Locomotion of a microorganism in weakly viscoelastic liquids

In the present work we study the motion of microorganisms swimming by an axisymmetric distribution of surface tangential velocity in a weakly viscoelastic fluid. The second-order fluid constitutive equation is used to model the suspending fluid, while the well-known “squirmer model” [M. J. Lighthill, Comm. Pure Appl. Math. 5, 109 (1952); J. R. Blake, J. Fluid Mech. 46, 199 (1971)] is employed to describe the organism propulsion mechanism. A regular perturbation expansion up to first order in the Deborah number is performed, and the generalized reciprocity theorem from Stokes flow theory is then used, to derive analytical formulas for the squirmer velocity. Results show that “neutral” squirmers are unaffected by viscoelasticity, whereas “pullers” and “pushers” are slowed down and hastened, respectively. The power dissipated by the swimming microorganism and the “swimming efficiency” are also analytically quantified.

Figure 1

Comoving frame streamlines generated by microorganisms in a Newtonian fluid: (a) neutral swimmer, (b) pusher with $\beta =-3$, (c) puller with $\beta =3$. (Somehow unrealistic high values of $\beta$ are chosen, merely to highlight the flow field features.) The solid dot represents the position of a stagnation point.

Figure 2

Squirmer swimming velocity as a function of $\text{De}$. Solid lines represent the analytical solution, Eq. (31), for $b=-1.8$; solid symbols are numerical solution for a Giesekus fluid with $b=-1.8$.

Figure 3

Total swimming power dissipated as a function of $\beta$, for $\text{De}=0.02$.

Figure 4

Swimming efficiency normalized with the Newtonian one, as a function of $\beta$, for $\text{De}=0.02$.

Figure 5

Comparison of (a) Newtonian and (b) order $\text{De}$ streamlines generated, in the swimming frame, by a puller with $\beta =3$, for $b=-2$. The solid circle represents the position of the stagnation point.

Figure 6

Comparison of (a) Newtonian and (b) order $\text{De}$ streamlines generated, in the swimming frame, by a pusher with $\beta =-3$, for $b=-2$. The solid circle represent the position of the stagnation point.