Dynamics of a helical swimmer crossing viscosity gradients

We experimentally and theoretically study the dynamics of a low-Reynolds number helical swimmer moving across viscosity gradients. Experimentally, a double-layer viscosity is generated by superposing two miscible fluids with similar densities but different dynamic viscosities. A synthetic helical magnetically driven swimmer is then made to move across the viscosity gradients along four different configurations: either head-first (pusher swimmer) or tail-first (puller), and through either positive (i.e., going from low to high viscosity) or negative viscosity gradients. We observe qualitative differences in the penetration dynamics for each case. We find that the swimming speed can either increase or decrease while swimming across the viscosity interface, which results from the fact that the head and the tail of the swimmer can be in environments in which the local viscosity leads to different relative amounts of drag and thrust. In order to rationalize the experimental measurements, we next develop a theoretical hydrodynamic model. We assume that the classical resistive-force theory of slender filaments is locally valid along the helical propeller and use it to calculate the swimming speed as a function of the position of the swimmer relative to the fluid-fluid interface. The predictions of the model agree well with experiments for the case of positive viscosity gradients. When crossing across a negative gradient, gravitational forces in the experiment become important, and we modify the model to include buoyancy, which agrees with experiments. In general our results show that it is harder for a pusher swimmer to cross from low to high viscosity, whereas for a puller swimmer it is the opposite. Our model is also extended to the case of a swimmer crossing a continuous viscosity gradient.

Figure 1

Experimental setup. (a) Schematic representation of the helical swimmer. The dimensions of the device are $2r_H=4.5$ mm; $L_H=16$ mm; $L_T=16$ mm; $\lambda =5.3$ mm; $2R_T=4.5$ mm; and the pitch angle $\psi ={45}^{\circ }$. The thickness of the wire is $2r_T=0.9$ mm. (b) In this example, the swimmer moves head-first from a high-to-low viscosity fluid through a sharp gradient and here gravity is pointing downwards. Note that in all figures the dark and light gray denote high- and low-viscosity regions, respectively.

Figure 2

(a) The viscosity gradients at $t=0$ and 16 h. (b) Normalized viscosity, $\mu \left(z\right)/\mu$, as a function of normalized distance from the interface, $z^{*}=z/L_H$. Data points represent the pixel intensity, which serves as a proxy for the viscosity profile, soon after the initial setup ($\delta =\mathrm{\Delta }/L_H=0.2735$) and 16 h afterwards ($\delta =1.1384$). Dashed lines represent the best fit of the Arrhenius equation $\mu \left(C\right)=A{e}^{BC/C_0}$ with $C/C_0$ given by Eq. (2), and $A={\mu }_{-}$ and $B=ln({\mu }_{+}/{\mu }_{-})$.

Figure 3

The four swimmer-viscosity interaction configurations for motion across the viscosity gradient. In all cases ${\mu }_{-}<{\mu }_{+}$ and the motion is in the upwards direction; note that gravity points upwards in (a) and (b) and downwards in (c) and (d). Notice as well the change in the sense of rotation of the tail, depending on its orientation. All conditions are described in Table 1.

Figure 4

Case I: time sequence of the head-first (pusher) swimmer crossing a positive viscosity gradient, for $\delta =0.274$ (narrow gradient). Images have been flipped so that the swimmer appears to move upwards.

Figure 5

Case I dynamics, head-first (pusher) swimmer crossing a positive viscosity gradient: (a) dimensionless position, $z^{*}$, as function of dimensionless time, $t^{*}$; (b) normalized speed $U/U_{+}$ as a function of dimensionless position $z^{*}$. At $t^{*}\approx 0$ the swimmer reaches the interface, located at $z^{*}\approx 0$.

Figure 6

Case II: time sequence of the tail-first (puller) swimmer crossing a positive viscosity gradient, for $\delta =0.274$ (narrow gradient). Images have been flipped so that the swimmer appears to move upwards.

Figure 7

Case II dynamics, tail-first (puller) swimmer crossing a positive viscosity gradient: (a) dimensionless position, $z^{*}$, as function of dimensionless time, $t^{*}$; (b) normalized speed $U/U_{+}$ as a function of dimensionless position $z^{*}$. At $t^{*}\approx 0$ the swimmer reaches the interface, located at $z^{*}\approx 0$.

Figure 8

Case III: time sequence of the head-first (pusher) swimmer crossing a negative viscosity gradient, for $\delta =0.274$ (narrow gradient). The sketch on the right depicts the crossing dynamics schematically.

Figure 9

Case III dynamics, head-first swimmer (pusher) crossing a negative viscosity gradient: (a) dimensionless position, $z^{*}$, as function of dimensionless time, $t^{*}$; (b) normalized speed $U/U_{+}$ as a function of dimensionless position $z^{*}$. At $t^{*}\approx 0$ the swimmer reaches the interface, located at $z^{*}\approx 0$.

Figure 10

Case IV: Time sequence of the tail-first swimmer crossing the viscosity gradients from high to low viscosity. The sketch on the right shows the crossing dynamics schematically.

Figure 11

Case IV dynamics, tail-first (puller) swimming crossing negative viscosity gradient: (a) dimensionless position, $z^{*}$, as function of dimensionless time, $t^{*}$; (b) normalized speed $U/U_{+}$ as a function of dimensionless position $z^{*}$. At $t^{*}\approx 0$ the swimmer reaches the interface, located at $z^{*}\approx 0$.

Figure 12

Helical swimmer crossing a sharp viscosity gradient.

Figure 13

Continuous viscosity gradient with a transition region of size $\mathrm{\Delta }$. Left: a linear viscosity profile between two fluid layers of viscosities ${\mu }^{\prime }$ and $\mu$. Right: a diffuse layer with $\mathrm{\Delta }\sim \sqrt{Dt}$, where $D$ is the diffusivity of $\mu \left(z\right)$.

Figure 14

Swimming speed in a linear viscosity gradient for different widths of the transition region. (a) Case I, (b) case III. The viscosities of the semi-infinite fluids on either side of the interface are given by the experimental values, i.e., ${\mu }^{\prime }={\mu }_{-}=0.55\text{Pa}\text{s}$ and $\mu ={\mu }_{+}=2.74\text{Pa}\text{s}$. The length of the tail and head are the same, i.e., $\lambda =1$, and the swimmer is neutrally buoyant in both fluids. The diagrams on the right indicate the direction of motion, with dark and light gray representing high ($\mu$) and low viscosity (${\mu }^{\prime }$), respectively. The values are normalized by the terminal speed $U_0$.

Figure 15

Swimming speed in a diffusive viscosity gradient for different widths of the transition region. (a) Case I, (b) case III. The parameters $\{\mu /{\mu }^{\prime },\lambda \}$ are the same as in Fig. 14.

Figure 16

Swimming speed and position in a diffusive viscosity gradient for different values of the viscosity ratio: ${\mu }_i/{\mu }^{\prime }=1.0$, 1.1, 2.0, 5.0, 10.0, 100.0, and 1000.0. (a), (c) Speed and position, Case I; (b), (d) speed and position, case III. The width of the transition region is $\delta =\mathrm{\Delta }/L_H=0.25$, the viscosity of the top fluid is ${\mu }^{\prime }={\mu }_{-}=0.55\text{Pa}\text{s}$ and $\lambda =1$.

Figure 17

Swimming speed and position in a diffusive viscosity gradient for different values of the tail-to-head size ratio, $\lambda$. (a), (c) Speed and position, case I; (b), (c) speed and position, case III. The width of the transition region is $\delta =\mathrm{\Delta }/L_H=0.25$, and the viscosity ratio is the same as in Fig. 14 (experimental values).

Figure 18

Comparison between experiments and data for case I [i.e., head-first (pusher) swimmer crossing a positive viscosity gradient]: Speed of the swimmer as a function of the dimensionless position of the head $h$ with respect to the interface located at $h=0$. The speed is normalized by the speed in the fluid of high viscosity, $U_{+}=U(h\to \infty )$. We display the measurements early after the viscosity gradient has been set up ([N]arrow gradient): triangles (experiment) and blue solid line (model), and 16 h after ([W]ide gradient): circles (experiment) and red dashed line (model).

Figure 19

Comparison between experiments and data for case II [tail-first (puller) swimmer in a positive viscosity gradient]. The speed of the swimmer is plotted as a function of the dimensionless position of the tail $h$ with respect to the interface located at $h=0$, and the speed is normalized by that in the high-viscosity fluid, $U_{+}=U(h\to \infty )$. We show the measurements soon after the viscosity gradient has been set up ([N]arrow gradient case): triangles (experiment) and blue solid line (model), as well as 16 h after the start of the experiment ([W]ide gradient case): circles (experiment) and red dashed line (model).

Figure 20

Comparison between experiments and data for case III [i.e., head-first swimmer (pusher) crossing a negative viscosity gradient]. We plot the dimensionless speed of the swimmer as a function of the dimensionless position of the head $h$ measured relative to the interface located at $h=0$. The speed of the swimmer is normalized by the speed in the high-viscosity fluid, $U_{+}=U(h\to -\infty )$. We display the measurements early after the viscosity gradient has been set up ([N]arrow gradient): triangles (experiment) and blue solid line (model), and 16 h after ([W]ide gradient): circles (experiment) and red dashed line (model). The figure on the left shows the predictions of the original model, not taking into account entrainment, while the plot on the right the modified model with variable buoyancy and ${\alpha }_{\max }=0.1$.

Figure 21

Viscous entrainment of the high-viscosity fluid by the swimmer. (a) Experimental picture showing the swimmer moving down the gradient and entraining some of the high-viscosity fluid as it crosses the interface, regardless of its orientation relative to the interface. (b) The entrained fluid accounts for an increase in the apparent density of the swimmer from ${\rho }_{\mathrm{swimmer}}=M_{\mathrm{swimmer}}/V_{\mathrm{swimmer}}$ to $\overline{\rho }=[1+\alpha \left(h\right)]{\rho }_{\mathrm{swimmer}}$, where $\alpha$ depends on the mass of fluid dragged along with the swimmer, $M_f=\rho V_f$, as given in Eq. (45).

Figure 22

Comparison between experiments and data for case IV (tail-first swimmer (puller) crossing the viscosity gradients from high to low viscosity): Speed of the swimmer as a function of the dimensionless position of the tail $h$ with respect to the interface located at $h=0$. Values for the speed are normalized by the speed in the high-viscosity fluid $U_{+}=U(h\to -\infty )$. We display the measurements early after the viscosity gradient has been set up ([N]arrow gradient): triangles (experiment) and blue solid line (model), and 16 h after ([W]ide gradient): circles (experiment) and red dashed line (model). The figure on the left shows the predictions of the original model, while the plot on the right the modified model with variable buoyancy and ${\alpha }_{\max }=0.1$.