Effect of solid boundaries on swimming dynamics of microorganisms in a viscoelastic fluid

Abstract

We numerically study the effect of solid boundaries on the swimming behavior of a motile microorganism in viscoelastic media. Understanding the swimmer-wall hydrodynamic interactions is crucial to elucidate the adhesion of bacterial cells to nearby substrates which is precursor to the formation of the microbial biofilms. The microorganism is simulated using a squirmer model that captures the major swimming mechanisms of potential, extensile, and contractile types of swimmers, while neglecting the biological complexities. A Giesekus constitutive equation is utilized to describe both viscoelasticity and shear-thinning behavior of the background fluid. We found that the viscoelasticity strongly affects the near-wall motion of a squirmer by generating an opposing polymeric torque which impedes the rotation of the swimmer away from the wall. In particular, the time a neutral squirmer spends at the close proximity of the wall is shown to increase with polymer relaxation time and reaches a maximum at Weissenberg number of unity. The shear-thinning effect is found to weaken the solvent stress and therefore, increases the swimmer-wall contact time. For a puller swimmer, the polymer stretching mainly occurs around its lateral sides, leading to reduced elastic resistance against its locomotion. The neutral and puller swimmers eventually escape the wall attraction effect due to a releasing force generated by the Newtonian viscous stress. In contrast, the pusher is found to be perpetually trapped near the wall as a result of the formation of a highly stretched region behind its body. It is shown that the shear-thinning property of the fluid weakens the wall-trapping effect for the pusher squirmer.

Appendix: Verification of the numerical method

A.1 Rotation of a single sphere in an Oldroyd-B shear flow

In an Oldroyd-B fluid, we simulate the rotation of a single sphere in a shear flow to verify our numerical platform. Simulation is conducted in a rectangular domain of [−2a,2a] × [−2a, 2a] × [−4a, 4a] where a is the radius of the sphere and the sphere is centered at (0,0,0). The flow is driven by two parallel plates at z = −4a and z = 4a moving opposite in x-direction with the same speed U. Periodic boundary conditions are applied in x and y directions. The mesh size is Δ=a/16 and the time step is Δt = 10⁻³ a/U. The shear rate of the flow is $\dot {\gamma }=U/4a$, the Weissenberg number $Wi=\lambda \dot {\gamma }$, the Reynolds number $Re=\rho \dot {\gamma } a^{2}/\mu =0.025$ and the viscosity ratio β _s = 0.5. Figure 19a shows the time evolution of the angular velocity of the sphere at different W i. It is seen that for the Newtonian case, the sphere asymptotically reaches to its steady state of ${\Omega }_{y}=0.5\dot {\gamma }$ while for viscoelastic cases, overshoots can be observed around $t\dot {\gamma }=0.2$. In Fig. 19b, the steady angular velocity as a function of W i is compared with previous experimental (Snijkers et al. 2011) and numerical (Goyal and Derksen 2012) results. It is evident that our simulation results are in good agreement with the previous results.

Fig. 19

(Color online) a Transient behaviour of a rotating sphere in an Oldroyd-B shear flow b Comparison of the steady angular velocity as a function of W i with the results of Snijkers et al. (2011) and Goyal and Derksen (2012)

A.2 Free swimming of a squirmer in a Giesekus fluid in an unbounded domain

The simulation is performed on a non-uniform structured grid with the smallest mesh size of Δ = D/40 near the squirmer, where D is the diameter of the spherical squirmer. The computational domain is [−40a, 40a] × [−40a, 40a] × [−40a, 40a] and the squirmer is initially placed at (0,0,0). The time step is Δt = 10⁻⁵. The Reynolds number, defined as R e = U ₀ a/ν, is 0.01 in all the simulations, and U ₀ = 2B ₁/3. According to the analysis of a squirmer in a Newtonian fluid at finite Reynolds number, the swimming speed of a squirmer is determined by U/U ₀ ≃ 1 − 0.15β R e (Wang and Ardekani 2012a), thus the effects of the inertia on the swimming speed can be neglected in our simulation. The viscosity ratio is β _s = 0.5 and mobility factor is α _m = 0.2. The Weissenberg number is defined as W i = λ B ₁/a. The swimming speed of the squirmer U is plotted in Fig. 20 for squirmers with β = −5, 0, and 5. Our results show good agreement with the results obtained by Zhu et al. (2012).

Fig. 20

(Color online) Swimming speed U as a function of the Weissenberg number W i, for the neutral squirmer β = 0 (solid line: (Zhu et al. 2012) and circles: present results), pusher β = −5 (dashed line: (Zhu et al. 2012) and squares: present results) and puller β = 5 (dashdot line: (Zhu et al. 2012) and triangles: present results). The Reynolds number is R e = 0.01 and the swimming speed is scaled by the squirmer’s speed U ₀ in a Newtonian fluid

A.3 Convergence study

Convergence studies have been performed for the near-wall motion of squirmers with β = 0 and −3 under different grid sizes and different time steps. Figure 21 shows the time history of the distance h away from the wall, orientation angle α and the swimming speed U of the squirmer. The results from these different computations agree well with each other. It is confirmed that the computed results are independent of the mesh size and the time step.

Fig. 21

(Color online) Time history of a vertical distance h and orientation angle α and b swimming speed U of the neutral squirmer calculated using different grid sizes, different time steps and different values of the parameter 𝜖. The corresponding parameters are W i = 6 and R e = 0.1 and the squirmer is initialized at h ₀ = 2 and α ₀ = −π/4

A.4 Repulsive force

When the squirmer lies in the close proximity of the surface, due to the lubrication effect and other non-hydrodynamic phenomena such as electrostatic charges, a repulsive force is developed which prevents intrusion of the swimmer’s body into the wall. To capture the associated hydrodynamic squeezing effect, exceedingly fine grid resolutions are needed which make the corresponding simulations computationally highly demanding. In addition, as indicated by Spagnolie and Lauga (2012), hydrodynamic interactions are inadequate to prevent the swimmer-wall interference in some settings. Hence, in order to avoid overlapping of the squirmer’s body and the nearby wall, we impose a short-range repulsive force (Glowinski et al. 2001) defined as,

$$ \mathbf{F}_{r} = \frac{C_{m}}{\epsilon} \left( \frac{h-h_{\min}-h_{r}}{h_{r}} \right) \mathbf{e}, $$

(7)

where h _min = a is the minimum possible distance from the wall and h _r represents the range over which the force is acting and is normally set to be the smallest grid size Δ in the computational domain (Glowinski et al. 2001). The direction of the repulsive force e is considered to be perpendicular to the wall. The parameters $C_{m}=M_{p} {U_{0}^{2}} /a$ and 𝜖 = 10⁻⁴ denote a scaling factor and a small positive number, respectively, with M _p being the mass of the squirmer. As demonstrated in Fig. 21, changing the value of 𝜖 have a negligible impact on the simulation results.