Point force singularities outside a drop covered with an incompressible surfactant: Image systems and their applications

In this work we derive the image flow fields for point force singularities placed outside a stationary drop covered with an insoluble, nondiffusing, and incompressible surfactant. We assume the interface to be Newtonian and use the Boussinesq-Scriven constitutive law for the interfacial stress tensor. We use this analytical solution to investigate two different problems. First, we derive the mobility matrix for two drops of arbitrary sizes covered with an incompressible surfactant. In the second example, we calculate the velocity of a swimming microorganism (modeled as a Stokes dipole) outside a drop covered with an incompressible surfactant.

Figure 1

Point force outside a surfactant covered drop and the associated coordinate system. All the distances are scaled by the drop radius.

Figure 2

Schematic used for deriving the mobility functions $\left(y_{12}^a,y_{22}^a\right)$ for the transverse motion of the drops. Here an external force ${\mathbf{F}}_2^e$ is acting on the small drop, while no external force is applied on the large drop. Note that unlike the analysis in Secs. 2 and 3, all the distances, in this section, are dimensional (i.e., not scaled by the radius of the drop).

Figure 3

Variation of $\left(6\pi b{y}_{22}^a-1\right)/\left(6\pi b{y}_{22}^a-1\right){|}_R$ with the viscosity ratio ${\lambda }_1$ and the interfacial viscosity ${\beta }_1$ of the large drop. Different symbols represent the distinct characteristics of the large drop. Here $★$ $(*)$ denotes the situation of the large drop being a clean drop without any interfacial viscosity (rigid sphere). The other symbols denote the situation of the large drop being a surfactant covered drop with an interfacial viscosity as mentioned in the figure. Also, $x=a/R=0.8,\delta =b/a=0.1$.

Figure 4

Schematic of a spheroidal swimmer of aspect ratio $\gamma =b/c$ outside a stationary spherical drop of radius $a$. The drop has a viscosity of ${\mu }_i$, while the viscosity of the surrounding fluid is ${\mu }_e$. The swimmer is oriented along $\mathbf{f}$. The distance between the center of the drop and the swimmer is denoted by $R$. The position vectors of the center of the drop and the swimmer are denoted by ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$, respectively. The origin of the coordinate system is located at the center of the drop. Note that unlike the analysis in Secs. 2 and 3, all the distances, in this section, are dimensional (i.e., not scaled by the radius of the drop).

Figure 5

Variation of (a) the dimensionless velocity of the swimmer that is normal to the line of centers and (b) dimensionless angular velocity of the spherical swimmer with the viscosity ratio $\lambda$ and dimensionless interfacial viscosity $\beta$ of the drop. Here $★$ $(*)$ denotes the situation of the swimmer lying outside a clean drop without any interfacial viscosity (rigid sphere). The other symbols denote the situation of swimmer moving outside the surfactant-laden drop.