Squirming in a viscous fluid enclosed by a Brinkman medium

Cell motility plays important roles in a range of biological processes, such as reproduction and infections. Studies have hypothesized that the ulcer-causing bacterium Helicobacter pylori invades the gastric mucus layer lining the stomach by locally turning nearby gel into sol, thereby enhancing its locomotion through the biological barrier. In this work, we present a minimal theoretical model to investigate how heterogeneity created by a swimmer affects its own locomotion. As a generic locomotion model, we consider the swimming of a spherical squirmer in a purely viscous fluid pocket (representing the liquified or degelled region) surrounded by a Brinkman porous medium (representing the mucus gel). The use of the squirmer model enables an exact, analytical solution to this hydrodynamic problem. We obtain analytical expressions for the swimming speed, flow field, and power dissipation of the swimmer. Depending on the details of surface velocities and fluid properties, our results reveal the existence of a minimum threshold size of mucus gel that a swimmer needs to liquify in order to gain any enhancement in swimming speed. The threshold size can be as much as approximately $30\%$ of the swimmer size. We contrast these predictions with results from previous models and highlight the significant role played by the details of surface actuations. In addition to their biological implications, these results could also inform the design of artificial microswimmers that can penetrate into biological gels for more effective drug delivery.

Figure 1

Setup of a minimal model for a spherical swimmer in a viscous solution (representing the liquified or degelled region) enclosed by a Brinkman porous medium (representing the mucus gel). The swimmer of radius $a$ propels at a speed $\mathbf{U}$ via a distribution of surface velocities. The boundary of the degelled region is at a distance $b$ from the center of the swimmer, with $\lambda =b/a$.

Figure 2

(a) Swimming speed of a squirmer $U$ with tangential surface velocities, normalized by its speed in the Stokes limit $U_S$, as a function of the size of the degelled region $\lambda$ at different values of resistance $\delta$. Inset: for a given resistance (shown results for $\delta =4$), the swimming speed displays a local minimum at the critical size of the degelled region ${\lambda }_C$. The swimmer needs to liquify a minimum threshold size ${\lambda }_T$ in order to gain any enhancement in speed. (b) The value of the critical size of the degelled region ${\lambda }_C$ (red dotted line, right axis) and the threshold size ${\lambda }_T$ (blue line, left axis) as a function of resistance $\delta$.

Figure 3

Swimming speed $U$ of a squirmer with radial surface velocities, normalized by its speed in the Stokes limit $U_S$, as a function of the size of of the degelled region $\lambda$ with different values of resistance $\delta$. The swimming speed displays local maxima as the size of the degelled region varies. In contrast to the case of a squirmer with tangential surface velocities (Fig. 2), as the size of the degelled region increases beyond certain thresholds, the swimmer experiences speed reduction compared with the case without any degelation ($\lambda =1$).

Figure 4

The decay of the magnitude of flow velocity $\left|\mathbf{u}\right|$ around a (a) neutral squirmer (${\beta }_2=0$) and (b) a pusher-puller (${\beta }_2=\pm 1$) as a function of distance from the origin $r$ along $\theta =\pi /2$, with various values of resistance $\delta$. The Brinkman medium ($r>\lambda$) is shaded in gray; here $\lambda =3$. The dash-dotted lines representing different algebraic decay scalings are shown for comparison.