Fig. 1: Visualizing hydrodynamic modes and spatiotemporal evolution of pairs.

a, Schematic showing the body-frame, \(({L},\,{T})\) − axes, for a colloid-pair used to reveal hydrodynamic correlations. b, Colormap of \(P(\Delta {{{{{{{{\bf{r}}}}}}}}}_{i}({{{{{{{{\bf{r}}}}}}}}}^{{\prime}},\; t) | {{{{{{{{\bf{r}}}}}}}}}_{j}={{{{{{{{\bf{r}}}}}}}}}^{{\prime}}+{{{{{{{\bf{r}}}}}}}})\) and \(P(\Delta {{{{{{{{\bf{r}}}}}}}}}_{j}({{{{{{{{\bf{r}}}}}}}}}^{{\prime} }+{{{{{{{\bf{r}}}}}}}},\; t) | {{{{{{{{\bf{r}}}}}}}}}_{i}={{{{{{{{\bf{r}}}}}}}}}^{{\prime} })\) for particles in pairs separated by \(r=2.0 \sigma\). Conditional \(P(\Delta {{{\bf{r}}}}(t))\) measured for the particle on the right of the pair when the particle on the left displaces by \(\geq 1.0\sigma\) c, along \({L}\) and \(r=2.0 \sigma\) d, along \({T}\) and \(r=2.5\sigma\) and e, along \({T}\) and \(r=1.1\sigma\), depicted by yellow arrows. Solid red and white circles represent the mean positions of the particles at \(t_0\) and \((t_0+t)\), respectively. The displacement color map for b−e are normalized by the maximum displacements in each case. f, Polar colormap, \({{{\bf{r}}}}(r,\, \theta)\), of displacement field when the colloid at origin (solid red circle) moves towards the right (open red circle). The dashed radial circles represent \(r=\{2,\, 3, \,4\}\sigma\). Representative red arrows, with their head and length represent the direction and strength (shown also in the background), respectively, of the field. The measurements for b−f were performed at \(\phi=0.15\) and \(t=0.5 \, {{{\rm{s}}}}\). g, \(H_L\) and \(H_T\) versus \({r}\) for quasi-2D experimental cell. The inset shows \(H_L\) and \(H_T\) versus \({r}\) for open (3D) experimental cell. h, \(Z_{rel}\) versus r at \(\phi=0.61\). The inset shows typical schematics depicting configurations corresponding to the peak position in \(Z_{rel}\). i, “Most-probable” schematic construction of spatiotemporal evolution of a pair of particles due to near-field hydrodynamics. Note, the relative magnitudes and directions of the yellow arrows correspond to expectation for each configuration.