Displacement flows are common in hydraulic fracturing, as fracking fluids of different composition are injected sequentially in the fracture. The injection of an immiscible fluid at the centre of a liquid-filled fracture results in the growth of the fracture and the outward displacement of the interface between the two liquids. We study the dynamics of the fluid-driven fracture, which is controlled by the competition between viscous, elastic and toughness-related stresses. We use a model experiment to characterize the dynamics of the fracture for a range of mechanical properties of the fractured material and fracturing fluids. We form the liquid-filled pre-fracture in an elastic brittle matrix of gelatin. The displacing liquid is then injected. We record the radius and aperture of the fracture, and the position of the interface between the two liquids. In a typical experiment, the axisymmetric radial viscous flow is accommodated by the elastic deformation and fracturing of the matrix. We model the coupling between elastic deformation, viscous dissipation and fracture propagation, and recover the two fracturing regimes identified for single-fluid injection. For the viscous-dominated and toughness-dominated regimes, we derive scaling equations that describe the crack growth due to a displacement flow and show the influence of the pre-existing fracture on the crack dynamics through a finite initial volume and an average viscosity of the fluids in the fracture.

Biological filtration systems offer a sustainable alternative to existing engineered solutions. In this computational work, we seek to optimize the surface coverage by an array of hairs to capture particles in channels. A variety of aquatic organisms rely on arrays of hairs to interact with their fluidic environments. The hair functionality can vary from sensing to smelling, filtration to flow control depending on the species considered. Among those organisms are filter-feeders that rely on suspension-feeding, one of the most widespread feeding mechanisms and one of the oldest. Baleen whales are filter-feeders that catch their prey by using the baleen, a complex structure composed of plates and bristles in their mouth. The hairs are hollow cylindrical structures with a diameter of a few hundred micrometers that can extend over tens of centimeters. The baleen filters out the prey while letting the seawater through. The baleen is composed of flexible and elongated structures whose properties fit the feeding habits of the whale. The porosity of the structure depends on the flow feature. Effectively, the flow can tune the filter properties, which sets biological filters apart from their engineered counterpart. Previous mechanical studies have shown that an array of hairs can either act as a sieve, allowing all the fluid to flow through it, or as a rake, forcing the fluid to flow around it instead. As the speed increases, the behavior shifts from rake to sieving for a given hair spacing. From a filtration perspective, the rake regime is not favorable as particles do not enter the array. For a fixed fluid velocity, the flow transitions from rake to sieve as the spacing between the hairs in the array increases. Our recent work has also demonstrated that the confinement of the channel influences the sieve to rake transition. The filtration mechanisms that filter-feeder organisms use to capture food particles exhibit complex fluid-structure interactions that have yet to be leveraged in engineered systems. To guide the development of hair-covered surfaces capable of trapping particles in channel flows, we investigate how different geometric factors affect the fluid transport and capture of particles by the array. In previous work, a small number of hairs, typically 25, were considered. Here, we vary the array geometry, the Reynolds number of the flow, and the surface coverage to study the transport through this confined porous structure. We compare arrays based on their optimal efficiency and the (sub-optimal) operating conditions which make the filter versatile.

During hydraulic fracturing, the injection of a pressurized fluid in a brittle elastic medium leads to the formation and growth of fluid-filled fractures. A disc-like or penny-shaped fracture grows radially from a point source during the injection of a viscous fluid at a constant flow rate. We report an experimental study on the dynamics of fractures propagating in the viscous regime. We measure the fracture aperture and radius over time for varying mechanical properties of the medium and fluid and different injection parameters. Our experiments show that the fracture continues to expand in an impermeable brittle matrix, even after the injection stops. In the viscous regime, the fracture radius scales as t 4 / 9 during the injection. Post-shut-in, the crack continues to propagate at a slower rate, which agrees well with the predictions of the scaling arguments, as the radius scales as t 1 / 9 . The fracture finally reaches an equilibrium set by the toughness of the material. The results provide insights into the propagation of hydraulic fractures in rocks.