# Hydrodynamic interactions of two nearly touching Brownian spheres in a stiff potential: Effect of fluid inertia

## Abstract

The hydrodynamic interaction of two closely spaced micron-scale spheres undergoing Brownian motion was measured as a function of their separation. Each sphere was attached to the distal end of a different atomic force microscopy cantilever, placing each sphere in a stiff one-dimensional potential (0.08 Nm{sup −1}) with a high frequency of thermal oscillations (resonance at 4 kHz). As a result, the sphere’s inertial and restoring forces were significant when compared to the force due to viscous drag. We explored interparticle gap regions where there was overlap between the two Stokes layers surrounding each sphere. Our experimental measurements are the first of their kind in this parameter regime. The high frequency of oscillation of the spheres means that an analysis of the fluid dynamics would include the effects of fluid inertia, as described by the unsteady Stokes equation. However, we find that, for interparticle separations less than twice the thickness of the wake of the unsteady viscous boundary layer (the Stokes layer), the hydrodynamic interaction between the Brownian particles is well-approximated by analytical expressions that neglect the inertia of the fluid. This is because elevated frictional forces at narrow gaps dominate fluid inertial effects. The significance is that interparticle collisions andmore »

- Authors:

- Department of Chemical Engineering, Virginia Tech, Blacksburg, Virginia 24060 (United States)
- Department of Mechanical Engineering, Virginia Tech, Blacksburg, Virginia 24060 (United States)

- Publication Date:

- OSTI Identifier:
- 22403211

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physics of Fluids (1994); Journal Volume: 27; Journal Issue: 2; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 42 ENGINEERING; ATOMIC FORCE MICROSCOPY; BOUNDARY LAYERS; BROWNIAN MOVEMENT; COLLISIONS; DRAG; FLUID MECHANICS; FLUIDS; INTERACTIONS; MIXTURES; MOMENT OF INERTIA; OSCILLATIONS; PARTICLES; RESONANCE

### Citation Formats

```
Radiom, Milad, E-mail: milad.radiom@unige.ch, Ducker, William, E-mail: wducker@vt.edu, Robbins, Brian, and Paul, Mark.
```*Hydrodynamic interactions of two nearly touching Brownian spheres in a stiff potential: Effect of fluid inertia*. United States: N. p., 2015.
Web. doi:10.1063/1.4908295.

```
Radiom, Milad, E-mail: milad.radiom@unige.ch, Ducker, William, E-mail: wducker@vt.edu, Robbins, Brian, & Paul, Mark.
```*Hydrodynamic interactions of two nearly touching Brownian spheres in a stiff potential: Effect of fluid inertia*. United States. doi:10.1063/1.4908295.

```
Radiom, Milad, E-mail: milad.radiom@unige.ch, Ducker, William, E-mail: wducker@vt.edu, Robbins, Brian, and Paul, Mark. Sun .
"Hydrodynamic interactions of two nearly touching Brownian spheres in a stiff potential: Effect of fluid inertia". United States.
doi:10.1063/1.4908295.
```

```
@article{osti_22403211,
```

title = {Hydrodynamic interactions of two nearly touching Brownian spheres in a stiff potential: Effect of fluid inertia},

author = {Radiom, Milad, E-mail: milad.radiom@unige.ch and Ducker, William, E-mail: wducker@vt.edu and Robbins, Brian and Paul, Mark},

abstractNote = {The hydrodynamic interaction of two closely spaced micron-scale spheres undergoing Brownian motion was measured as a function of their separation. Each sphere was attached to the distal end of a different atomic force microscopy cantilever, placing each sphere in a stiff one-dimensional potential (0.08 Nm{sup −1}) with a high frequency of thermal oscillations (resonance at 4 kHz). As a result, the sphere’s inertial and restoring forces were significant when compared to the force due to viscous drag. We explored interparticle gap regions where there was overlap between the two Stokes layers surrounding each sphere. Our experimental measurements are the first of their kind in this parameter regime. The high frequency of oscillation of the spheres means that an analysis of the fluid dynamics would include the effects of fluid inertia, as described by the unsteady Stokes equation. However, we find that, for interparticle separations less than twice the thickness of the wake of the unsteady viscous boundary layer (the Stokes layer), the hydrodynamic interaction between the Brownian particles is well-approximated by analytical expressions that neglect the inertia of the fluid. This is because elevated frictional forces at narrow gaps dominate fluid inertial effects. The significance is that interparticle collisions and concentrated suspensions at this condition can be modeled without the need to incorporate fluid inertia. We suggest a way to predict when fluid inertial effects can be ignored by including the gap-width dependence into the frequency number. We also show that low frequency number analysis can be used to determine the microrheology of mixtures at interfaces.},

doi = {10.1063/1.4908295},

journal = {Physics of Fluids (1994)},

number = 2,

volume = 27,

place = {United States},

year = {Sun Feb 15 00:00:00 EST 2015},

month = {Sun Feb 15 00:00:00 EST 2015}

}