Swimming sheet near a plane surfactant-laden interface

In this work we analyze the velocity of a swimming sheet near a plane surfactant-laden interface by assuming the Reynolds number and the sheet's deformation to be small. We observe a nonmonotonic dependence of the sheet's velocity on the Marangoni number $\left(\text{Ma}\right)$ and the surface Péclet number $\left({\text{Pe}}_s\right)$. For a sheet passing only transverse waves, the swimming velocity increases with an increase in $\text{Ma}$ for any fixed ${\text{Pe}}_s$. When ${\text{Pe}}_s$ is increasing, on the other hand, the swimming velocity of the same sheet either increases (at large Ma) or it initially increases and then decreases (at small Ma). This dependence of the swimming velocity on $\text{Ma}$ and ${\text{Pe}}_s$ is altered if the sheet is passing longitudinal waves in addition to the transverse waves along its surface.

Figure 1

A schematic showing a swimming sheet located near a plane surfactant-laden interface. The sheet propels by passing waves along its surface. The distance between the midplane of the sheet and the interface is $h$. We denote the fluid in which the swimmer is suspended as “Fluid 1” while the fluid above the interface is denoted as “Fluid 2.” In the frame moving with the swimming velocity of the sheet, a uniform streaming flow exists in fluid 2 far away from the sheet with the velocity that is negative of the sheet's swimming velocity. The origin of the coordinate system is located at the midplane of the sheet.

Figure 2

The variation of the swimming velocity with the distance between the interface and (the mid plane of) the sheet for a sheet passing only transverse waves (blue solid line) and that passing both longitudinal and transverse waves (red dashed line). Here the swimming velocity is normalized with the swimming velocity of a sheet passing only transverse waves in an unbounded fluid $\left(U_{2T\infty }\right)$. The viscosity ratio $\lambda =0$, Marangoni number $\text{Ma}=1$, and the surface Péclet number ${\text{Pe}}_s=1$.

Figure 3

For a sheet passing transverse waves near a plane surfactant-laden interface, the variation of the leading order swimming velocity with (a) $\text{Ma}$ for ${\text{Pe}}_s=1$ and (b) ${\text{Pe}}_s$ for $\text{Ma}=1$, 10, and 100. Here the swimming velocity is normalized with the swimming velocity of the same sheet in an unbounded fluid $\left(U_{2T\infty }\right)$. The distance between the midplane of the sheet and the interface is $h=1$, the viscosity ratio $\lambda =0$, and the amplitudes of the longitudinal waves $a=d=0$.

Figure 4

For a sheet passing both longitudinal and transverse waves near a plane surfactant-laden interface, the variation of the leading order swimming velocity with (a) $\text{Ma}$ for ${\text{Pe}}_s=0.1,1,10,100$ and with (b) ${\text{Pe}}_s$ for $\text{Ma}=1,10,100$. Here the swimming velocity is normalized with the swimming velocity of a sheet passing transverse waves in an unbounded fluid $\left(U_{2T\infty }\right)$. The distance between the midplane of the sheet and the interface is $h=1$, the viscosity ratio $\lambda =0$, and the amplitudes of the waves $a=0$, and $d/b=2$.

Figure 5

The variation of apparent viscosity ratio with ${\text{Pe}}_s$ at $\text{Ma}=1$ (blue solid line), 10 (red dashed line), and 100 (yellow dash-dotted line). Here, the sheet is passing only transverse waves, hence $a=d=0$. The actual viscosity ratio of the surfactant-laden interface $\lambda =0$ while $h=1$.

Figure 6

For a sheet passing transverse waves near a plane clean interface, the variation of the (a) leading order interface slip and (b) leading order swimming velocity with the viscosity ratio $\lambda$. Here the swimming velocity is normalized with the swimming velocity of the same sheet in an unbounded fluid $\left(U_{2T\infty }\right)$. The values of the other parameters $h$, $a$, and $d$ are kept the same as those of Fig. 3.

Figure 7

Comparison of interface slip of a clean interface (blue solid line) with that of a surfactant-laden interface (blue dotted line). Also plotted is the surfactant concentration on the surfactant-laden interface (red dash-dotted line). The vertical dashed lines are just for reference. The blue (upper) and red (lower) arrows denote, respectively, the vector field of clean interface slip and the direction of Marangoni induced slip. The axis for the interface slip is on the left while that for surfactant concentration is on the right. Here, $\text{Ma}=10$ and ${\text{Pe}}_s=1$ for the surfactant-laden interface. Also, $\epsilon =0.1$ is used to calculate $\mathrm{\Gamma }$ while the values of the other parameters $h$, $\lambda$, $a$, and $d$ are kept the same as those of Fig. 3.

Figure 8

The variation of (a) the Marangoni stress and (b) the leading order interface slip with the Marangoni number $\text{Ma}$ for a fixed surface Péclet number ${\text{Pe}}_s=1$. Here $\epsilon =0.1$ is used to calculate the Marangoni stresses while the values of the other parameters $\lambda$, $h$, $a$, and $d$ are kept the same as those of Fig. 3.

Figure 9

The variation of (a) the surfactant concentration and (b) the leading order interface slip with the surface Péclet number ${\mathrm{Pe}}_s$ for a fixed Marangoni number $\mathrm{Ma}=100$. Here $\epsilon =0.1$ is used to calculate surfactant concentration while the values of the other parameters $\lambda$, $h$, $a$, and $d$ are kept the same as those of Fig. 3.

Figure 10

Comparison of the interface slip of surfactant-laden interfaces with different ${\text{Pe}}_s$ but with a fixed $\text{Ma}=1$. In each plot, the interface slip at low and high ${\text{Pe}}_s$ are denoted, respectively, by blue solid and blue dotted lines. Also plotted is the change in the surfactant concentration at high ${\text{Pe}}_s$ in comparison to that at low ${\text{Pe}}_s$ (red dash-dotted line). The dashed lines are just for reference. In (a) the low and high ${\text{Pe}}_s$ are 0.1 and 1, while in (b) they are 1 and 10, respectively. The blue (upper) and red (lower) arrows denote, respectively, the vector field of the surfactant-laden interface's slip at low ${\text{Pe}}_s$ and the direction of the Marangoni induced slip as ${\text{Pe}}_s$ is increased from its low to high value. The axis for the interface slip is on the left while that for the change in the surfactant concentration is on the right. The values of the other parameters $h$, $\lambda$, $a$, and $d$ are kept the same as those of Fig. 3.