Direct Measurement of Shear-Induced Cross-Correlations of Brownian Motion

Shear-induced cross-correlations of particle fluctuations perpendicular and along streamlines are investigated experimentally and theoretically. Direct measurements of the Brownian motion of micron-sized beads, held by optical tweezers in a shear-flow cell, show a strong time asymmetry in the cross-correlation, which is caused by the non-normal amplification of fluctuations. Complementary measurements on the single particle probability distribution substantiate this behavior and both results are consistent with a Langevin model. In addition, a shear-induced anticorrelation between orthogonal random displacements of two trapped and hydrodynamically interacting particles is detected, having one or two extrema in time, depending on the positions of the particles.

Figure 1

A cell cross section (lithographic mask) is shown with $150\mu \mathrm{m}$ depth in the $z$ direction having opposite flow directions in its upper and lower channel and a linear shear profile at its center: see inset with PIV data ($\Delta$) and linear fit. One or two particles at a distance $d=4\mu \mathrm{m}$ were held by optical tweezers in the center of the linear velocity profile $\mathbf{u}(y)$.

Figure 2

Shear-induced cross-correlations between orthogonal random displacements: $⟨x(t)y(0)⟩$ (triangles) and $⟨x(0)y(t)⟩$ (squares). Lines are fits according to Eq. (6, 7) and open circles represent $⟨x(0)y(t)⟩$ in the absence of flow.

Figure 3

The particle’s distribution in the shear plane is shown in a) without flow and for a shear flow with $\dot{\gamma }=49{\mathrm{s}}^{-1}$ in b). The angle between the major and the $x$ axis is $\varphi \approx 38°$ and the ratio between the two principal axes is $R\approx 0.75$.

Figure 4

Correlations $⟨x_1(t)y_2(0)⟩$ (triangles) and $⟨x_1(0)y_2(t)⟩$ (squares) between random displacements of two particles. Colored lines are fits according to Eq. (11, 12). Circles represent the same correlations in the absence of flow.