3. Flow structure within the boundary layer

We now look at the flow structure within the near-wall boundary layer. Figure 2 $(a)$ schematically illustrates the relevant parameters, including the outer-edge velocity $U_e$ , the tangential flow velocity $u_{r}$ , the flow velocity at the wall $u_{t}$ , the boundary-layer thickness $\delta$ , the linear near-wall region of thickness $\varepsilon$ and the slip length $l$ . For a no-slip boundary, the tangential velocity increases from zero at the wall ( $u_t=0$ ), initially varying linearly with wall-normal distance before approaching the outer-edge velocity $U_e$ through a nonlinear profile. In contrast, under a slip boundary condition the velocity at the wall is finite and satisfies the Navier relation:

(3.1) \begin{align} u_{t}=l\bigg (\frac {\partial u_{r}}{\partial n}\bigg )_{y=0} , \end{align}

where $u_r$ denotes the tangential velocity and $n$ is the wall-normal unit vector.

Figure 2(b) compares the computed near-wall $u_r$ profiles for three cases with $l$ = 0, 400 and 800 nm at $\gamma =0.83$ . Each profile is extracted at the time $t$ and radial location $r$ where the maximum shear rate occurs for that case. The results show that the outer velocity $U_e$ increases with $l$ , reflecting the reduced wall friction. Correspondingly, the wall velocity $u_r$ rises from zero in the no-slip case to approximately 35 and 55 m s⁻¹ for $l=400$ and 800 nm, respectively. The increases in both $U_e$ and $u_{t}$ are nonlinear with respect to $l$ .

Figure 2.

(a) Schematic of the tangential velocity $u_r$ profile near a slip wall. (b) Simulated $u_r$ profiles along the wall-normal coordinate $y$ for slip lengths $l=$ 0, 400 and 800 nm for $\gamma =0.83$ . In all cases, the near-wall region exhibits a linear variation of the radial velocity $u_r$ with $y$ , consistent with a locally constant shear rate, but with slopes that depend on $l$ . The horizontal dashed line at $y=0.2\,\unicode{x03BC} \text{m}$ indicates the upper bound of the linear region common to all cases.

In all cases, the tangential velocity $u_r$ varies linearly with the wall-normal coordinate within $y \lt 0.2\,\unicode{x03BC} \text{m}$ , consistent with the Navier slip boundary condition. The extrapolated dashed lines intersect the vertical axis $y$ at the virtual origin defined by the slip length. The associated shear rate decreases from 9.5 $\,\times\, 10^7\,\text{s}^{-1}$ for the no-slip wall to 8.5 $\,\times\, 10^7\,\text{s}^{-1}$ for $l=400\,$ nm and to 6.9 $\,\times\, 10^7\,\text{s}^{-1}$ for $l=800\,$ nm, demonstrating a substantial reduction in wall shear stress with increasing slip length.

4. Wall shear stress

Within the linear near-wall region ( $y\leqslant \varepsilon$ ), $u_r$ varies approximately linearly with $y$ . The Navier slip relation (3.1) then gives the wall shear stress at $y=0$ as

(4.1) \begin{align} \boldsymbol{\tau } ={{\left . \mu \frac {\partial {{\boldsymbol{u}}_{r}}}{\partial y} \right |}_{y=0}}\approx {{\left . \mu \frac {{{\boldsymbol{u}}_{r}}(y)}{y +l} \right |}_{y\leqslant \varepsilon }} . \end{align}

In practice, $\tau$ is evaluated at $y = 0.1\,\unicode{x03BC} \text{m}$ , well within the linear viscous sublayer ( $y\lt 0.2\,\unicode{x03BC}$ m) identified in figure 2.

Figure 3.

Flow structure and wall shear stress $\tau$ at $\gamma =1.03$ for selected times before and after jet impact. (a) Liquid velocity for $l=0$ . (b) Liquid velocity for $l=4\,\unicode{x03BC} \mathrm{m}$ , with the bubble contours for $l=0$ superimposed in blue. (c) Wall shear stress distributions along the boundary for corresponding time. Blue dashed and red solid lines denote the result of $l=0$ and $l=4\,\unicode{x03BC} \mathrm{m}$ , respectively. Arrows indicate flow direction, and colour denotes velocity magnitude. The maximum outward stresses are approximately 100 kPa for $l=0$ and 20 kPa for $l=4\,\unicode{x03BC}$ m (see (c) $t=109\,\unicode{x03BC}$ s). Therefore, increasing slip length from zero to $l=4\,\unicode{x03BC}$ m leads to a virtually indistinguishable change to the bubble shape, whereas wall shear decreases by a factor of 5.

Figure 3. Long description

The image contains three panels labeled (a), (b), and (c), each depicting different aspects of the flow structure and wall shear stress at selected times before and after jet impact. Panel A: Three heat maps show liquid velocity at different times (105 microseconds, 107 microseconds, and 109 microseconds). The color gradient represents velocity magnitude, with darker colors indicating higher velocities. The red lines outline the bubble contours, and arrows indicate flow direction. Panel B: Three heat maps similar to Panel A but with an additional set of blue dashed lines superimposed, representing bubble contours for a different condition. Panel C: Three line graphs display wall shear stress distributions along the boundary for corresponding times. The x-axis represents the radial distance (r) in micrometers, and the y-axis represents the wall shear stress (τ) in kilopascals. Blue dashed lines and red solid lines denote the results for two different conditions, with labels l = 0 and l = 4 micrometers, respectively. The graphs show how wall shear stress changes over time and distance, with notable peaks and troughs indicating areas of high and low stress.

We next examine how slip length modifies the spreading flow following microjet impact. Figure 3 shows the key stages of jet formation, impact and radial spreading for two representative cases $l=0$ and $l=4\,\unicode{x03BC}$ m, at the same stand-off distance $\gamma =1.03$ . The simulations indicate that the overall bubble morphology and jet profile remain nearly indistinguishable between the two cases until the jet has propagated a finite radial distance at $t=109\,\unicode{x03BC}$ s. At $t=109\,\unicode{x03BC}$ s, although the toroidal bubble shapes are globally similar, the $l=4\,\unicode{x03BC}$ m case exhibits a further radial spread of the near-wall flow, accompanied by enhanced stretching of the toroidal bubble and increased daughter-bubble formation.

While the slip length $l$ only marginally changes dynamics at the bubble scale, the resulting wall shear stress is profoundly affected. During jet formation ( $t=105\,\unicode{x03BC}$ s), bubble shrinkage generates an inward shear stress of approximately −1.5 kPa under no-slip conditions, which decreases to about −0.5 kPa for $l=4\,\unicode{x03BC}$ m. At the early spreading stage ( $t=107\,\unicode{x03BC}$ s), the no-slip case produces a peak shear stress of approximately 70 kPa at $r\approx 70\,\unicode{x03BC}$ m, together with a secondary hump near $r\approx 100\,\unicode{x03BC}$ m. For $l=4\,\unicode{x03BC}$ m, the peak shifts outward to $r\approx 95\,\unicode{x03BC}$ m and decreases to 15 kPa, while the secondary hump disappears. As spreading continues and the jet is narrowed by the toroidal bubble ( $t=109\,\unicode{x03BC}$ s), the no-slip boundary yields a maximum outward shear stress of 100 kPa together with an inward stress of approximately −20 kPa caused by the adverse pressure gradient and local flow reversal. In contrast, for $l=4\,\unicode{x03BC}$ m the maximum $\tau$ drops to around 20 kPa and the reversed-flow-induced inward stress disappears.

Figure 4.

Spatio-temporal evolution of $\tau$ along the boundary at $\gamma =1.03$ for (a) $l=0$ and (b) $l=4\,\unicode{x03BC}$ m. Green and blue regions indicate outward and inward shear stresses, respectively, with stress magnitude shown on a logarithmic scale. The solid black line shows the bubble equivalent radius for the corresponding case, and the dashed line that of the comparison case. (c) Time evolution of the maximum outward and inward $\tau$ over the entire boundary.

Figure 4 further illustrates the spatio-temporal distribution of $\tau$ for the two cases $l=0$ and $l=4\,\unicode{x03BC}$ m. Each panel spans a time range of $0{-}112\,\unicode{x03BC} \text{s}$ (horizontal direction) and a radial range of $0{-}700\,\unicode{x03BC} \text{m}$ (vertical direction). A base-10 logarithmic scale is employed to capture the wide dynamic range of $\tau$ , with green and blue regions denoting outward (positive) and inward (negative) shear stresses, respectively. The nearly indistinguishable superimposed $R(t)$ curves confirm that the slip length has a negligible influence on the temporal evolution of the bubble equivalent radius. This insensitivity reflects the fact that bubble oscillation is governed by the bulk pressure field, whereas slip modifies only the thin near-wall shear layer.

As $l$ increases to $4\,\unicode{x03BC}$ m, reduced wall friction strongly modifies the shear rate, leading to a restructured spatio-temporal stress distribution and significantly lower magnitudes.

The first prominent effect is the shift in both location and magnitude of the peak outward shear stress. For $l=0$ , the maximum outward stress is $102\,\text{kPa}$ , occurring at $r = 116.2\,\unicode{x03BC} \text{m}$ and $t = 109.1\,\unicode{x03BC} \text{s}$ . When the slip length increases to $l=4\,\unicode{x03BC}$ m, this peak decreases to $28\,\text{kPa}$ and shifts outward to $r = 167.8\,\unicode{x03BC} \text{m}$ at $t = 110\,\unicode{x03BC} \text{s}$ . The inward shear stress is likewise strongly attenuated. For $l=0$ , the peak inward stress reaches −35 kPa within the flow separation region at $r = 164.8\,\unicode{x03BC} \text{m}$ and $t=108.8\,\unicode{x03BC} \text{s}$ following jet spreading. In contrast, for $l=4\,\unicode{x03BC}$ m, the peak inward stress is reduced to only −0.5 kPa and shifts to an earlier bubble contraction stage at $r=290.6\,\unicode{x03BC} \text{m}$ and $t=106.3\,\unicode{x03BC} \text{s}$ , indicating the suppression of separation-driven inward shear during jet spreading.

A second notable feature is the enlarged spatial extent of positive (outward) shear stress in the slip case. For $l=0$ , inward stress develops before maximum bubble expansion, as the far-field radial flow reverses once the bubble pressure falls below the ambient pressure $p_\infty$ . With finite slip, reduced friction delays flow deceleration and reversal, so inward stress emerges later in the oscillation cycle. The positive and negative stress regions are separated by a stagnation ring (white contour). In the no-slip case, this ring migrates gradually towards the origin. For $l=\,4\,\unicode{x03BC}$ m, however, its inward motion is interrupted by the rapidly spreading jet, causing it to stabilise near $r\approx \,100\,\unicode{x03BC}$ m and thereby maintain positive stress in the inner region $r\lt 100\,\unicode{x03BC}$ m.

Finally, after jet impact ( $t\gt 106\,\unicode{x03BC}$ s), the rapid radial spreading drives the shear stress from a few pascals to several tens of kilopascals in both cases. For $l=4\,\unicode{x03BC}$ m, the separation region ahead of the spreading jet vanishes, leaving the wall shear stress positive throughout the domain.

Figure 4(c) details the maximum positive and negative $\tau$ along the boundary at different time instants, namely $\tau _{pm}$ and $\tau _{nm}$ . The results show that increasing $l$ strongly suppresses both extrema. For $\tau _{pm}$ , the peak value decreases from 102 kPa for $l=0$ to only 9 kPa for $l=16\,\unicode{x03BC}$ m. For $\tau _{nm}$ , the peak negative stress due to post-spreading flow separation, initially of −35 kPa for $l=0$ , decreases progressively with $l$ . When $l\gt 2\,\unicode{x03BC}$ m, flow separation is fully suppressed and the inward stress after jet spreading vanishes.

5. Maximum stress distribution

We now study the maximum outward stress $\tau _m$ over the parameter space $(l,\gamma )$ for $10\,\text{nm}\leqslant l\leqslant \,16\,\unicode{x03BC} \text{m}$ and $0.63\leqslant \gamma \leqslant 1.53$ . Figure 5(a) shows the distribution of $\tau _m$ . In general, increasing slip length leads to a monotonic reduction in $\tau _m$ . However, the distribution exhibits non-trivial structure due to the influence of the stand-off distance $\gamma$ , which governs the jet profile and hence the local outer-edge velocity $U_e$ . To rationalise these coupled effects, we next derive a scaling law for $\tau _m$ .

Figure 5.

(a) Maximum outward wall shear stress on the $(l,\gamma )$ parameter plane. (b) Inverse normalised maximum wall shear stress $1/\tau _{m}^*$ , plotted against the composite parameter $k+l^*$ . Symbols denote various stand-off distances $\gamma$ spanning from 0.63 to 1.53. The dashed line represents the linear relation $1/\tau _{m}^*=k+l^*$ .

Figure 5. Long description

The image contains two graphs analyzing wall shear stress. Panel A: A contour plot shows the maximum outward wall shear stress on a parameter plane. The x-axis represents the parameter l in nanometers, ranging from 10 to 16000 nanometers. The y-axis represents the parameter gamma, ranging from 0.7 to 1.5. The color gradient indicates the magnitude of the maximum wall shear stress in kilopascals, with darker reds representing higher stress values. Panel B: A scatter plot shows the inverse normalized maximum wall shear stress plotted against the composite parameter k plus l star. The x-axis is labeled k plus l star and ranges from 10^0 to 10^1. The y-axis is labeled 1 over tau m star and ranges from 10^0 to 10^1. Different symbols denote various stand-off distances, ranging from 0.63 to 1.53. The dashed line represents the linear relation.

To begin, we define the Reynolds number as ${Re}={{\rho }_{l}}{{U}_{0}}{{R}_{max }}/\mu$ , where ${{U}_{0}}=\sqrt {{{p}_{\infty }}/{{\rho }_{l}}}$ is the characteristic velocity of Rayleigh collapse of a spherical bubble (Brennen Reference Brennen2014), with ${p}_{\infty }$ the ambient pressure and ${\rho }_{l}$ the liquid density. The following scaling analysis is performed at the spatio-temporal location where $\tau _m$ occurs. According to classical boundary-layer theory (Schlichting & Gersten Reference Schlichting and Gersten2016; Zeng et al. Reference Zeng, An and Ohl2022), the boundary-layer thickness $\delta$ satisfies

(5.1) \begin{align} \delta \sim \sqrt {\frac {\mu }{{{\rho }_{l}}}\frac {1}{\partial {{U}_{e}}/\partial r}} . \end{align}

Since the characteristic variation length of the outer velocity scales with $R_{\textit{max}}$ , i.e. $\partial {{U}_{e}}/\partial r\propto U_e/R_{\textit{max}}$ , we obtain

(5.2) \begin{align} \delta \sim \sqrt {\frac {\mu R_{\textit{max}}}{{{\rho }_{l}}{{U}_{e}}}}=\sqrt {\frac {\mu }{{{\rho }_{l}}{{U}_{0}}R_{\textit{max}}}{R_{\textit{max}}^2}\frac {{{U}_{0}}}{{{U}_{e}}}}={{{Re}}^{-0.5}}R_{\textit{max}}{{\left ( \frac {{{U}_{0}}}{{{U}_{e}}} \right )}^{0.5}} . \end{align}

We therefore introduce a characteristic length scale $\lambda = {Re}^{-0.5}R_{\textit{max}}(U_0/U_e)^{0.5}$ , of the order of $\delta$ . The ratio $U_0/U_e$ physically compares the characteristic velocity of the collapsing bubble with the velocity at the outer edge of the boundary layer. By looking into the constant-shear-region geometry in figure 2(a), we expect the maximum wall shear stress to follow

(5.3) \begin{align} \tau _m \approx \mu \frac {U_e}{l+k\lambda }, \quad k\lt 1, \end{align}

where $k$ accounts for the effective velocity-gradient correction within the boundary layer. We introduce the normalised slip length:

(5.4) \begin{align} {{l}^{*}}=\frac {l}{\lambda }=\frac {l}{R_{\textit{max}}}{{{Re}}^{0.5}}{{\left ( \frac {{{U}_{e}}}{{{U}_{0}}} \right )}^{0.5}} , \end{align}

and normalise $\tau _m$ using the classical order of shear stress $\mu U_e/\lambda$ :

(5.5) \begin{align} \tau _{m}^{*}=\frac {{{\tau }_{m}}}{\mu {{U}_{e}}/\lambda }=\frac {{{\tau }_{m}}R_{\textit{max}}}{\mu {{U}_{0}}{{{Re}}^{0.5}}{{\left ( {{U}_{e}}/{{U}_{0}} \right )}^{1.5}}} . \end{align}

Substituting the proposed stress relation (5.3) into the definition of $\tau _{m}^*$ yields

(5.6) \begin{align} \tau _{m}^{*}=\frac {{{\tau }_{m}}}{\mu {{U}_{e}}/\lambda }\approx \mu \frac {{{U}_{e}}}{l+k\lambda }/\left ( \mu {{U}_{e}}/\lambda \right )=\frac {\lambda }{l+k\lambda }=\frac {1}{k+{{l}^{*}}} . \end{align}

Figure 5(b) validates this model by plotting $1/\tau _{m}^*$ against $k+l^*$ . All data points spanning multiple $\gamma$ and $l$ collapse onto a single straight line for $k=0.3$ , which demonstrates $1/\tau _{m}^* \approx k+l^*$ , supporting the anticipated stress approximation using (5.3).

The scaling relation (5.6) is first validated by substituting the simulation-extracted values of $U_e$ into (5.4) and (5.5). Next we develop a semi-empirical expression for $U_e$ using the more accessible parameters $\gamma$ and the liquid-film thickness $h_{\textit{jet}}$ between the bubble bottom and the wall. As shown in figures 3 and 4, the slip length $l$ has negligible influence on the macroscopic bubble morphology and jet structure. Therefore, once the microjet penetrates the bubble bottom, both the jet velocity $v_{\textit{jet}}$ and the liquid-film thickness $h_{\textit{jet}}$ are primarily determined by $\gamma$ .

Given the macroscopic bubble morphologies are insensitive to slip length, one can follow the empirical scaling established previously (Zeng et al. Reference Zeng, Zhang, Tan, An and Ohl2024) for which the jet velocity scales as $v_{\textit{jet}}\propto \gamma ^{0.5}$ , with the prefactor depending on the ambient and geometric conditions. For the present dataset, this gives

(5.7) \begin{align} v_{\textit{jet}}=83.88\,\gamma ^{0.5}\;(\mathrm{m\,s^{-1}}) . \end{align}

Meanwhile, the liquid-film thickness follows

(5.8) \begin{align} h_{\textit{jet}} = \begin{cases} 29.2\,\gamma ^{4.86}+4.74\;(\unicode{x03BC} \mathrm{m}),\quad \mathrm{for}\, \gamma \leqslant 1.05,\quad \\[4pt] 832.8\,\gamma -867.9\;(\unicode{x03BC} \mathrm{m}),\quad \mathrm{for}\, \gamma \gt 1.05, \end{cases} \end{align}

consistent with previous measurements and simulations (Reuter & Kaiser Reference Reuter and Kaiser2019; Zeng et al. Reference Zeng, An and Ohl2022). These relations indicate that $v_{\textit{jet}}$ and $h_{\textit{jet}}$ are not independent empirical inputs requiring separate measurements, but are governed primarily by the stand-off distance $\gamma$ , the key control parameter for jetting bubbles near rigid boundaries. The prefactors are set by the global physical conditions, such as ambient pressure, liquid density and wall geometry, which are known a priori in experiments and practical applications. Therefore, once $\gamma$ and the global conditions are specified, both $v_{\textit{jet}}$ and $h_{\textit{jet}}$ can be estimated directly, allowing $U_e$ and hence the maximum wall shear stress to be predicted without introducing additional free fitting parameters.

For the spreading flow, the relevant velocity and length scales are $v_{\textit{jet}}$ and $h_{\textit{jet}}$ . Since boundary slip enhances the near-wall transport velocity, we decompose the normalised outer-edge velocity as

(5.9) \begin{align} \frac {U_e}{v_{\textit{jet}}} \sim \left .{\left (\frac {U_e}{v_{\textit{jet}}}\right )}\right |_{l=0} + f(\gamma ,l/h_{\textit{jet}}), \end{align}

where the first term denotes the no-slip contribution and the second term represents the slip-induced enhancement. Because $\gamma$ controls the jet profile and $l/h_{\textit{jet}}$ characterises the relative slip strength, we assume

(5.10) \begin{align} f(\gamma ,l/h_{\textit{jet}}) \sim \gamma ^m (l/h_{\textit{jet}})^n. \end{align}

Figure 6.

(a) Normalised outer-edge velocity $U_e/v_{\textit{jet}}$ as a function of $\eta =\gamma l/h_{\textit{jet}}$ . The red solid line represents $U_e/v_{\textit{jet}} = 0.925 + 0.8\,\eta ^{0.62}$ and the blue one represents $U_e/v_{\textit{jet}} = 0.463 + 0.4\,\eta ^{0.62}$ . (b) Collapse of all scattered data onto a solid line, $U_e/(k(\gamma )v_{\textit{jet}}) = 1.16 + \eta ^{0.62}$ , by rescaling with $k(\gamma )$ . Symbols for different $\gamma$ follow the same marker and colour scheme as in (a). The insets depict jet velocity and liquid-film thickness as functions of $\gamma$ .

Figure 6(a) plots $U_e/v_{\textit{jet}}$ against $\eta =\gamma l/h_{\textit{jet}}$ for all 16 stand-off distances and slip lengths ranging from 10 nm to 16 $\unicode{x03BC}$ m. The data for different $\gamma$ exhibit a similar dependence on $\eta$ , suggesting $m=n=0.62$ in (5.10). Within the range of $\gamma$ considered here, three regimes of $U_e/v_{\textit{jet}}$ can be identified. In the first regime, $\gamma \leqslant 0.98$ , the data collapse onto the red solid line:

(5.11) \begin{align} U_e/v_{\textit{jet}}=0.925+0.8\,\eta ^{0.62}. \end{align}

The second regime, $0.98\lt \gamma \lt 1.28$ , shows a non-trivial variation: $U_e/v_{\textit{jet}}$ first increases, exceeding the values in the first regime and then decreases with increasing $\gamma$ . In the third regime, $1.28\leqslant \gamma \leqslant 1.53$ , the data collapse onto the blue solid line:

(5.12) \begin{align} U_e/v_{\textit{jet}}=0.463+0.4\,\eta ^{0.62}. \end{align}

This regime-dependent behaviour indicates that, although the slip-induced growth follows a common exponent, the baseline and prefactor of the outer-edge velocity are controlled by the stand-off-dependent jet-spreading dynamics.

Figure 7.

Bubble morphology and velocity fields immediately following jet impact for representative stand-off distances: (a) $\gamma$ = 0.93, (b) $\gamma$ = 1.03, (c) $\gamma$ = 1.13, (d) $\gamma$ = 1.23 and (e) $\gamma$ = 1.33. The images have a common width of 150 $\unicode{x03BC} \mathrm{m}$ , while the heights are 100, 125, 175, 225 and 325 $\unicode{x03BC} \mathrm{m}$ for (a–e), respectively. Arrows indicate the flow direction, colour denotes the velocity magnitude and blue contours mark the bubble interface. The bottom of each box is the substrate $y=0$ .

Figure 7. Long description

Panel A: A heat map showing bubble morphology and velocity fields for a stand-off distance of 0.93. The width of the image is 150 micrometers, and the height is 100 micrometers. Arrows indicate the flow direction, color denotes the velocity magnitude, and blue contours mark the bubble interface. The bottom of the box is the substrate. Panel B: A heat map showing bubble morphology and velocity fields for a stand-off distance of 1.03. The width of the image is 150 micrometers, and the height is 125 micrometers. Arrows indicate the flow direction, color denotes the velocity magnitude, and blue contours mark the bubble interface. The bottom of the box is the substrate. Panel C: A heat map showing bubble morphology and velocity fields for a stand-off distance of 1.13. The width of the image is 150 micrometers, and the height is 175 micrometers. Arrows indicate the flow direction, color denotes the velocity magnitude, and blue contours mark the bubble interface. The bottom of the box is the substrate. Panel D: A heat map showing bubble morphology and velocity fields for a stand-off distance of 1.23. The width of the image is 150 micrometers, and the height is 225 micrometers. Arrows indicate the flow direction, color denotes the velocity magnitude, and blue contours mark the bubble interface. The bottom of the box is the substrate. Panel E: A heat map showing bubble morphology and velocity fields for a stand-off distance of 1.33. The width of the image is 150 micrometers, and the height is 325 micrometers. Arrows indicate the flow direction, color denotes the velocity magnitude, and blue contours mark the bubble interface. The bottom of the box is the substrate.

Next, we explain the distinct distributions of $U_e/v_{\textit{jet}}$ by examining the jet-impact dynamics for representative stand-off distances, as shown in figure 7. For relatively small stand-off distances, e.g. $\gamma =0.93$ , the thin liquid film constrains the downward motion of the inner rim of the toroidal bubble, and no pronounced secondary splitting occurs. In the intermediate regime, represented by $\gamma =1.03$ , 1.13 and 1.23, splitting of the inner toroidal rim becomes significant. Depending on $\gamma$ , one or two small toroidal bubbles are shed and subsequently move towards the wall, further narrowing the near-wall flow and modifying $U_e/v_{\textit{jet}}$ . For larger stand-off distances, e.g. $\gamma =1.33$ , the liquid film is sufficiently thick that the shed toroidal bubble remains too small and too distant from the wall to significantly influence the formation of $U_e/v_{\textit{jet}}$ .

The dataset in figure 6(a) yields the following relations:

(5.13) \begin{align} \frac {U_e}{v_{\textit{jet}}} &= 0.925 + 0.8\,\eta ^{0.62}, \qquad 0.63 \leqslant \gamma \leqslant 0.98, \\[-12pt] \nonumber \end{align}

(5.14) \begin{align} \frac {U_e}{v_{\textit{jet}}} &= \big(1.26-0.08\gamma ^{9.84} \big) \big(1.16+\,\eta ^{0.62} \big), \qquad 0.98 \lt \gamma \lt 1.28, \\[-12pt] \nonumber \end{align}

(5.15) \begin{align} \frac {U_e}{v_{\textit{jet}}} &= 0.463 + 0.4\,\eta ^{0.62}, \qquad 1.28 \leqslant \gamma \leqslant 1.53. \\[10pt] \nonumber \end{align}

These three regimes can be unified through a stand-off-dependent prefactor $k(\gamma )$ , as shown in figure 6(b):

(5.16) \begin{align} k(\gamma ) = \begin{cases} 0.8, & 0.63 \leqslant \gamma \leqslant 0.98 , \\[4pt]1.26 - 0.08 \gamma ^{9.84}, & 0.98 \lt \gamma \lt 1.28 , \\[4pt]0.4, & 1.28 \leqslant \gamma \leqslant 1.53 , \end{cases} \end{align}

satisfying $U_e/(k(\gamma )v_{\textit{jet}}) = 1.16 + \eta ^{0.62}$ . In (5.16), the non-monotonic intermediate regime originates from secondary toroidal-bubble shedding after jet penetration. Accordingly, the outer-edge velocity $U_e$ can be expressed in the unified form

(5.17) \begin{align} U_e \approx v_{\textit{jet}}{k(\gamma )}\left [1.16 + \left (\frac {\gamma l}{h_{\textit{jet}}}\right )^{0.62}\right ] . \end{align}

Figure 8.

Comparison between the simulated maximum wall shear stress $\tau _m$ and the semi-empirical prediction $\tau _{m,pr}$ . Symbols denote different stand-off distances $\gamma$ . The dashed line indicates the linear scaling $\tau _{m}=\tau _{m,pr}$ . Opacity from dark to light indicates increasing $l$ . The light-green shaded region marks $0.75\tau _{m,pr}\leqslant \tau _{m}\leqslant 1.5\tau _{m,pr}$ .

Incorporating the characteristic length $\lambda$ and the established coefficient $k=0.3$ into (5.3), the predicted maximum wall shear stress $\tau _{m,pr}$ is introduced as

(5.18) \begin{align} \tau _{m,pr} = \mu \frac {U_e}{l+0.3\lambda } , \end{align}

where the outer-edge velocity $U_e$ is evaluated using (5.17) based on the empirical scalings for $v_{\textit{jet}}$ from (5.7) and $h_{\textit{jet}}$ from (5.8) which both only depend on $\gamma$ .

Figure 8 compares the simulated stress $\tau _m$ against the semi-empirical prediction $\tau _{m,pr}$ . The data follow the one-to-one trend over more than one order of magnitude in $\tau _m$ , confirming that the scaling captures the dominant dependence on slip length and stand-off distance. Quantitative deviations remain, but most data fall within the shaded uncertainty band $0.75\tau _{m,pr}\leqslant \tau _m \leqslant 1.5\tau _{m,pr}$ , indicating that the semi-empirical model captures the maximum stress within approximately $-25\,\%$ to $+50\,\%$ over the present parameter range. These deviations are expected because the prediction uses a reduced description of the outer-edge velocity and does not resolve the detailed bubble deformation, toroidal-bubble shedding and local flow rearrangement during jet impact and spreading. The agreement is better in the low-stress regime, particularly for $\tau _m \lesssim 40\,\mathrm{kPa}$ , whereas larger deviations occur mainly in the high-stress regime, where the near-wall flow is most sensitive to the complex post-impact bubble morphology. Thus, the model should be interpreted as a semi-empirical scaling framework rather than an exact pointwise prediction, with an explicit approximation for the maximum stress as

(5.19) \begin{align} \tau _m \approx \frac {\mu U_{0} {Re}^{0.5}(U_e/U_{0})^{1.5}}{0.3R_{\textit{max}}+l(U_e {Re}/U_{0})^{0.5}}, \end{align}

with $U_e$ predicted by (5.17) with the empirical scalings for $v_{\textit{jet}}$ and $h_{\textit{jet}}$ (5.7) and (5.8), which only depend on $\gamma$ . This provides a semi-empirical scaling framework linking the maximum wall shear stress $\tau _m$ to the two more accessible parameters, $\gamma$ and $l$ .

Although the present study considers a canonical single-bubble configuration, the mechanism is relevant to cavitating flows over microstructured, coated, superhydrophobic and soft surfaces. In many such situations, the characteristic size of the cavitation bubble is much larger than the surface texture, coating thickness or trapped-gas structure, so that the detailed surface geometry may be represented, to leading order, by an effective slip length. This reduced description is relevant to cavitation-assisted cleaning of microstructured electronic or optical surfaces, where near-wall shear controls particle removal, and to coated or soft materials, where excessive cavitation-induced shear may promote delamination, scouring or damage. The present framework therefore isolates a local fluid-mechanical mechanism by which modified interfacial momentum transfer regulates the intense transient shear generated by cavitation-driven wall-jet spreading.