Session J01: Focus Session: Fluid Dynamics in a Deformable Porous Medium (8:00am - 8:45am CST)

Boundary conditions on a hydrogel-fluid interface

Hydrogels are an important class of soft material for biological, biomedical and Micro-Electro-Mechanical Systems (MEMS) applications. Hydrogels are often deployed alongside fluids, thus the interfacial dynamics of a gel-fluid system becomes an interesting question. Given the wide range in length scales, from the nanometer pore size in the gel to the dimension of a MEMS device (millimeters or more), it seems appropriate to model the hydrogel as a two-phase mixture of a (deformable) skeleton and a liquid that permeates the gel. In such a poroelastic framework, the boundary conditions on the gel-fluid interface are extremely tedious to derive from first principles of coarse-graining, and must be postulated. In this talk, we describe an energy dissipation formalism that suggests two sets of boundary conditions. We will compare the flow profiles predicted by these conditions with those of published models.   

Viscous Transport in Eroding Porous Media

In groundwater flow, erosion leads to a deforming porous media. Erosion creates conduits or channels of high porosity, and these are directly related to the transport properties of particulates and concentrations through the eroded geometry. This talk will use high-order simulations of both erosion and transport to analyze the flow through eroded porous media.   

Cell nucleus as a microrheological probe to study the rheology of the cytoskeleton

Mechanical properties of the cell are important biomarkers for probing its architectural changes caused by cellular processes and/or pathologies. Recent advancements in microfluidic and high-speed imaging technologies have enabled probing the cell cortex deformations in microchannels, while also tracking different intracellular components in high-throughputs. Most previous studies of cell mechanics using microchannels only measure the cell stiffness, and do not disentangle the rheology of different cellular components, including the cortex, the cytoplasm and the nucleus. We present a novel method that utilizes the correlation between the cortical deformations that are induced by external microfluidic flows, and the nucleus displacements, induced by those cortical deformations, to decouple the cell cortex and the cytoplasm mechanics. As a proof of concept, we consider a rigid spherical nucleus centered in a spherical cell. We obtain analytical expressions for the time-dependent nucleus velocity vs the cell deformations, when the interior cytoplasm is modeled as a viscous, viscoelastic, porous and poroelastic material, and demonstrate how the nucleus velocity can be used to characterize the linear rheology of the cytoplasm over a wide range of forces and timescales/frequencies.   

Growth of an isotropic cluster of activated fractures in the presence of stress anisotropy

Hydraulic stimulation of low permeability rocks via hydroshearing of pre-existing fractures is widely used in the oil/gas and geothermal industries. Optimization of this process requires deep understanding of the interplay between fluid transport and permeability changes due to the growth of a cluster of activated fractures. The effects of network connectivity, pressure-driven flow of the fracturing fluid, and stress anisotropy on the cluster growth are analyzed. To delineate the effects of the network's connectivity, pre-existing fractures are modeled as line segments of uniform length that are randomly oriented. To capture the effects of fluid transport, we use discrete network simulations of the stimulation process. We show that the effects of stress anisotropy, which tend to produce a quasi-one-dimensional network, can be mitigated by fracturing in such a way that the viscous pressure drop required to drive the injected fluid becomes important over a length scale that is comparable with the pre-existing fractures' correlation length. This result provides a criterion to increase the ramification of the induced hydraulically conductive flow paths by optimal design of the fracturing process.   

A phase-field model for capillary bulldozing

The invasion of non-wetting gas into a horizontal, liquid-filled tube or Hele-Shaw cell is a classical problem in fluid mechanics that has been studied extensively from a variety of perspectives. However, the addition of a sedimented granular material to the defending liquid phase can fundamentally change the mechanics of the problem by introducing friction, leading to a class of ``multiphase frictional flows'' that remain relatively poorly understood. For example, recent experiments [Dumazer et al., 2016, PRL] show that, in a capillary tube, the motion of the gas-liquid interface will bulldoze the granular material, accumulating a pile of grains on the liquid side of the interface that will grow until it forms a plug and clogs the tube. Here, we present a thermodynamically consistent phase-field model for capillary bulldozing. The model involves three phases -- gas, liquid, and liquid-solid mixture -- and takes the form of a coupled pair of nonlinear conservation laws and a linear elliptic equation for the velocity of liquid-solid mixture. We solve our model numerically for a variety of different scenarios to develop insight into the roles of sliding friction, rearrangement, capillarity, viscosity, and plug formation during capillary bulldozing.   

Catch and release of bubbles in a soft granular medium

A liquid-saturated packing of soft particles can behave like a complex fluid or like a porous solid, depending on the confining stress. In the fluid-like state, a gas bubble can rise through the packing due to buoyancy with a rise velocity that decreases as the solid fraction increases. In the solid-like state, gas bubbles cannot rise unless their buoyancy overcomes the failure stress of the packing or the capillary entry pressure of the pore space. Here, we use laboratory experiments, discrete-element simulations, and theoretical modelling to study these two states and the transition between them. Specifically, we study the rise velocity of bubbles as a function of bubble size and solid fraction in order to identify the solid fraction at which bubbles are immobilised. We also study the poromechanics of changing the solid fraction by compressing the packing with a fluid-permeable piston. We combine these ingredients to show that active manipulation of the solid fraction can therefore be used for the on-demand catch and release of gas bubbles.   

Large-Deformation Poroelasticity with Friction

Fluid impinging on a soft porous material may cause the material to deform, which alters the behavior of the fluid flow in turn. Flows exhibiting this so-called poroelastic coupling are not generally found in isolation; frictional contacts between the solid phase and any confining boundaries strongly influence the nature of the coupling. For the tractable geometry of uniaxial flow through a cylindrical porous medium, we develop both analytic and numerical models to study the effects of wall friction on the properties of large-deformation poroelastic flows in steady state. The presence of friction leads to a measurable hysteresis in the volume flux, solid deformation, and pore pressure gradient of the flow. Using data from a mechanically-compressed latex foam to parameterize the frictional input to our theoretical models, we present direct quantitative comparisons between model predictions and experimental measurements of volume flux, deformation, and pressure of the foam subject to flow-driven compression.   

Relaxation of a fluid-filled blister on a porous substrate

The relaxation dynamics of a fluid-filled blister between an elastic sheet and a porous substrate are controlled by the deformation of the elastic sheet, the viscous stresses in the pores, and the capillary pressure at the fluid-air interface. We present experiments where fluid is injected between a porous substrate and an elastic sheet. First, fluid invades the pores and subsequently, the elastic sheet is peeled and uplifted from the substrate resulting in a fluid-filled blister. Further injection causes both the fluid front in the pores and the fracture front of the blister to propagate radially. After injection is stopped, the fluid continues to advance into the pores as the elastic stresses in the overlying sheet drive drainage of the blister. We conduct experiments and develop a mathematical model to study the effects of blister size, permeability of the porous substrate, and bending stiffness of the elastic sheet on the relaxation dynamics. We find that the bending stiffness of the elastic sheet and the permeability of the substrate largely control the dynamics, but for thin sheets and low permeability substrates, the capillary pressure at the fluid-air interface significantly affects the dynamics.   

Law of mass action (LMA) with energetic variational approaches (EnVarA) with applications

h $-abstract-$\backslash $pard In this talk, we will present a derivation to generalize the mass-action kinetics of chemical reactions using an energetic variational approach. Our general framework involves the energy dissipation law for a chemical reaction system, which carries all the information about the dynamics. The dynamics of the system is determined through the choice of the free energy, the dissipation (the entropy production), as well as the kinematics (conservation of species). The method enables us to capture the coupling and competition of various mechanisms, including mechanical effects such as diffusion, viscoelasticity in polymeric fluids and muscle contraction, as well as the thermal effects. We will also discuss several applications under this approach, in particular, the modeling of wormlike micellar solutions. This is joint work with Bob Eisenberg, Pei Liu, Yiwei Wang, and Tengfei Zhang.$\backslash $pard-/abstract-$\backslash $\tex   

Variational Lagrangian Scheme for the Porous Medium Equation and Phase-field models: A Discrete Energetic Variational Approach

In this talk, we present a systematic framework of deriving variational schemes for generalized diffusions and gradient flows, by a discrete energetic variational approach, which performs an energetic variational approach (EnVarA) at a semi-discrete level. The resulting semi-discrete equation Inherits the variational structures from the continuous energy-dissipation law directly. In particular, we apply such an approach to construct variational Lagrangian schemes to the porous medium type generalized diffusion and the Allen-Cahn type phase-field models. Numerical examples show the advantages of our schemes in capturing singularities, thin diffuse interfaces, and free boundaries.~ This is joint work with Professor Chun Liu (IIT).   

Dynamics of swimmers in fluids with resistance

Micro-swimmers such as spermatozoa are able to efficiently navigate through viscous fluids that contain a sparse network of fibers or other macromolecules. We utilize the Brinkman equation to capture the fluid dynamics of sparse and stationary obstacles that are represented via a single resistance parameter. The method of regularized Brinkmanlets is utilized to solve for the fluid flow and motion of the swimmer in 2-dimensions when assuming the flagellum (tail) propagates a curvature wave. For a single swimmer, we determine that increased swimming speed occurs for smaller cell body radius and smaller fluid resistance. Progression of swimmers exhibits complex dynamics when considering hydrodynamic interactions; attraction of two swimmers is a robust phenomenon for smaller beat amplitude of the tail and smaller fluid resistance. Wall attraction is also observed, with a longer time scale of wall attraction with larger resistance parameter.   

A poro-elasto-visco-plastic model of the dewatering of a two-phase suspension

A poro-elasto-visco-plastic model for the consolidation of a two-phase suspension is presented, motivated by the compaction and dewatering of wood-fibre pulp. For that material, traditional two-phase models of particulate porous media based upon plastic yielding of the particle network prove insufficient to capture the observed dynamics. The incorporation of viscous effects stemming from the compaction of the wood-fibre-network assists the model in reproducing experimental dewatering tests at moderate rates of compaction. However, during more rapid dewatering there is clear emergence of an elastic behaviour in the wood-fibre network. We present a poro-elasto-visco-plastic extension of the model, its calibration for wood-pulp using quasi-static cycles of loading and unloading, and demonstrate its improved representation of the rapid dewatering experiments.   

Fluid-structure interactions in a soft-walled Hele-Shaw cell

The interaction of viscous and interfacial flows with soft materials has recently attracted substantial interest from a variety of different perspectives. Here, we study these interactions in the context of a model problem: Flow in a deformable Hele-Shaw cell, where one wall is rigid and the other is soft. Combining experiments with mathematical modelling, we consider the coupling of flow and deformation as viscous fluid is injected into an initially empty cell. We then discuss the implications of these results for hydrodynamic instabilities such as viscous fingering.   

Viscous backflows from elastic reservoirs

The dynamics of fluid removal from an elastic reservoir is related to the practice of waste water migration following hydraulic fracturing, surface morphology control during geotechnical engineering and skin fluid collection for medical use. We report a series of reduced-order modelling studies and scaling results for the viscous backflow process from elastic reservoirs, including that from a hydraulic fracture and from beneath a stretched membrane or a bending plate. The backflow is generated once the exit of the elastic cavities is exposed to the atmosphere, or a fluid bath with a lower pressure. A couple of first-order ODEs are derived to describe the time evolution of the length and height of the fracture/cavity. The rate of fluid removal can also be estimated accordingly. The flow dynamics is found to depend on three nonlinear competing terms that correspond to the process of fluid propagation within a fracture or an elastic cavity, the viscous flow in the exiting channel, and the induced flow from the low pressure at the exit.   

Deformation-driven solute transport in soft porous media

Solute transport plays an important role in many soft porous materials, including the movement of contaminants in soils and the movement of nutrients and waste in living tissues and tissue-engineering scaffolds. These systems are also often exposed to large, periodic loading and deformation, which drives nontrivial fluid motion and changes in pore structure. Here, we study the strong coupling between fluid flow and mechanical stimulation during periodic deformations using a 1D continuum model based on large-deformation poroelasticity. We show that these reversible deformations lead to non-reversible spreading and mixing, even in a homogeneous medium. We analyse the three primary mechanisms of solute transport (advection, molecular diffusion, and mechanical dispersion) and study their separate impacts on the solute distribution. We also identify the key dimensionless parameters that govern deformation-driven transport, and we study their qualitative and quantitative impacts on solute spreading and mixing.   

Gas-liquid phase separation in a soft porous medium

Various biological and chemical processes can lead to the nucleation and growth of gas bubbles within the pore space of an otherwise liquid-saturated granular medium, such as in lake beds and waste ponds. The gas is typically non-wetting and, as the bubbles approach the pore size, it is energetically costly for them to invade narrow pore throats. If the solid skeleton is sufficiently soft, it is favourable for the bubbles to displace the solid grains and form macroscopic cavities. Here, we consider this process through the lens of phase separation, where thermomechanics govern the separation of a gas phase from a gas-liquid-solid mixture. We construct a phase-field model informed by large-deformation poromechanics, in which two immiscible fluids interact with a poroelastic solid skeleton. Our model captures the competing effects of elasticity and gas-liquid-solid interactions. As a model problem, we consider an initial distribution of gas in the pore space that separates into multiple gas cavities, which then coarsen over time as the smaller cavities collapse and the larger cavities expand. We identify the key parameters that control phase separation, the conditions that favour the formation of gas cavities, and the characteristic size of the resulting gas cavities.   

Deformations of the cell cytoplasm through the lens of poro-viscoelastic materials

The cell cytoplasm is the largest part of the cell by volume and its rheological properties dictate the material response of the whole cell. In particular, the cytoskeleton behaves like a porous elastic solid on timescales of seconds, but a viscous fluid on timescales longer than minutes. On intermediate timescales, the actin network behaves like an elastic material that exhibits stress relaxation due to the reorganization of the cytoskeleton. We consider a poroelastic immersed boundary method in which a fluid permeates a porous, elastic structure of negligible volume fraction. Then, we extend this method to describe a poro-viscoelastic material in a computationally efficient way, and thus include stress relaxation of the material. Lastly, we demonstrate how this computational method can be used, together with fast cytometry measurements, to infer material properties of the cell.   

Unraveling cytoplasmic streaming using a coarse-grained model of microtubule hydrodynamics

During the development of the fruit fly oocyte, flows with short-ranged correlations transition to a dramatic cell-spanning vortex, accompanied by coherent deformations in the microtubule cytoskeleton. Using a coarse-grained model for the hydrodynamics of ordered fibers, we show that sufficiently dense microtubule arrays, forced only by molecular motors transporting cargo, undergo a ``swirling transition'' that is fundamentally different than the buckling transition which leads to the flapping motion of isolated filaments. Our model produces streaming velocities consistent with \emph{in vivo} measurements, and allows us to place bounds on the number density of kinesin-1 motors transporting cargo within the microtubule array.