence followed by convergence is observed. Furthermore, the divergence gap widens as the magnitude of pz increases.

Pulsatile Flow Under Sinusoidal Wave Pressure Gradient

Enact a sinusoidal wave pressure gradient $P_z(t)=\hat{p}^{iwt}$ when run simulation, the suspension will therefore sloshing like the video shows below:

New bifurcation found with different Reynolds number:

Bode_diagram_for_the_amplitude_of_central_line_velocity

video 2, the concentration and velocity distribution responses around bifurcation frequency of $P_z$ at the boundary of the divergence area

* Refer to pdf for more details

  1. John O Kessler. Individual and collective fluid dynamics of swimming cells. Journal of Fluid Mechanics, 173:191–205, 1986. ↩︎ ↩︎

  2. Lloyd Fung, Rachel N Bearon, and Yongyun Hwang. Bifurcation and stability of downflowing gyrotactic micro-organism suspensions in a vertical pipe. arXiv preprint arXiv:2001.08072, 2020. ↩︎