Electrochemical wall shear rate microscopy of collapsing bubbles

An electrochemical high-speed wall shear raster microscope is presented. It involves chronoamperometric measurements on a microelectrode that is flush-mounted in a submerged test specimen. Wall shear rates are derived from the measured microelectrode signal by numerically solving a convection-diffusion equation with an optimization approach. This way, the unsteady wall shear rates from the collapse of a laser pulse seeded cavitation bubble close to a substrate are measured. By planar scanning, they are resolved in high spatial resolution. The wall shear rates are related to the bubble dynamics via synchronized high-speed imaging of the bubble shape.

Figure 1

Experimental setup.

Figure 2

Illustration of convection source term modeling. The squares to the left represent bulk liquid, the rectangles above the electrode characterize the diffusion layer.

Figure 3

Time series of a single bubble $\left(\gamma \cong 1\right)$ collapsing at the substrate with the flush-mounted electrochemical sensor. The substrate extends horizontally at the bottom of the image (see the dashed line along the boundary in the first frame). The substrate surface can be also identified by the mirror image of the bubble at the bottom of the frames. Normalized times $\tau$ are given in the left top of each frame (the time of first collapse is $\tau =1.00$ corresponding to $T_{\mathrm{L}}=88\mu \mathrm{s})$. The (equivalent) bubble radius at maximum volume reaches a value of $R_{\max }=425\mu \mathrm{m}$ (measured at $\tau =0.47)$.

Figure 4

Time series of a single bubble that collapses in vicinity to a substrate as in Fig. 3, but now imaged in bottom view through a transparent substrate $\left(R_{\max }=475\mu \mathrm{m},T_{\mathrm{L}}=97.5\mu \mathrm{s}\right)$. Normalized times $\tau$ are indicated in the top left corner of each frame.

Figure 5

Electrochemical wall shear microscopy of a single bubble collapsing in the vicinity of a rigid boundary $(R_{\max }=425\mu \mathrm{m}$; $\gamma \cong 1)$. Wall shear rates are height-coded over the substrate that extends in the x/y plane. To present the entire time evolution in one figure, we use the same height scale for all frames but adjust the color scale to each frame individually. The respective values of the maximum wall shear rates (corresponding to the darkest colors in each frame) are indicated in the second row of the top left corner of each frame (the brightest coding always corresponds to $0{\mathrm{s}}^{-1})$. Normalized times are given in the upper row of each top left corner of each frame. The bubble is generated at $\tau =0;$ the time series presentation starts at $\tau =0.36\left(=32\mu \mathrm{s}\right)$ and ends at $\tau =2.26\left(=200\mu \mathrm{s}\right)$. Note the different time intervals between the frames. The corresponding time series of the bubble shape is presented in Figs. 3 and 4. For comparison, $R_{\max }$ is depicted (see the second frame); note that the initial contact radius is much smaller than $R_{\max }$. For the dynamics of the entire time series at high resolution, see Video A of the Supplemental Material [57].

Figure 6

Kinetic energy in the near-wall liquid layer in percentage of the initial bubble energy.

Figure 7

Maximum wall shear rate $G_{\max }\left(x,y\right)={max}_t\left[G\left(x,y,t\right)\right]$ occurring during the bubble collapse in the measured region. The values are shown both height coded and in planar projection.

Figure 8

Time of occurrence of maximum wall shear rate. The substrate surface can be divided into four sections, A–D (according to the time of maximum wall shear rate). The $\tau$ scale is clipped at 1.6 for best contrast.

Figure 9

Lagrangian ink map of a collapsing bubble, illustrating the liquid displacement a long time after the collision of flows (at $\tau =10.7)$. Perspective as in Fig. 3 with the boundary extending horizontally at $z=0$. The circle indicates the bubble shape at maximum expansion. Liquid from the boundary layer (dark blue) is ejected and colliding flows have detached and formed a ring vortex. Experimental data obtained by particle imaging velocimetry; for details, see Ref. [56].

Figure 10

Left: Comparison between measured current density at the electrode and simulated one during the convection event produced by the cavitation bubble. Right: Respective calculated shear rates.