Locomotion of microorganisms near a no-slip boundary in a viscoelastic fluid

Locomotion of microorganisms plays a vital role in most of their biological processes. In many of these processes, microorganisms are exposed to complex fluids while swimming in confined domains, such as spermatozoa in mucus of mammalian reproduction tracts or bacteria in extracellular polymeric matrices during biofilm formation. Thus, it is important to understand the kinematics of propulsion in a viscoelastic fluid near a no-slip boundary. We use a squirmer model with a time-reversible body motion to analytically investigate the swimming kinematics in an Oldroyd-B fluid near a wall. Analysis of the time-averaged motion of the swimmer shows that both pullers and pushers in a viscoelastic fluid swim towards the no-slip boundary if they are initially located within a small domain of “attraction” in the vicinity of the wall. In contrast, neutral swimmers always move towards the wall regardless of their initial distance from the wall. Outside the domain of attraction, pullers and pushers are both repelled from the no-slip boundary. Time-averaged locomotion is most pronounced at a Deborah number of unity. We examine the swimming trajectories of different types of swimmers as a function of their initial orientation and distance from the no-slip boundary.

Figure 1

Schematic of the geometry of the system. The red circle indicates the orientation of the squirmer's thrust, which is defined by the polar angle $\theta =\varphi$ relative to the positive $\widehat{x}$ axis. The swimmer's translational and rotational locomotion velocity vectors are denoted by $\mathbf{U}$ and $\mathit{\Omega }$, respectively.

Figure 2

Time-averaged locomotion kinematics of a neutral squirmer at De = 10.

Figure 3

Time-averaged locomotion kinematics of a pusher with $\beta =-2$ in an Oldroyd-B fluid with $\text{De}=10$ as a function of the squirmer's distance from the wall for different swimming directions.

Figure 4

Time-averaged locomotion kinematics of a puller with $\beta =2$ in an Oldroyd-B fluid with $\text{De}=10$ as a function of the squirmer's distance from the wall for different swimming directions.

Figure 5

The size of the attraction layer (${\delta }_a$) for a pusher ($\beta =-2$) and a puller ($\beta =2$) near a no-slip boundary in an Oldroyd-B fluid. The value of ${\delta }_a$ indicates the distance from the wall ($d/R$) at which $\langle {\mathbf{U}}_{2,y}\rangle$ vanishes.

Figure 6

Computed squirmer trajectories along the wall (first column) and swimming orientation versus normalized distance from the wall (second column) for a pusher with $\beta =-2$ initially positioned at $d_0/R=1.05$ (a), $d_0/R=1.1$ (b), and $d_0/R=4.0$ (c) with initial orientation ${\varphi }_0=0$, $\pi /6$, $\pi /3$, or $\pi /2$.

Figure 7

Computed squirmer trajectories along the wall (first column) and swimming orientation versus normalized distance from the wall (second column) for a puller with $\beta =2$ initially positioned at $d_0/R=1.05$ (a), $d_0/R=1.1$ (b), and $d_0/R=4.0$ (c) with initial orientation ${\varphi }_0=0$, $\pi /6$, $\pi /3$, or $\pi /2$.

Figure 8

Streamlines (in the swimmer frame of reference) color-labeled by the value of the stream function for the flow fields generated by (a) a puller ($\beta =4$), (b) a pusher ($\beta =-4$), and (c) a neutral squirmer ($\beta =0$) at a distance $d/R=4$ away from a no-slip surface in a Newtonian fluid. The red dot on the surface of the cylindrical swimmer indicates the swimming orientation ($\varphi =\pi /6$).