Mutual interaction of a collapsing bubble and a nearby viscoelastic solid

The interaction between a microscale collapsing bubble and a nearby viscoelastic solid is investigated numerically. The bubble initiates rapid expansion with high initial pressure, and the stand-off parameter is set to be near 1.0. The viscoelastic properties of the solid are described using the Kelvin-Voigt model, with its elastic modulus $E$ and viscosity $\eta$, varying several orders of magnitude ($E=20–2000$ kPa and $\eta =2–1000 \mathrm{Pa}\mathrm{s}$). The influence of solid viscoelasticity on bubble behavior and solid deformation are analyzed throughout the entire life cycle of the bubble, from its initial expansion to the moment of liquid jet impingement. The Deborah number, which quantifies the relative timescales of solid deformation and bubble expansion, is employed to characterize the bubble–solid interaction. As the Deborah number increases from the order of ${10}^{-2}$ to ${10}^2$, the dynamics of the bubble converge toward those observed in the cases of a rigid solid; while the solid deformation displacement reduces to a value close to zero, the maximum expansion radius of the bubble increases by up to 8%, and the speed of the liquid jet decreases by approximately 80%. The temporal distribution of energy components within the fluid domain reveals that the maximum expansion radius of the bubble and the kinetic energy of the liquid jet are inversely related to the energy transferred from the fluid to the solid. The imbalance in pressure surrounding the contracting bubble and the narrow region of the liquid jet are responsible for the enhancement of the liquid jet speed for small solid viscosity. Upon the impingement of the liquid jet onto the solid surface, the width and depth of the crater formed by the jet become greater for large jet speed and small solid viscosity.

Figure 1

(a) Schematic of a bubble immersed in water over a viscoelastic solid. (b) Grid layout of axisymmetric two-dimensional fluid domain.

Figure 2

Grid convergence test results for (a) maximum radius $R_{\max }$ of the expanding bubble near a viscoelastic solid ($E=1000\mathrm{kPa}, \eta =10\mathrm{Pa}\mathrm{s}$) and (b) maximum downward displacement ${\delta }_{\max }$ of the viscoelastic solid during bubble expansion. (ci) Temporal change in pressure at solid center, $p_c$, for different grid sizes; the graph is zoomed in for the moments (cii) when the shock wave reaches the solid surface and (ciii) when the bubble expands above the surface. The number of grid points $N=69-788$.

Figure 3

(a) Comparison of bubble evolution near a rigid solid between experiment [43] and simulation. (b) Comparisons of bubble evolution near an elastic solid between experiments [51] (upper) and simulations (lower): (i) $E=124\mathrm{kPa}, \gamma =0.87$; (ii) $E=1040\mathrm{kPa}, \gamma =1.22$. Frame interval is 40 $\mu \mathrm{s}$, and frame width is 3.5 mm.

Figure 4

(a) Sequential profiles of solid surface at initial solid deformation stage: (i) purely elastic, $E=20\mathrm{kPa}$; (ii) $E=20\mathrm{kPa}, \eta =10\mathrm{Pa}\mathrm{s}$; (iii) $E=20\mathrm{kPa}, \eta =1000\mathrm{Pa}\mathrm{s}$. The profiles are captured every $t_{{\delta }_{\max }}/5$ up to the time corresponding to maximum deformation, $t=t_{{\delta }_{\max }}$. (b) Surface profiles normalized by maximum displacement ${\delta }_{\max }$ at the solid center and (c) bubble shapes at $t=t_{{\delta }_{\max }}$ for one elastic case and five viscoelastic cases. All cases have the same elastic modulus, $E=20\mathrm{kPa}$.

Figure 5

Temporal strain responses for (a) an elastic material and (b) a Kelvin–Voigt material under constant stress.

Figure 6

(a) Normalized maximum displacement ${\delta }_{\max ,VE}/{\delta }_{\max ,E}$ and (b) normalized time at maximum displacement, $t_{{\delta }_{\max ,VE}}/t_{{\delta }_{\max ,E}}$, with respect to the Deborah number De.

Figure 7

Graphical description of the special numerical simulation. A viscoelastic solid with $E=100\mathrm{kPa}$ and $\eta =2\mathrm{Pa}\mathrm{s}$ is switched to a rigid solid after it reaches the maximum downward deformation (vertical dashed line). $z_c$ is the $z$ coordinate of the solid surface center.

Figure 8

(a) Maximum expansion radius $R_{\max }$ and (b) time at maximum expansion, $t_{R_{\max }}$, of the bubble for different solid properties.

Figure 9

Dimensionless maximum expansion radius $R_{\max }/R_{\mathrm{ref}}$ of the bubble with respect to the dimensionless energy transferred to the solid domain, $E_{S,R_{\max }}/E_{P0}$, up to the maximum expansion phase. The dashed horizontal line denotes the value for the rigid solid case.

Figure 10

(a) Temporal change in the $z$ coordinate $z_c/h_0$ of the solid surface at $r=0$ for a purely elastic solid and five viscoelastic solids. All cases have the same elastic modulus, $E=20\mathrm{kPa}$. (b) Evolution of bubble shape during contraction for a purely elastic solid and two viscoelastic solids: (i) purely elastic, $E=20\mathrm{kPa}$; (ii) $E=20\mathrm{kPa}, \eta =2\mathrm{Pa}\mathrm{s}$; (iii) $E=20\mathrm{kPa}, \eta =1000\mathrm{Pa}\mathrm{s}$.

Figure 11

Formation of liquid jet in elastic solid cases (a)–(e), viscoelastic solid cases (g)–(k), and a rigid solid case (f). Black and grey denote the gas bubble and water areas, respectively. The number on the top right corner of each panel indicates the time at which each case is captured. See Supplemental Material movie 1 [84].

Figure 12

(a) Normalized liquid jet speed $U_{j,VE}/U_{j,E}$ and (b) normalized bubble collapse time $t_{\mathrm{collapse},VE}/t_{\mathrm{collapse},E}$ with respect to the Deborah number De.

Figure 13

Temporal change in pressure at (a) topmost point $p_{\mathrm{top}}/p_{\infty }$ and (b) bottommost point $p_{\mathrm{bot}}/p_{\infty }$ of the bubble for different solid properties. (c) Pressure imbalance between the top and bottom points, $(p_{\mathrm{top}}-p_{\mathrm{bot}})/p_{\infty }$. All cases have the same elastic modulus, $E=20\mathrm{kPa}$.

Figure 14

Temporal change in energy components normalized by initial potential energy $E_{P0}$ of the bubble for different solid properties: (a) purely elastic solid with $E=20\mathrm{kPa}$; (b) viscoelastic solid with $E=20\mathrm{kPa}$ and $\eta =10\mathrm{Pa}\mathrm{s}$; (c) rigid solid. Time $t$ along the $x$ axis is normalized by the collapse time $t_{\mathrm{collapse}}$.

Figure 15

(a) Relationship between fluid kinetic energy at bubble collapse time, $E_{K,\mathrm{collpase}}$, and total energy transferred to the solid up to collapse time, $E_{S,\mathrm{collapse}}$, for viscoelastic solids. (b) Relationship between liquid jet speed $U_j$ and $E_{K,\mathrm{collapse}}$. (c) Contours of velocity magnitude for viscoelastic cases at bubble collapse time: (i) $E=20\mathrm{kPa}, \eta =1000\mathrm{Pa}\mathrm{s}$; (ii) $E=20\mathrm{kPa}, \eta =10\mathrm{Pa}\mathrm{s}$; (iii) $E=20\mathrm{kPa}, \eta =2\mathrm{Pa}\mathrm{s}$. See Supplemental Material movie 2 for (c).

Figure 16

Sequential profiles of solid surface after liquid jet impact for (a)–(c) purely elastic solids and (d)–(i) viscoelastic solids: (a) $E=20\mathrm{kPa}$; (b) $E=100\mathrm{kPa}$; (c) $E=2000\mathrm{kPa}$; (d) $E=20\mathrm{kPa}, \eta =2\mathrm{Pa}\mathrm{s}$; (e) $E=100\mathrm{kPa}, \eta =2\mathrm{Pa}\mathrm{s}$; (f) $E=2000\mathrm{kPa}, \eta =2\mathrm{Pa}\mathrm{s}$; (g) $E=20\mathrm{kPa}, \eta =5\mathrm{Pa}\mathrm{s}$; (h) $E=20\mathrm{kPa}, \eta =100\mathrm{Pa}\mathrm{s}$; (i) $E=20\mathrm{kPa}, \eta =1000\mathrm{Pa}\mathrm{s}$. The profiles are captured every 5 $\mu \mathrm{s}$.

Figure 17

Comparison of pressure $p$, velocity $u$, and volume fraction $\alpha$ fields at $t=0.25$ in one-dimensional shock tube problem: Simulation results (black lines) and analytical solutions from Miller et al. [71] (red lines).

Figure 18

Comparison of temporal variations in normal strain between simulation results (black dashed lines) and analytical solution (red dashed lines) for five different viscosities.

Figure 19

(a) Maximum displacement ${\delta }_{\max }$ and (b) time at maximum displacement $t_{{\delta }_{\max }}$ for viscoelastic solids with different properties.

Figure 20

(a) Liquid jet speed $U_j$ and (b) bubble collapse time $t_{\mathrm{collapse}}$ for different solid properties. The dashed horizontal line denotes the value for the rigid solid case.