Fig. 2: Mechanism of hydraulic fracturing.
From: Patterning and dynamics of membrane adhesion under hydraulic stress
a Schematic of the physical ingredients controlling the formation of a pattern of hydraulic cracks. Mechanical force balance (left) and osmolyte and water transport (right) allow us to solve for hydraulic and osmotic pressures inside the vesicle Pi(t) and Πi(t) and in the interstice P(x, y, t) and Π(x, y, t), for the membrane velocity v(x, y, t), for bare membrane tension, for bond concentration and for the vesicle shape given by R(t), θ(t) (left inset) and height z(x, y, t). The right inset shows the non-monotonic relation between bond traction Tbonds and z, along with the effective adhesion potential U(z). Other symbols are described in the text and Supplementary Note 2. b Snapshots of various fields at the adhesion patch and 3D view of the membrane. The blue map is permeation velocity, proportional to the water chemical potential across the membrane, P(x, y, t) − Π(x, y, t) − (Pi − Πi). The green map is the osmotic pressure relative to that of the external medium Π(x, y, t) − Πe. The red map is the hydraulic pressure relative to the vesicle pressure, P(x, y, t) − Pi, whose gradient is proportional to Darcy flow. The purple map is the membrane tension, σ − kBTc, and the gray map is the bond concentration. c Zoom of water transport in the interstice, with water permeation (color map) and Darcy flow (arrows); pockets are outlined in red color. Model parameters are reported in Supplementary Table 1.
