Abstract:

This dataset contains analytical expressions for the: (i) mobility functions of two surfactant covered drops where one drop is much smaller than the other; and, (ii) velocity of a micro-organism (modeled as a Stokes dipole) outside a stationary clean and surfactant-laden drop. This dataset supports the manuscript: Shaik, V.A. & Ardekani, A.M. 2017. Point force singularities outside a drop covered with an incompressible surfactant: Image systems and their applications. Phys. Rev. Fluids, 2(11): 113606. doi: 10.1103/PhysRevFluids.2.113606 As the analytical expressions are too complicated to deduce the dependence of the aforementioned quantities on the viscosity ratio or the interfacial viscosity of the drops, we provide the data (by evaluating the analytical expressions) for: (a) increase in the speed of a small surfactant-laden drop (due to an external force acting on it) near a large force-free surfactant-laden drop; (b) velocity of a spherical micro-swimmer that is normal to the line joining the center of the swimmer and the surfactant-laden drop; and, (c) angular velocity of a spherical micro-swimmer outside a surfactant-laden-drop.

Suggested Citation:

Vaseem A. Shaik, Arezoo M. Ardekani. 2018. Dataset for: Point force singularities outside a drop covered with an incompressible surfactant: Image systems and their applications. Distributed by: GRIIDC, Harte Research Institute, Texas A&M University–Corpus Christi. doi:10.7266/N75M645B

Publications:

Shaik, V. A., & Ardekani, A. M. (2017). Point force singularities outside a drop covered with an incompressible surfactant: Image systems and their applications. Physical Review Fluids, 2(11). doi:10.1103/physrevfluids.2.113606

Purpose:

One part of the data was generated to derive an expression for the mobility matrix for two drops of arbitrary sizes covered with an incompressible surfactant. The mobility matrix helps determine the velocity of the two drops, if the force acting on them (e.g., due to buoyancy) is prescribed. A second part of the data was generated to calculate the velocity of a model micro-swimmer (a Stokes dipole) outside a drop covered with an incompressible surfactant. This velocity of the micro-swimmer can be used to analyse its motion in the vicinity of the drop.

Data Parameters and Units:

All the analytical expressions are dimensional. If a specific unit system, (for instance SI) is used for the parameters (such as length or viscosity ratio), the velocities are obtained in that specific unit system. However, the data used for plotting (i.e., the numerical data being provided) is non-dimensional. The non-dimensionalization scheme is summarized below: (a) The increase in the speed of a small surfactant-laden drop outside a large force-free surfactant-laden drop is non-dimensionalized with the increase in the speed of a similar small surfactant-laden drop outside a large rigid sphere. We report the variation of this quantity with: (i) the viscosity ratio of the large surfactant-laden drop, \lambda_1 = \mu_{i,1}/\mu_e, where \mu_{i,1} and \mu_e denote the viscosity inside and outside a large surfactant-laden-drop; and, (ii) the dimensionless interfacial viscosity of the large surfactant-laden drop, \beta_1 = \mu_{s,1}/(\mu_e*a, where \mu_{s,1} and a are the interfacial viscosity and the radius of the large surfactant-laden drop, respectively. (b) The translational velocity of a spherical micro-swimmer that is normal to the line joining the centers of the swimmer and the drop is non-dimensionalized using the velocity scale P/(8*\pi*\mu_e*R^2). Here 'P' is the dipole strength of the microorganism, 'R' is the distance between the centers of the swimmer and the drop. (c) The angular velocity of a spherical micro-swimmer outside a drop is non-dimensionalized using P * sin(2*\theta) / (8*\pi*\mu_e*R^3). Here '\theta' is the orientation of the swimmer as measured with respect to the normal to the line joining the centers. We report the variation of the translational and angular velocity of a micro-swimmer outside a surfactant-laden drop with the viscosity ratio (\lambda) and the dimensionless interfacial viscosity of the drop (\beta).

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