next up previous
Next: Pair diffusion near a Up: Hydrodynamic Interactions Previous: Normal mode diffusivity

Hydrodynamic coupling to a wall

The flow field around a moving sphere spans an unbounded system. A wall's no-flow boundary condition modifies this flow, breaking its symmetry and increasing the drag on the sphere. Faxén introduced the method of reflections in 1927 to address this question and obtained the diffusivity for a sphere at height $ h$ above a wall moving parallel ($ \Vert$) to the surface:

$\displaystyle D_\Vert(h) = D_0 \, \left[ 1 - \frac{9}{16} \, \frac{a}{h} + \fra...
...\, \frac{a^5}{h^5} + \ensuremath{{\cal O} \left(\frac{a^6}{h^6}\right)}\right].$ (35)

The method of reflections is challenging even for this configuration and essentially unworkable for more complex systems.

Figure 6: Left: The hydrodynamic image of a stokeslet in a flat surface is a combination of an oppositely directed stokeslet, a stokes doublet and a source doublet. Right: The in-plane diffusivity of a 1  $ \mathrm{\mu m}$-diameter silica sphere is suppressed by proximity to a surface. The solid curve shows the stokeslet prediction (Eq. (39)) while the essentially indistinguishable dashed curve is Faxén's prediction (Eq. (35)).
\includegraphics[height=2.25in]{figures/one_stks} \includegraphics[height=2in]{figures/faxen}

As early as 1906, Lorentz reported the Green's function for flow near a flat surface. Seventy years later, Blake [62] recognized that Lorentz's result could be reinterpreted by analogy to electrostatics. He suggested that the flow due to a stokeslet could be canceled on a bounding surface by conceptually placing its hydrodynamic image on the opposite side. Solutions to the Stokes equations being unique, the resulting flow must be Lorentz's Green's function for bounded flow.

The electrostatic image needed to cancel a charge distribution's field on a surface is just an appropriately scaled mirror image of the initial source. In hydrodynamics, the image of a stokeslet is not simply another stokeslet, but rather a more complicated construction including sources which Blake dubbed a stokeslet doublet (D) and a source doublet (SD). This combination is depicted schematically in Fig. 6. The flow due to the entire image system is described by the Green's function [62]

$\displaystyle \ensuremath{\mathsf{G}}\xspace ^W_\ensuremath{{\alpha\beta}}(\vec...
...nsuremath{\mathsf{G}}\xspace ^{SD}_\ensuremath{{\alpha\beta}}(\vec r - \vec R),$ (36)

where $ \vec R = \vec r - 2h \hat z$ is the position of the image and

$\displaystyle \ensuremath{\mathsf{G}}\xspace ^D_\ensuremath{{\alpha\beta}}(\vec x)$ $\displaystyle = \frac{1 - 2 \delta_{\beta z}}{8\pi\eta} \, \frac{\partial}{\partial x_\beta} \, \left( \frac{x_\alpha}{x^3} \right)$   and (37)
$\displaystyle \ensuremath{\mathsf{G}}\xspace ^{SD}_\ensuremath{{\alpha\beta}}(\vec x)$ $\displaystyle = (1 - 2 \delta_{\beta z}) \, \frac{\partial}{\partial x_\beta} \, \ensuremath{\mathsf{G}}\xspace ^S_\ensuremath{{\alpha\beta}}(\vec x)$ (38)

are Green's functions for a source doublet and a stokeslet doublet, respectively.

Applying Faxén's first law and identifying $ \ensuremath{\mathsf{b}}\xspace ^e = \ensuremath{\mathsf{G}}\xspace ^W$ leads to

$\displaystyle D^S_\Vert(h) = D_0 \, \left[1 - \frac{9}{16} \, \frac{a}{h} + \ensuremath{{\cal O} \left(\frac{a^3}{h^3}\right)}\right],$ (39)

where the $ S$ superscript distinguishes this result from Eq. (35). As before, we are left wondering about the higher-order terms in Faxén's more specialized analysis.

Figure 6 shows typical data obtained with optical tweezers and digital video microscopy for a silica sphere's diffusion above a wall. The particular sphere for this data set was one of the pair studied in the previous section. The sphere's height $ h$ above the wall was repeatedly reset by the optical tweezer at $ \tau = 83~\mathrm{msec}$ intervals. During this period of free motion, it could diffuse out of plane only $ \Delta z = \sqrt{2 D_0 \tau} = 0.3~\ensuremath{\mathrm{\mu m}}\xspace $, on average. Advancing the microscope's focus in $ 1.0 \pm 0.3~\ensuremath{\mathrm{\mu m}}\xspace $ steps allowed us to sample the dynamics' dependence on $ h$. The measured height-dependent diffusivity agrees well with both Eqs. (35) and (39).

next up previous
Next: Pair diffusion near a Up: Hydrodynamic Interactions Previous: Normal mode diffusivity
David G. Grier 2001-01-16