Cracks and fingers: Dynamics of ductile fracture in an aqueous foam

Gas-Liquid Foam Dynamics: From Structural Elements to Continuum Descriptions

“…In the anomalous regime, finger width λ is expected to decay in a universal way with an increase in the parameter 1/B, proportional to the product of the capillary number and the square of the Hele-Shaw cell aspect ratio W/b (Rabaud et al 1988). While this decay is confirmed by measurements over a range of parameters, Lindner et al (2002) found that λ saturates at finite 1/B in non-Newtonian fluids, attributed to a breakdown of the 2D Hele-Shaw modeling because the curvature scale of the finger becomes comparable to b. Analogously, experimental data and simulations of foam fingering (Hilgenfeldt et al 2008, Stewart & Hilgenfeldt 2017 have found a saturated finger width as the tip radius of curvature becomes comparable to the size of individual bubbles, again invalidating lower-dimensional approximations. Thus, the discreteness of foam structure provides automatic imperfections inducing anomalous fingering but simultaneously limits the thinning of the fingers.…”

“…Famously, the morphology of the finger is set by a singular perturbation that for low surface tension selects (out of a family of finger shapes) the single solution with λ = 1/2 (McLean & Saffman 1981). By contrast, flowing foam exhibits fingers with Saffman-Taylor shape but with widths of λ = 1/2 in both experiments (Hilgenfeldt et al 2008) (Figure 4d) and network model simulations (Stewart & Hilgenfeldt 2017) (Figure 4e (2005). Panel d adapted with permission from Stewart et al (2015).…”

“…In some flow configurations the yield stress can be neglected entirely (Ben Salem et al 2013b, Stewart & Hilgenfeldt 2017.…”

“…Panel g adapted with permission from Rabaud et al (1988). Panel h adapted with permission from Stewart & Hilgenfeldt (2017). Panel i adapted with permission from Ben Salem et al (2013b).…”

“…Thus, the discreteness of foam structure provides automatic imperfections inducing anomalous fingering but simultaneously limits the thinning of the fingers. Empirical data on the plasticity around a foam finger show a well-defined pattern of T1 transitions localized around the tip (Arif 2010, Stewart & Hilgenfeldt 2017, and experiments observe irregular propagation of the finger with tip-splitting events as the driving pressure crosses a critical threshold (Arif 2010), where strain from previous T1 events cannot be fully relieved and accumulates to alter subsequent neighbor exchanges (Stewart & Hilgenfeldt 2017). Figure 4h shows how cracks for P < P c closely follow the Saffman-Taylor family of anomalous finger shapes (Rabaud et al 1988), while crack tips for P > P c exhibit a blunted morphology.…”