# Schedule for: 24w5230 - Non-Newtonian Flows in Porous Media

Beginning on Sunday, July 14 and ending Friday July 19, 2024

All times in Banff, Alberta time, MDT (UTC-6).

Sunday, July 14 | |
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16:00 - 17:30 | Check-in begins at 16:00 on Sunday and is open 24 hours (Front Desk - Professional Development Centre) |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building. (Vistas Dining Room) |

19:30 - 21:30 |
Informal gathering ↓ Presentation of the presenters of Thursday/Friday
(3mn talk each)
This will be in the BIRS lounge, above the reception area where you checked in (Other (See Description)) |

Monday, July 15 | |
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07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

08:45 - 09:00 |
Introduction and Welcome by BIRS Staff ↓ A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions. (TCPL 201) |

09:00 - 09:30 |
Arezoo Ardekani: Stability and dispersion of viscoelastic flows through porous media ↓ Viscoelastic flows are common in many natural and industrial applications, such as biofilm transport, drug delivery, and enhanced oil recovery. The stretching of polymeric chains in viscoelastic flows induces elastic instability, which manifests in symmetry-breaking, time-dependent flows, and anomalous transport properties. The knowledge of the polymeric stress field is essential for understanding transport in viscoelastic flows because the topology of the polymeric stress field controls the flow states and dynamics in viscoelastic flows. However, the experimental measurements of the stress field are challenging. Through analytical and numerical analyses, we obtain a relationship between the polymeric stress field and the Lagrangian stretching field. The Lagrangian stretching field depends solely on the flow kinematics, which is relatively easy to measure in the experiment. Thus, our result establishes a simple framework to unveil the topology of the polymeric stress field directly from readily measurable flow field data, even for strongly viscoelastic and unstable flows. (TCPL 201) |

09:30 - 10:00 |
Vedad Dzanic: Numerical characterisation and control of viscoelastic instabilities through porous media ↓ A remarkable feature of viscoelastic fluids is their ability to transition the flow into strong spatiotemporal “turbulent-like” fluctuations at the micro-scale in the absence of inertia. These instabilities are notoriously sensitive to their surrounding geometry. Consequently, understanding the onset and general behaviour of viscoelastic instabilities in geometrically complex applications, such as porous media, is far from a trivial task and has large-scale potential to further improve enhanced-oil-recovery processes. The purpose of this presentation is to discuss recent works [Dzanic et al. Phys Fluids, 35 023105 (2023); Dzanic et al. Phys Fluids, 35 093108 (2023)], which have numerically demonstrated the geometric dependence and exploitation of viscoelastic instabilities through porous media. More specifically, we first shed light on the numerical hybrid lattice Boltzmann model used to conduct the simulations. This model is then applied to simulate viscoelastic fluids in confined one-dimensional channel arrays of circular pore contractions with varying levels of pore asymmetry. At low elasticity, we demonstrate that the viscoelastic instability in all geometries conform to the same flow behaviour, characterised by a loss of velocity streamline symmetry, as well as the buildup of secondary vortices. However, at higher elasticity, we observe a transition into strong transient behaviour, whereby the viscoelastic instability response in each pore geometry adheres to the Pakdel–McKinley criterion for elastic instability, specifically the streamline curvature and elastic stress anisotropy. Following this, we demonstrate the existence of an additional viscoelastic mobilization mechanism, beyond a favourable mobility ratio, which aids in the recovery of trapped droplets in a two-dimensional porous media. When compared to the purely Newtonian displacing fluid, the additional viscoelastic response provides a considerable mobility enhancement across a range of competing capillary, wettability, and elastic conditions. It is revealed that the mobilization mechanism is purely elastic, being attributed to the growth of polymer stresses near the fluid–solid contact regions, generating a “pinch-off” mechanism. We conclude by addressing future perspectives, specifically an extension to three-dimensional heterogenous porous media, as well as elastoviscoplastic fluids. (TCPL 201) |

10:00 - 10:30 |
Emad Chaparian: Percolation of yield-stress fluids in porous media: yield limit and a general Darcy-law ↓ Yield-stress fluid flows through porous media are inherent to many industries including filtration, oil \& gas and mining and also biomedical treatments. The challenge in finding a Darcy-type law for bulk transport properties of yield-stress fluids is twofold: firstly, due to the yield stress, a finite pressure gradient should be applied to initiate the flow, otherwise, the fluid remains quiescent and the flow rate is zero. Moreover, the flow rate is not a linear function of the imposed pressure gradient in the flowing regime mainly due to the non-linear rheological behaviour of this type of fluids. Firstly, we investigate the onset of the flow when the yield-stress fluids percolate inside porous media (i.e. the yield limit). A mathematical model is presented that can successfully predict the critical yield number (i.e. the ratio of the yield stress to the critical pressure gradient) which reduces to purely geometrical features of the media - a universal scale [1]. We validate our model with previously published data [2,3] where the porous media are mimicked with mono-dispersed circular obstacles. Also, we validate our mathematical model further with exhaustive simulations on more complex cases: bi-dispersed and various shapes of obstacles. Moreover, far from the yield limit, we propose a general Darcy-type law that can predict the bulk transport properties of yield-stress fluid flows in porous media for all the regimes. Again, the proposed law is extensively validated by the computational results.
[1] E. Chaparian, J. Fluid Mech. 980 (2024) A14.
[2] E. Chaparian & O. Tammisola, J. Fluid Mech. 911 (2021) A17.
[3] D. Fraggedakis, E. Chaparian & O. Tammisola, J. Fluid Mech. 911 (2021) A58. (TCPL 201) |

10:30 - 11:00 | Coffee Break (TCPL Foyer) |

11:00 - 11:30 |
Seyed Mohammad Taghavi: Rheological and geometric interactions in viscoplastic fluid transport through grooved channels ↓ The study of non-Newtonian flows in porous media is crucial for understanding fluid behavior in various environmental, biomedical, and energy contexts. In particular, the coupling of complex rheological properties with the heterogeneity of porous structures presents significant challenges and opportunities for advancing our knowledge of such flows. Our research addresses this by investigating the transport of viscoplastic fluids through channels with grooved superhydrophobic walls, using a comprehensive modeling approach and numerical simulations. Our goal is to understand the interactions between the fluid and the superhydrophobic surface and identify factors that influence flow behavior. Groove orientation, defined by angle theta, can be longitudinal, transverse, or oblique with respect to the main flow stream. We assume that the interface between the viscoplastic fluid and trapped air in the Cassie state is flat and we model it using the Navier slip law and Bingham model for viscoplastic rheology. A range of channel thicknesses, characterized by the ratio of groove period to half channel height, and flow parameters including Bingham number, slip number, groove periodicity length, slip area fraction, and groove orientation angle are considered. Perturbation theory is used to derive semi-analytical and closed-form solutions for velocity fields in thick channels, while numerical simulations are employed for both thick and thin channels using the Papanastasiou regularization method. These solutions are developed for all groove orientations, with the oblique case being unique due to the nonlinear effects of viscoplastic rheology, a feature absent in corresponding Newtonian flows. We obtain closed-form relations for the flow mobility tensor and effective slip length, highlighting the strong nonlinear effect of viscoplastic rheology. Linear stability analysis of the homogeneous slip condition for a particular flow configuration reveals stabilizing/destabilizing effects of the streamwise/spanwise slip condition on the Poiseuille-Bingham flow. (TCPL 201) |

11:30 - 12:00 |
Paul Grassia: Three Bubbles Good, Two Bubbles Better: A Foam Hadron Collider ↓ How readily foam flows when driven along a narrow channel under pressure depends on how the foam is structured. However it turns out that the converse is also true: how a foam is structured can depend in turn upon how it flows along a channel. Here this idea is illustrated using a ''foam'' consisting of just three bubbles. This is analysed using the viscous froth model, which incorporates driving pressure and capillary forces, but also viscous drag between moving foam films and the channel wall. For small imposed pressures, the foam structure stays together. For larger imposed pressure, viscous effects cause the bubbles to rearrange topologically, possibly leaving one or more bubbles behind. Different types of topological transformations can occur depending on bubble sizes relative to channel size and on imposed pressure. Moreover states that are formed after a first transformation are often unstable and hence short-lived. Thus it is necessary to explore not just single transformations but instead entire topological transformation paths. (TCPL 201) |

12:00 - 12:30 |
Miguel Beneitez: Instabilities in rectilinear flows and their link to viscoelastic turbulence with and without inertia. ↓ Many liquids in the modern world possess both elastic and viscous properties (e.g. paints, saliva and DNA suspensions among many others polymers). Understanding their behaviour is paramount in many industrial processes where turbulence is commonplace but may be highly undesirable (e.g. pharmaceutical, chemical, and plastic). Little is known about how and why this viscoelastic turbulence occurs, mainly due to the complexity and variety of the mathematical models used to describe it. In particular, it has only recently been realised that there may be three distinct types: a polymer-adjusted Newtonian turbulence (NT) which predominantly exists due to inertial effects, Elastic turbulence (ET) which exists in the absence of inertia and is entirely driven by the elasticity, and a third form called Elasto-inertial turbulence (EIT) which requires a balance of inertia and elasticity to exist. Many questions exist about these two forms of viscoelastic turbulence: in particular, whether they are dynamically connected, what triggers them and how EIT interacts with the presence of Newtonian turbulence at low levels of elasticity. In this talk, I will attempt to review recent work seeking some answers largely stimulated by a recently-discovered viscoelastic centre mode instability [1] and a newly-discovered ``polymer diffusive instability'' [2].
[1] Garg et al. Phys. Rev. Lett., 121, 024502, 2018
[2] Beneitez et al. Phys. Rev. Fluids 8, L101901, 2023. (TCPL 201) |

12:30 - 14:30 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

13:40 - 14:00 |
Group Photo ↓ Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo! (TCPL Foyer) |

14:00 - 14:30 |
Pedro Ponte Castañeda: Variational Linear Comparison Estimates for the Flow of Yield-Stress Fluids through Porous Media ↓ Modeling the flow of complex fluids through porous media is of crucial importance in many industrial and naturally occurring processes, including in the oil, cosmetics, food, and pharmaceutical industries, as well as in soil remediation and the manufacturing of polymer- infiltrated composite materials. For this reason, there have been numerous attempts to generalize Darcy's law, relating the flow velocity to the pressure drop by the permeability of the porous medium, for various types of non-Newtonian fluids. This work is concerned with the flow of yield-stress (viscoplastic) fluids through rigid porous media under Stokes flow conditions. The porosity distribution may be periodic, or random and statistically uniform, but not necessarily isotropic. Variational estimates are obtained for the nonlinear permeability or mobility of the viscoplastic fluid under macroscopically uniform flow conditions through a generally anisotropic porous medium, with given microstructure, in terms of the porosity and the permeability tensor for the flow of a Newtonian fluid through a porous medium with the same microstructure. Applications of the method are given for special types of generalized Newtonian fluids, including Herschel-Bulkley and Carreau fluids, by means of known estimates for the permeability of Newtonian fluids, including the Blake-Carman-Kozeny formula and Weissberg- Prager bounds for isotropic random porous media, as well as numerical and asymptotic estimates for anisotropic periodic fibrous microstructures. (TCPL 201) |

14:30 - 15:00 |
Federico Lanza: Dynamic instability of a temperature-dependent viscous fluid in a Hele-Shaw Cell ↓ Our study investigates a dynamic instability hypothesized to occur when hot fluids such as magma flow into confined spaces like fractures and cool down due to the surrounding environment. This instability arises from the significant increase in viscosity that such fluids undergo as their temperature decreases, and leads to the formation of finger-shaped regions of hot fluid with enhanced flow velocity. To investigate this phenomenon, we employ a model system where hot fluid is injected into a Hele-Shaw cell and subjected to a small perturbation in the inlet flow. Numerical simulations are performed to solve the equations governing the flow of the temperature-dependent viscous fluid. We find that, if the temperature dependance of the fluid viscosity is sufficiently strong, a hot finger will grow from the perturbation and stabilize after some time. In the growing stage, a dispersion relationship is identified that links the finger growth rate to the wavelength of the imposed perturbation, allowing us to determine an optimal wavelength. In the stationary state, we derive and characterize expressions for the temperature and local flow rate of the finger as functions of the Peclet number, which represents the ratio between advection and diffusion rate of the temperature, and the cooling rate through the fluid surfaces. (TCPL 201) |

15:00 - 15:30 |
Vivek Narsimhan: Microhydrodynamics of spheroids in weakly viscoelastic fluids - lift forces, orientation dynamics, and effective stress ↓ When particles are in a pressure-driven flow of a non-Newtonian fluid, the particles can acquire lift forces due to the imbalance of normal stresses on the particle surface. This phenomenon has been well-studied for spherical particles, but the role of particle shape is still in its early stages. In this work, we develop a theory to describe the rigid body motion of a non-spherical particle in a polymeric fluid. The theory is based on a retarded expansion in the Deborah number (i.e., second order fluid model), for the case when the particle is in a quadratic (i.e., pressure-driven) flow. We find that for particles in a circular tube flow, spherical particles move to the center of the tube faster than prolate and oblate particles of the same volume, due to the unique orientation dynamics of the spheroids in the polymeric fluid. We also find that prolate particles move slower than oblate particles of the same aspect ratio. These trends are verified by performing microfluidic experiments where we visualize polystyrene particles of various shapes moving through circular capillaries in a Boger fluid with weak viscoelasticity (De=O(0.01)) and vanishing inertia (Re = O(0.0001)).
In the last part of the talk, we will discuss preliminary results for how non-spherical particles alter the effective stress in a viscoelastic fluid (i.e., shear viscosity, extensional viscosity, normal stress differences). We will compute the averaged stress in a dilute suspension of spheroids, akin to Einstein viscosity, but in viscoelastic medium. We will also discuss some experimental methods to track 3D particle position and orientation in such fluids using holographic techniques (TCPL 201) |

15:30 - 16:00 | Coffee Break (TCPL Foyer) |

16:00 - 16:30 |
Marie-Laure Olivier: PoreLab - 7 years of experience in the CoE Program ↓ PoreLab, acronym for ``Porous Media Laboratory'', has thrived under the Center of Excellence (CoE) scheme financed by the Research Council of Norway (RCN), the Norwegian University of Science and Technology (NTNU) and the University of Oslo (UiO) for the past 7 years. We introduce in this presentation an overview of PoreLab's mission, organization, management, challenges, and future goals. We present its main outcomes in terms of publications, collaboration, networking, staffing plan and further projects through additional funding. The project ``Non-Newtonian Flow in Porous Media'' is one of these additional projects funded by the INTPART program of the Research Council of Norway. The INTPART program is aimed at supporting international partnerships in education and research. ``INTPART'' stands for ``International Partnerships for Excellent Education and Research''. The actual workshop ``Non-Newtonian Fluids in Porous Media'' is partly organized and financed through the INTPART program. (TCPL 201) |

16:30 - 17:00 |
Christopher Bowers: Modeling generalized Newtonian fluid flow in porous media at the macroscale: New models based on averaging theory ↓ Generalized Newtonian fluid (GNF) flow in porous media is typically modeled at the macroscale, defined here as the porous medium continuum scale, due to computational limitations. To bring macroscale modeling of GNFs into parity with Newtonian flow modeling, a new model for hydraulic resistance has been developed based on an analysis using thermodynamically constrained averaging theory (TCAT). The hydraulic resistance, which is related to the inverse of hydraulic conductivity, can be modeled using a function of similar form to the rheological model of the GNF being investigated. This GNF-like resistance model uses fluid rheology, as well as system parameters that can be computed from observation of a single Newtonian fluid flowing through the medium. The system properties used in the resistance model include porosity, permeability, specific interfacial area, and a new length scale parameter related to the average shear stress at the fluid-solid interface. All of these system properties are independent of the fluid being used, and can be tabulated in advance of macroscale simulation of any GNF in a similar manner to the Newtonian fluid case.
The GNF hydraulic resistance model has been extensively validated for both Cross and Carreau model fluids, in a variety of both isotropic and anisotropic media. Averaged microscale simulation data has also been collected for flow of a Newtonian fluid through a large set of 1000 porous media, and was used to propose a relationship between the new length scale parameter and more readily available system parameters. Taking the derivative of the resistance model with respect to average fluid velocity through the system can also give an indication of the degree of GNF behavior that can be expected at each flow rate through the system. Microscale species transport simulations during GNF flow through porous media have also been carried out, and it was shown that the resistance model can be used to predict the degree of dispersivity through a system at different flow rates a priori. The resistance model, and corresponding species transport model, represent a significant improvement in the state of macroscale GNF modeling compared to past efforts. (TCPL 201) |

17:00 - 17:30 |
Ilaria Beechey-Newman: Hierarchical Drying Patterns of Colloidal Suspensions Under Confinement ↓ Human nature draws us in to patterns that exist all around us the natural world, from honeycomb to constellations[1]. Here we present the pattern formation of colloidal systems allowed to dry slowly in a horizontal confinement [2]. The pattern left behind is a distinct labyrinth of colloidal fingers, showing multiple length scales. While some similar patterns have been documented before, the driving forces behind this process are very different to frictional finger formation observed in granular systems[3].
[1] Mattson Mark P., Superior pattern processing is the essence of the evolved human brain, Frontiers in Neuroscience 8, 2014. https://doi.org/10.3389/fnins.2014.00265
[2] Beechey-Newman et al., in preparation
[3] Eriksen, Fredrik K.; Toussaint, Renaud; Turquet, Antoine Leo et al., Pressure evolution and deformation of confined granular media during pneumatic fracturing, Phys. Rev. E, 2018. https://doi.org/10.1103/PhysRevE.97.012908 (TCPL 201) |

17:30 - 18:00 |
Rob Poole: Viscoelastic fluid flow in microporous media ↓ An experimental investigation of the flow of flexible polymer solutions through a distinctive micro-porous glass structure (typical pore sizes of 500 microns) has been conducted encompassing both global flow-rate pressure drop and pore-scale velocity field measurements.
For the global measurements, we are able to relate the bulk flow properties to measurable rheological parameters, demonstrating that a key parameter in estimating the pressure drop through the porous media (for a variety of polymer types, concentrations, solvents, molecular weights, and states of degradation) is the extensional relaxation time (or, more correctly, a characteristic time for extensional stress growth). A Weissenberg number (Wi) is calculated as a product of this extensional relaxation time and the nominal shear rate in the flow. Results, suitably normalised with Newtonian pressure-drop data, show a critical Wi of roughly 0.01 where all working fluids in two different pore structures reveal the onset of elastic dominance over viscous forces as the flowrate increases. Such a low critical value of Wi is due to the estimate of a nominal shear rate based on pore size, which severely underestimates the maximum shear rates within the complex pore structure.
Significant deviation from the universal behaviour is observed for high concentrations of the polyacrylamide, which is thought to be a result of shear-thinning for these systems. Systematic degradation of the polyethylene oxide solutions caused an exponential decay in elasticity, which is reflected in the pressure-drop measurements. The results imply that, although the flow through porous media is known to be a complex combination of both shear and extensional flow, the extensional effects maybe of primary importance in this context (or that transient stretching in the media is well represented by this relaxation time).
These global results are complemented with micro-PIV (micro-particle image velocimetry) to measure mean and fluctuating velocities within individual pores. The velocity field measurement includes the velocity magnitude and fluctuation intensity in several different pores within the porous material across a Weissenberg number (Wi) range of approximately 0.01 to 5 for each of the test fluids. The global averaged fluctuation intensity increased with Wi but the critical value, which indicates the onset of unsteadiness within the flow at pore scale gives an approximately constant value of Wi≈0.4, which is significantly higher than the value that is observed in the pressure-drop measurements for the data to rise above the Newtonian base line. In essence, this suggests that the enhanced pressure-drop behaviour of the bulk flow may not be due to the local velocity fluctuations within the pores but due to mean (steady) flow effects, at least over a significant portion of the data (up to Wi≈0.4). (TCPL 201) |

18:00 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building. (Vistas Dining Room) |

Tuesday, July 16 | |
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07:00 - 09:00 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

09:00 - 09:30 |
Alex Hansen: A New Kind of Thermodynamics for Two-Phase Flow in Porous Media ↓ Homogenization is the standard approach to upscaling immiscible two-phase flow in porous media from the pore scale to the Darcy scale. The trouble with homogenization techniques is that they can only produce averages of existing variables and not new types of variables. Statistical mechanics does produce new types of variables when scaling up thermal systems from the molecular scale to the continuum scale. Temperature is an example of such a variable. It connects the mechanistic description at the molecular level with a thermodynamic description at the continuum level. The trouble with statistical mechanics is that it demands equilibrium. Immiscible two-phase flow in porous media is not an equilibrium process. It is, however, possible to map immiscible two-phase flow in porous media onto an equivalent equilibrium process through a trick. This makes it possible to formulate a version of statistical mechanics for this problem [1,2,3]. This leads to a thermodynamics-like description at the Darcy scale where the fluid velocities play the roles of internal energy and free energies [4-6]. New variables such as the agiture - a temperature equivalent - emerge. Another emergent variable at the Darcy scale is the co-moving velocity. The co-moving velocity has many interesting properties, many of which remain mysterious [7,8]. The biggest mystery is why it is linear in the natural variables, even though the flow is highly non-linear [9].
[1] J. Feder, E. G. Flekkøy and A. Hansen, Physics of Flow in Porous Media (Cambridge Univ. Press, Cambridge, 2022).
[2] A. Hansen, E. G. Flekkøy, S. Sinha and P. A. Slotte, A statistical framework for immiscible and incompressible two-phase flow in porous media, Adv. Water Res., 171, 104336 (2023) ; doi : 10.1016/j.advwatres.2022.104336.
[3] H. Fyhn, S. Sinha and A. Hansen, Local statistics of immiscible and incompressible two-phase flow in porous media, Physica A, 616, 128626 (2023) ; doi : 10.1016/j.physa.2023.128626.
[4] A. Hansen, S. Sinha, D. Bedeaux, S. Kjelstrup, M. Aa. Gjennestad, and M. Vassvik, Relations between seepage velocities in immiscible, incompressible two-phase flow in porous media, Transp. Porous Media, 125, 565 (2018) ; doi :10.1007/s11242-018-1139-6.
[5] S. Roy, S. Sinha and A. Hansen, Flow-area relations in immiscible two-phase flow in porous media, Front. Phys. 8, 4 (2020) ; doi :10.3389/fphy.2020.00004.
[6] H. Pedersen and A. Hansen, Parametrizations of immiscible two-phase flow in porous media, Front. Phys. 11, 1127345 (2023) ; doi :10.3389/fphy.2023.1127345.
[7] S. Roy, H. Pedersen, S. Sinha and A. Hansen, The co-moving velocity in immiscible two-phase flow in porous media, Transp. Porous Media, 143, 69 (2022) ; 10.1007/s11242-022-01783-7.
[8] F. Alzubaidi, J. E. McClure, H. Pedersen, A. Hansen, C. F. Berg, P. Mostaghimi and R. T. Armstrong, The impact of wettability on the co-moving velocity in two-fluid flow in porous media, arXiv :2309.00362
(2023) ; doi :10.48550/arXiv.2309.00362 (to appear in Transp. Porous Media).
[9] A. Hansen, Linearity of the co-moving velocity, arXiv :2402.13826 ; doi :10.48550/arXiv.2402.13825. (TCPL 201) |

09:30 - 10:00 |
Emily Chen: Viscoelastic flow instabilities in porous media: insights from pore-scale flow fields ↓ Viscoelastic fluid flows in porous media arise in many applications, from groundwater remediation to separations and chemical processing. The addition of small amounts of polymer or surfactant can impart fluids with substantial viscoelasticity. Above a threshold flow rate, the development of elastic stresses coupled with streamline curvature inherent to the tortuous pore space can generate an elastic flow instability characterized by persistent spatiotemporal flow fluctuations-despite negligible inertial effects. Our previous studies in disordered 3D media suggest that the instability is highly sensitive to medium geometry; however, how exactly geometry influences the flow instability remains unclear. Here, we first examine the influence of the pore space geometry on elastic instabilities by directly imaging flow in microfabricated 3D porous media consisting of body-centered cuboid or simple-cubic arrays of spheres. Unexpectedly, in both cases, the flow instability is generated upstream of the contact regions between spheres rather than at sphere surfaces-suggesting that the consolidation of solid grains, inherent in naturally-occurring media, may play a pivotal role in establishing the flow instability in field settings. Further, despite their shear thinning nature in bulk rheology, polymer solutions can exhibit anomalous flow thickening above a threshold flow rate in porous media, marked by a drastic increase in flow resistance. Previous studies have attributed this thickening to extensional viscosity, energy dissipation by elastic instabilities, and adsorption or pore clogging; however, direct quantification of these mechanisms remains lacking. We derive a mechanical power balance incorporating both the added viscous dissipation arising from an elastic flow instability and resistance from extensional viscosity that captures the macroscopic flow resistance of polymer solution flow in ordered porous media. Our model directly links pore-scale flow fields obtained using confocal microscopy to the macroscopic flow resistance. Finally, we examine the application of elastic instabilities towards the remediation of porous media soiled with microplastic deposits. We find that both viscoelasticity and instability-related mechanisms of particle removal contribute to enhanced cleaning efficiency for polymer solution flows compared to Newtonian fluids. Our work thus leverages access to in situ pore-scale flow fields to elucidate the physics underlying viscoelastic flow instabilities and provides exciting evidence that these instabilities can be used to enhance mass transport within porous media. (TCPL 201) |

10:00 - 10:30 |
Davide Picchi: Decoupling rheological and geometry effects on the description of flow in confined environments ↓ The study of flow in porous media finds applications in many fields, such as energy, biology, the food industry, and soil remediation. The most common strategy is to adopt a continuum description (i.e., Darcy's law), where the complexity of the medium (e.g., geometry of the pores) is embedded in effective parameters such as permeability. In many practical scenarios, the working fluid exhibits a shear-thinning behavior (e.g., biological fluids, polymers, suspensions), and its viscosity is a function of the imposed shear rate. This places an additional burden on the development of accurate flow models for flow in porous media.
The goal of this talk is to show that these two effects (i.e., the shape of the pores and non-Newtonian rheology) can be decoupled in the macroscopic description. To show that, we address the problem of steady laminar flow of a shear-thinning fluid in a rectangular duct with an arbitrary aspectv ratio, which can be seen as the simplest representation of a complex pore. An accurate solution for the case of a Carreau fluid is obtained to analyze the effects of the fluid rheology and the aspect ratio on the velocity field and pressure gradient. Interestingly, the analysis shows that (i) the fluid rheology and the aspect ratio have independent contributions to the integral flow characteristics; (ii) universal scaling laws for the effective viscosity and the effective channel size exist and generalize the classical formula for the friction factor of Newtonian flows to shear-thinning fluids flowing in a rectangular channel with an arbitrary aspect ratio.
These results clarify the effect of fluid rheology and channel aspect ratio on frictional pressure gradients and may inspire the modeling of the flow of shear-thinning fluids in complex geometries. In fact, the existence of a master curve for effective viscosity appears to be typical of a more general class of problems, including capillary imbibition (when a shear-thinning fluid invades a single pore) and the motion of a confined bubble. (TCPL 201) |

10:30 - 11:00 | Coffee Break (TCPL Foyer) |

11:00 - 11:30 |
Sarah Hormozi: Motility of Bacteria in elastoviscoplastic biological fluids ↓ Pathogenic bacteria are a persistent threat to human health incurring a heavy cost on the healthcare system. The motility of bacteria is an essential mechanism with which pathogens reach the membranes of susceptible cells and cause infections. A well known example is Helicobacterium Pylori, which moves through mucosal barriers in the gut to cause infections. Apart from pathogenesis, motility is also a crucial factor in formation of harmful bacterial biofilms on tissues and implants, the spatial distribution of microbiome etc. The majority of the cells and tissues prone to pathogenic infections in the human body are typically lined with a multi-scale complex biological fluid, e.g. mucus. These are complex fluids, meaning that they possess a microstructure (porous structure), exhibit non-Newtonian rheology, which is a function of length scale and also possess a yield stress. In this talk, we seek to gain a fundamental mechanistic understanding of how the microstructure and the non-Newtonian rheology including the yield stress of such complex biological fluids affect the motion of a swimming bacterium.
This talk primarily consists of two parts. In the first part, we present a continuum mathematical model for the complex fluid with a microstructure, where the complex fluid of interest is an entangled polymer solution. Assuming small rheological effects, we model the complex fluid to be made up of two interpenetrating continuum Newtonian fluids, the solvent and polymer, which have different viscosities and can interact independently with an object moving through it. This independent interaction results in a relative motion between the two fluids, and the spatial extent of this zone of relative motion could be controlled by a parameter that describes the extent of coupling between the two fluids, the screening length. We develop a slender body theory for a bacterium that moves through this two-fluid medium and from our analysis of the theory, we find that the presence of microstructure can result in an enhanced swimming velocity of the bacterium.
In the second part of this talk, we describe a numerical framework, with which the aforementioned two-fluid model can be extended to include the rheological effects of the polymer. This method combines a finite-difference solver for solving the equations of motion of an elastoviscoplastic polymer solution with the slender body theory described above in order to simulate a single bacterium moving through an elastoviscoplastic two-fluid medium, which models an entangled polymer solution, capturing the effects of both microstructure and rheology. (Online) |

11:30 - 12:00 |
Marco Edoardo Rosti: The effect of elasticity and plasticity on model porous media flows ↓ Elastoviscoplastic fluids are a distinct category of non-Newtonian fluids characterized by their ability to demonstrate both solid-like and liquid-like behaviors in response to applied stress. We use numerical simulations, validated by microfluidic experiments, to study the flow characteristics on the development of yielded and unyielded regions, flow regimes (steady or unsteady), and pressure drop. Our findings demonstrate that plasticity plays a critical role in such geometries, promoting a transition from steady to time-dependent behavior when compared to viscoelastic fluid under the same flow condition. We will also discuss the interplay of elasticity and plasticity, and the effect of confinement. (TCPL 201) |

12:00 - 12:30 |
Holger Stark: Swimming and Rheology of Active Suspensions in Non-Newtonian Fluids ↓ The natural habitat of microorganisms are non-Newtonian fluids, which besides shear viscosity also have an elastic response. Using a second-order model fluid, we present an analysis, how weak viscoelasticity affects the rheology of a dilute suspension of microswimmers [1]. Starting with modifications of the well-known Jeffery orbits and the orientational distribution due to tumbling and rotational diffusion, we show how the effective shear viscosity is influenced compared to an active suspension in a Newtonian fluid. In particular, for pusher swimmers such as E.coli bacteria, the shear viscosity is further reduced due to elastic stresses, which offers a further route towards realizing superfluids with vanishing shear viscosity [2]. In the end, we also comment on swimming in a viscoelastic fluid with a reciprocal stroke pattern, which is not possible in a Newtonian fluid due to the constraint of kinematic reversibility [3].
[1] A. Choudhary, S. Nambiar, and H. Stark, Comm. Phys. 6, 163 (2023).
[2] H.M. López, J. Gachelin, C. Douarche, H. Auradou, and E. Clément, Phys. Rev. Lett. 115, 028301(2015).
[3] M. Eberhard, A. Choudhary, and H. Stark, Phys. Fluids 35, 063119 (2023). (TCPL 201) |

12:30 - 13:30 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

14:00 - 14:30 |
Laurent Talon: On the statistical properties of yield stress fluid in porous media ↓ Because of the interaction between the medium's heterogeneity and non-linear rheology, the flow of yield stress fluids in porous media presents an interesting level of complexity. The non-linear Darcy law, for example, is caused by the increase of the number of flowing paths with the applied pressure difference.
In this presentation, we will talk about several statistical aspects of this problem. In particular, how the directed polymer problem -which minimizes the energy of a path in a random field- introduced by Kardar Parisi and Zhang (KPZ) in 1987 relates to the small flow rate limits. We will talk about how the KPZ universality class impacts the flow and the non-linear Darcy's law in different configurations. (TCPL 201) |

14:30 - 15:00 |
Isaac Pincus: Rheology and linear dichroism of dilute solutions of flexible and semiflexible polymers in shear flow ↓ Although the non-equilibrium behaviour of polymer solutions is generally well understood, there remain several unanswered questions for dilute solutions in simple shear flow. Experimental viscosity data exhibit qualitative differences in shear-thinning exponents, shear rate for onset of shear-thinning and high-shear Newtonian plateaus depending on polymer semiflexibility, contour length and solvent quality. While polymer models can incorporate all these effects through various spring force laws, bending potentials, excluded volume (EV) potentials, and hydrodynamic interaction (HI), the inclusion of each piece of physics has not been systematically matched to experimentally observed behaviour. Furthermore, attempts to develop multiscale models which can make quantitative predictions are hindered by the lack of ability to fully match the results of bead-rod models, which represent a polymer at the Kuhn step level, with bead-spring models, which consider the entropic elasticity. In light of these difficulties, this work aims to develop a general model based on the so-called FENE-Fraenkel spring, originally formulated by Larson and coworkers (Hsieh et al., 2006, J. Chem. Phys., 124(4)), which can span the range from rigid rod to traditional entropic spring, as well as include a bending potential, EV and HI. This model can reproduce, and smoothly move between, a wide range of previously observed polymer solution rheology in shear flow. By showing that one can correctly capture the solvent quality for semiflexible polymer models using Yamakawa's Quasi-Two-Parameter (QTP) theory, we develop a successive-fine-graining scheme for predicting polymer rheology and conformation, particularly focusing on capturing the Linear Dichroism (LD) of semiflexible polymers. Following the approach of earlier authors, our multiscale model can be used to relate the LD of each segment in our bead-spring chain to the extension of the spring, giving quantitative agreement with experimental data. Additionally, we show how this model may be useful in developing novel constitutive equations which can capture the complex behaviour of semiflexible polymers such as xanthan gum without resorting to full BD simulations. In this talk, I will also briefly discuss future work investigating bacterial dispersion in mixed-phase flow through porous media. (TCPL 201) |

15:00 - 15:30 |
Giovanniantonio Natale: Dynamics and microstructures of Pseudomonous aeruginosa biofilms at fluid-fluid interfaces ↓ Biofilms are heterogeneous communities of bacterial cells, encased in a self-secreted matrix of polymers. Fluid-fluid interfaces are a common venue for biofilm formation in nature and humans. In humans, biofilms are responsible for most microbial infections due to their resistance to antibiotics.
As a biofilm ages, its microstructure evolves and increasingly resembles a transient porous medium. This complex structure includes various pore lengths, which can be categorized into two main types: intercellular pores and inter-colony pores. Moreover, single cells' motility and directionality evolve as the biofilm matrix develops. This micromechanical and microstructural heterogeneity consequentially affects transport properties.
In this work, we use wild-type Pseudomonas aeruginosa PA-14 bacterium, a well-studied microbe, that secretes the PEL polysaccharide to facilitate biofilm formation. Using a combination of fluorescence microscopy, image analysis, and microrheology, we characterize the dynamics of PA-14 bacteria at a model fluid-fluid interface.
First, we explore the microstructural and micromechanical features of these 2D systems as a function of aging time. We measure single-cell dynamics and correlate them with the viscoelastic properties of the biofilm matrix. We also employ isogenic mutant strains to determine the role of PEL in determining the mechanical properties of these biofilms. Then, we explore the transport of a chemical stressor (e.g., antibiotic) in this active porous medium. We observe that the biofilm heterogeneity, viscoelasticity, and biochemical interactions with the matrix hinder antibiotic diffusion in the matrix. The microrheology results show that the antibiotic diffusion results in heterogeneous live and dead regions in the biofilm matrix and further modifies the viscoelasticity of the matrix.
Our results provide valuable insights into the physical and biochemical interactions at play in biofilm environments. This understanding can inform the development of more effective strategies for biofilm control, targeting the biofilm matrix's rheological properties to enhance antibiotic penetration and action. (TCPL 201) |

15:30 - 16:00 | Coffee Break (TCPL Foyer) |

16:00 - 16:30 |
Duncan Hewitt: Simple yield-stress fluids in slots and channels: clogging and channelling ↓ Insights from simple (ideal) modelling of viscoplastic fluids in confined geometries will be presented, including flows in Hele-Shaw cells and conduits. The impacts of obstacles and channel topography will be explored, together with their role in deflecting, channelling and clogging the fluid. Roughness in cell walls will be explored, as a tentative route towards flow in a more disordered cell (like a real porous medium). Results from numerical and analytical modelling will be presented, with some discussion of limiting behaviour and possible directions. (TCPL 201) |

16:30 - 17:00 |
Hossein Hassanzadeh: What happens when a jet meets viscoplasticity? Insights from mixing to fingering and fracturing regimes ↓ We explore the flow dynamics of a buoyant miscible jet (microjet) where a Newtonian fluid is injected through a circular nozzle (needle) into a large tank filled with a viscoplastic fluid. Using a combination of various non-intrusive experimental techniques, i.e., time-resolved tomographic particle image velocimetry (TR-Tomo PIV), high-speed imaging, planar laser-induced fluorescence (PLIF) and ultrasound Doppler velocimetry (UDV), we investigate the effects of the injection velocity, the density difference, the viscosity ratio, and the rheological parameters (in particular, the yield stress) on the jet flow behavior. To characterize the jet flow behavior, we employ different jet flow features, including the laminar length, the penetration length, the jet radius, and the turbulent kinetic energy. We also developed a mathematical model to predict the jet penetration length with the incorporation of the velocity profiles and the jet Reynolds stresses obtained from Tomo-PIV analysis. Our results demonstrate that increasing viscosity ratio and yield stress resist the jet evolution, leading to decreased penetration length and jet radius. Moreover, the yield stress can significantly influence the turbulent features and self-similarity behavior. Finally, we succeed in classifying various jet flow regimes versus the main governing dimensionless numbers. (TCPL 201) |

17:00 - 17:30 |
Hossein Rahmani: Network modeling of yield stress fluids in porous media: a Hele-Shaw cell flow analogy ↓ In this work, we explore the network modeling approach for the flow of yield stress fluids in porous media. Inspired by the analogous flow behaviour between porous media and a Hele-Shaw cell of varying aperture, we exploit a two-dimensional gap-averaged (2DGA) model developed for the flow of Herschel-Bulkley fluids in a Hele-Shaw cell to formulate the network model. The 2DGA model provides us with an explicit formulation for the gap-averaged streamfunction that depends on the pressure gradient, fluid rheology and the size of the gap. Assuming a 2D porous medium, the Hele-Shaw cell with irregular wall boundaries may also resemble a 2D network of nodes (pores) connected through the edges (throats). Considering a staggered mesh, with pressure nodes located at the cell corners and the streamfunction nodes placed at the cell center, the generated discretization can be considered in the network setting to represent primal and dual graphs. The flow-rate at each edge is calculated as a function of the pressure drop along the edge, the fluid rheology and a characteristic gap size of the edge. For the yield stress fluid to flow along an edge, the pressure drop should be above a threshold, which is a function of the yield stress, otherwise the edge is blocked. Conversely, a pore-throat network flow can be mapped to an analogous Hele-Shaw model. Employing the principle of virtual power while developing an augmented Lagrangian framework, the flow-rate in the edges and the pressure at the nodes are updated via numerical iteration until a converged solution is reached. Depending on the total pressure drop applied on the network system, the fluid rheology and the edge size characteristic, different scenarios of open and blocked edges are observed based on the network model. The developed network model can be used to explore the complex flow regimes of the yield stress fluid in a porous medium in various settings of interest. (TCPL 201) |

17:30 - 18:00 |
Hadi Mohammadigoushki: Magnetophoresis of Newtonian Metal Salt Fluids in Porous Media ↓ The phenomenon of magneto-migration, which involves the movement of solutes within a solution due to gradients in magnetic fields, offers potential for advancements in chemical separation techniques, environmental pollutant remediation, targeted drug delivery, and so on. Despite the considerable potential applications of magneto-migration, our understanding of this process remains limited. The extent and effectiveness of the migration of ions are contingent upon several factors including the strength of the magnetic field, the concentration of solute in the solution, and the properties of the surrounding medium. In this study, magneto-migration experiments were conducted on individual metal ions that included manganese (Mn+2) and zinc (Zn+2) ions, each at varying concentrations ranging from 1 mM to 1000 mM. For the manganese (Mn) experiments, enrichment was observed at the region closest to the magnet for all concentrations tested. Enrichment levels are in the range of 2.0 - 2.3 % for Mn+2. In contrast, the zinc (Zn+2) ions showed depletion in the range of 0.50 - 0.95 % at the region closest to the magnet. These results imply that while the paramagnetic ions are attracted, and the diamagnetic ions are repelled from high magnetic gradients as expected, the magnitude of the effect does not seem to scale with the difference between the susceptibility of the ion and that of the solvent. The diamagnetic ions migrate almost as much under the same circumstances as the strongly paramagnetic ions (Mn +2). We performed numerical simulations via COMSOL Multiphysics™ and compared the simulation results with experiments. Our numerical simulations suggest that both paramagnetic and diamagnetic ions may form clusters of sub-micrometer size. In addition, our simulations suggest that the cluster size is a function of time and initial concentration of solute. We provide an approximate form of cluster size by matching the simulations to experimental results. (TCPL 201) |

18:00 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building. (Vistas Dining Room) |

Wednesday, July 17 | |
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07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

09:00 - 09:30 |
Yohan Davit: Localized stress patterns control viscoelastic flows in porous media ↓ The flow of polymer solutions through porous media is often characterized by an increase of the apparent viscosity at Darcy-scale, even for solutions that are shear-thinning in a rotational rheometer. Despite continuous efforts, a clear picture of the pore-scale mechanisms involved - including the role of stress localization, low Reynolds numbers instabilities and elastic turbulence - and their link with Darcy-scale dissipation are still elusive. In this talk, we will show how we used high-performance computing to study the flow of Oldroyd and FENE-type fluids through model porous structures and what we learnt about the physics of this problem.
We will first present the simulation tool that we developed [1], which is based upon a new staggered projection scheme for viscoelastic flows that has good accuracy, keeps the conformation tensor symmetric positive definite, has a space semi-discretization that is consistent with a free-energy estimate, is well suited to high-performance computing and can be readily used for a variety of viscoelastic constitutive laws.
Focusing on 2D flows through arrays of cylindrical obstacles at moderate Weissenberg number, we will then show that localized zones of large polymeric stress, known as birefringent strands, form a complex pattern - a web - that guides the flow through the structure [2]. The reorganization of the flow is associated with an increase of stagnation zones and a reinforcement of preferential paths, which yield an increase in global dissipation. These results provide a mechanistic explanation for the increase of flow resistance at steady-state, before any transition to elastic turbulence and any increase in dissipation due to time fluctuations [3].
We will further demonstrate that the sticky interactions between strands and cylinders are a fundamental part of the transition to unsteady flow [4] and can be used to understand recent microfluidic experiments [5,6]. We will also discuss how these results may apply to realistic 3D structures and how the patterns of localized stress change with the dimensionality.
[1] O. Mokhtari, Y. Davit, J.-C. Latché and M. Quintard. A staggered projection scheme for viscoelastic flows. Mathematical Modelling and Numerical Analysis (ESAIM : M2AN), 57(3):1747–1793, 2023.
[2] O. Mokhtari, J.-C. Latché, M. Quintard and Y. Davit. Birefringent strands drive the flow of viscoelastic fluids past obstacles. Journal of Fluid Mechanics, 948:A2, 2022.
[3] C.A. Browne and S.S. Datta. Elastic turbulence generates anomalous flow resistance in porous media. Science advances 7.45 (2021):eabj2619.
[4] O. Mokhtari, M. Quintard and Y Davit. A web of sticky strands: how localized stress controls spatio-temporal fluctuations in viscoelastic flows through a lattice of obstacles. Journal of Fluid Mechanics 980, A7, 2024.
[5] D. M. Walkama, N. Waisbord, and J. S. Guasto. Disorder suppresses chaos in viscoelastic flows. Physical Review Letters, 124(16):164501, 2020.
[6] S. J. Haward, C. C. Hopkins, and A. Q. Shen. Stagnation points control chaotic fluctuations in viscoelastic porous media flow. Proceedings of the National Academy of Sciences, 118(38), 2021. (Online) |

09:30 - 10:00 |
Chandi Sasmal: Preferential paths and birefringent strands formation in viscoelastic fluid flow through porous media ↓ Viscoelastic fluids, such as polymer or micellar solutions, exhibit a unique combination of viscous and elastic responses when subjected to deformation. When these fluids flow through porous media, encountered in various applications from enhanced oil recovery to groundwater remediation, the behavior becomes notably intricate and rich in physical phenomena compared to simple Newtonian fluids. Particularly intriguing is the transition of the flow field from steady and laminar to unsteady, chaotic, or even turbulent-like states as flow rate or fluid elasticity increases within the porous system. While prior recent series of experiments have observed this transition in the flow of viscoelastic fluids through porous media, a comprehensive understanding of the underlying mechanism has been lacking. Initially, explanations from the experiments focused on the role of number of stagnation points in the porous media exposed to the flow. However, our recent extensive numerical studies have provided a different perspective. In particular, we have revealed that this flow transition is not solely determined by the number of stagnation points but rather by the formation of preferential paths or lanes within the porous media as viscoelastic fluids flow through it. These preferential paths or lanes further facilitate the formation of birefringent strands within the porous media which ultimately drives this chaotic flow inside the porous media. Understanding this mechanism holds significant promise for practical applications, notably in enhanced oil recovery. By gaining insights into how preferential paths influence the transition from steady to chaotic flow, we can better control the displacement of oil from reservoirs using viscoelastic fluids, thereby maximizing the recovery processes. (Online) |

10:00 - 10:30 |
Sibani Lisa Biswal: Bubble Trapping in Flowing Foam in Porous Media ↓ The rheology of foam plays a crucial role in various industrial applications, including enhanced oil recovery and carbon sequestration. Understanding the complex physicochemical processes governing foam behavior in porous media is essential for predicting their flow in natural environments. In this study, we utilize microfluidic devices mimicking natural sandstone porous media to investigate the impact of gas types on foam characteristics. Specifically, we focus on nitrogen foam with varying surfactant concentrations. By directly observing phenomena such as foam bubble trapping and lamellae division, we assess foam texture, stability, and phase mobility under quasi-steady-state conditions. We analyze pressure drops, apparent viscosities, and foam mobility, correlating them with variations in foam texture. Our results reveal that trapped foam significantly contributes to apparent viscosity, while the size of flowing foam bubbles also influences viscosity. Additionally, we combine high-speed imaging and image processing to provide a detailed understanding of bubble trapping and flow diversion in relation to foam quality and flow rate. (TCPL 201) |

10:30 - 11:00 | Coffee Break (TCPL Foyer) |

11:00 - 11:30 |
Amir Pahlavan: Bacterial chemotaxis and dispersion in porous media ↓ Chemotaxis is the ability of bacteria to sense and react to different chemicals present in their environment. This enhances their ability to bias their motility towards favorable environments and colonize new regions or move away from toxic environments. Here, using microfluidic experiments and numerical simulations, we study the chemotactic response of Escherichia coli to ephemeral nutrient plumes emerging from hydrogel sources in the presence of a heterogeneous flow field. We demonstrate how the interplay between flow and chemotaxis leads to the preferential accumulation of bacteria around the localized nutrient sources. We discuss the implications of our observations in the context of bacterial chemotaxis and colonization of dissolving organic matter in the oceans and soil. (Online) |

11:30 - 12:00 |
Marcel Moura: Linking anomalous diffusion and rheology for a droplet spreading in a corner: A rheometer for power-law liquids ↓ We investigate the dynamics of a power-law fluid spreading within a corner geometry, influenced by both capillary and viscous forces. Our study reveals that the droplet spreading follows an anomalous diffusion scaling, with the temporal evolution intricately tied to the rheological properties of the fluid. Leveraging these findings, we devise an experimental framework enabling the determination of the power-law coefficient, effectively establishing a simple rheometer tailored for power-law fluids. (TCPL 201) |

12:00 - 12:30 |
Outi Tammisola: Elastoviscoplastic fluid flow through randomized porous media ↓ A numerical study of yield-stress fluids flowing in porous media is presented. The porous media is randomly constructed by non-overlapping mono-dispersed circular obstacles. Two class of rheological models are investigated: elastoviscoplastic fluids (Saramito model) and viscoplastic fluids (Bingham model). A wide range of practical Weissenberg and Bingham numbers is studied at three different levels of porosities of the media. The emphasis is on revealing some physical transport mechanisms of yield-stress fluids in porous media when the elastic behaviour of this kind of fluids is incorporated. Thus, computations of elastoviscoplastic fluids are performed and are compared with the viscoplastic fluid flow properties. At a constant Weissenberg number, the pressure drop increases both with the Bingham number and the solid volume fraction of obstacles. However, the effect of elasticity is less trivial. At low Bingham numbers, the pressure drop of an elastoviscoplastic fluid increases compared to a viscoplastic fluid, while at high Bingham numbers we observe drag reduction by elasticity. At the yield limit (i.e. infinitely large Bingham numbers), elasticity of the fluid systematically promotes yielding: elastic stresses help the fluid to overcome the yield stress resistance at smaller pressure gradients. We observe that elastic effects increase with both Weissenberg and Bingham numbers. In both cases, elastic effects finally make the elastoviscoplastic flow unsteady, which consequently can result in chaos and turbulence. (TCPL 201) |

12:30 - 13:30 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

14:00 - 18:00 | Free Afternoon (Banff National Park) |

17:30 - 19:30 |
Dinner ↓ |

Thursday, July 18 | |
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07:00 - 09:00 |
Breakfast ↓ |

09:00 - 09:30 |
Ian Frigaard: Let me tell you about my problems? ↓ In the hope of both intellectual and cathartic relief, I will outline 5 areas of current and enduring interest, each involving yield stress fluids in porous media geometries.
A. From gas displacement to bubble percolation
B. Squeeze cementing flows along micro-annuli
C. Filling cavities with slurries: transitions from suspension flows to filtration
D. Continuum and network modelling of porous media: modelling and numerical methods
E. Pore space geometry and yield stress fluids: can we ever really have general closure expressions? (TCPL 201) |

09:30 - 10:00 |
James Hewett: Modelling and estimating the viscoelastic properties of blood clots ↓ Modelling the dynamic behaviour of blood clots using numerical simulations is a difficult task which includes multiphysics, multiscale, and complex fluid dynamics with fluid-structure interactions. Here, we focus on estimating the viscoelastic properties of a developed blood clot, using sensor data such as pressure along the blood vessel wall. A viscoelastic porous medium was assumed for the blood clot because of its fibrous network. The fluid-structure interaction between the clot and blood flow was modelled in the Eulerian reference frame using the Cahn-Hilliard and Navier-Stokes equations. Two viscoelastic properties of the blood clot, namely the viscosity and relaxation time, were estimated by using an inverse model with a twin experiment approach. Synthetic results were first generated with a prescribed viscosity and relaxation time, and then second an optimiser was employed for minimising the difference between the simulated and synthetic blood clot interface. (TCPL 201) |

10:00 - 10:30 |
Claudio Fonte: Network modelling of viscoplastic fluid flow in randomly disordered porous media ↓ Yield-stress fluids are widely used in various industrial applications. Understanding their flow behaviour in complex geometries such as porous media can offer solutions to challenges in processes such as enhanced oil recovery, and membrane emulsification in the food and pharmaceutical industries. However, the high computational cost associated with flow simulations using advanced tools like Computational Fluid Dynamics (CFD) presents a significant challenge for studying large-scale systems.
In this work, we address this challenge by developing a fully predictive network model for simulating the flow of fluids governed by the Hershel-Bulkley constitutive relation through 2D disordered porous media. The model employs Voronoi tessellation to generate a network of interconnected nodes and extracts additional geometric parameters that capture the domain's topology. We then obtain the pressure and velocity distributions in the network by numerically solving a system of nonlinear equations derived from 1D mass and momentum conservation equations. Moreover, the model is also able to account for wall slip, a common phenomenon in the flow of yield-stress fluids.
The results obtained with the network model show strong agreement with CFD simulations performed in-house and others reported in the literature. We are able to achieve solutions in minutes on a single CPU core, in stark contrast to the days or weeks required by CFD simulations utilising multiple CPU cores. Finally, we use the network model to study scaling laws for the relationship between the pressure drop across the network and the flow rate for different media porosities and Bingham numbers. (TCPL 201) |

10:30 - 11:00 | Coffee Break (TCPL Foyer) |

11:00 - 11:30 |
Prabir Daripa: Some aspects of modeling chemical enhanced oil recovery and fracturing instability in complex fluids ↓ In this talk, we will present models of shear-thinning polymer-surfactant flooding in porous media with and without non-uniform mixing and dispersion. Novel numerical methods and results, some surprising, with these models will be presented. Time permitting, we will present some analytical results on fracturing instability involving displacement of complex fluids in rectilinear Hele-Shaw cells. (TCPL 201) |

11:30 - 12:00 |
Vinicius Gustavo Poletto: Simulation of generalized Newtonian Fluid Flow in porous media applied to drilling fluid hydraulics and lost circulation control ↓ Well drilling hydraulics poses interesting challenges in modeling and simulating the non-Newtonian fluid flow to represent the drilling fluid in contact with the porous substrate. This works presents studies concerning the modeling and simulation of the invasion phenomena, also known as lost circulation, in which the drilling fluids get in contact with a highly permeable formation or a substrate containing discrete fractures, being drained to the formation and not returning to the surface as expected. First, LBM modeling of power law and Bingham fluids is modeled in a horizontal channel in contact with a heterogeneous porous medium. The effect of porosity, obstacles density, Reynolds number, power index and Bingham number is evaluated in terms of friction coefficient and velocity profiles. Second, the experimental characterization of power law and Herschel Bulkley fluids in a fractured channel and a partially-porous-fractured channel is numerically reproduced by CFD to visualize the velocity magnitude, pressure, and viscosity fields. A counter measure to lost circulation is the use of particulate material to seal fractures and porous regions. Third, the modeling and simulation of power law fluid with discrete particles for particle packing in fractures and heterogeneous porous medium elucidates useful combination of fluid rheology and particulate size to reduce the invasion flow rate. Finally, the numerical results also comprise a data base in which a machine learn-based code is under development to predict scenarios prone to severe lost circulation. (TCPL 201) |

12:00 - 12:30 |
Santanu Sinha: Non-linear growth of viscous fingers in two-phase flow of immiscible Newtonian fluids in porous media ↓ Immiscible displacement of a Newtonian fluid by a different fluid of lower viscosity inside a porous medium creates viscous instabilities, and produces a type of patterns known as viscous fingers. Growth of such fingers has widely been assumed to exhibit a linear Laplacian growth behavior where the velocities of advancing fronts depend linearly on local pressure gradient [1]. Viscous fingers in continuum Hele-Shaw cell observed by Saffman and Taylor [2], and diffusion limited aggregates exhibit such linear growth. However, in a recent computational study [3] with a dynamic pore-network model [4] we measured the local finger growth as a function of the local pressure drop in a network with unform distribution of pore-radii, and showed that the there exists a regime at lower capillary numbers where the two quantities show a non-linear power-law type relationship which then crosses over to a linear regime at high capillary numbers. Such a characteristics is also supported by an older experiment [5]. By assuming that the origin of the non-linear growth is the disorder in the capillary barriers at the pores, one may derive the non-linear relationship by integrating the local viscous-to-capillary pressure difference over the distribution of capillary thresholds which is related to the pore-radii distribution. In order to investigate this further, we therefore simulated finger growth in networks with different distributions of pore-radii and measured the growth. We in this talk will illustrate how the exponent related to the fingers' growth law depends on the functional form of that distribution and will present some counter-intuitive results.
[1] L. Paterson, Phys. Rev. Lett. 52, 1621 (1984).
[2] P. G. Saffman and G. I. Taylor, Proc. R. Soc. London 245, 312 (1958).
[3] S. Sinha, Y. Méheust, H. Fyhn, S. Roy and Alex Hansen, Phys. Fluids 36, 033309 (2024).
[4] G. Løvoll, Y. Méheust, R. Toussaint, J. Schmittbuhl and K. J. Måløy, Phys. Rev. E 70, 026301 (2004). (Online) |

12:30 - 13:30 |
Lunch ↓ |

14:00 - 14:30 |
Erika Eiser: Microrheology of Hydrogels ↓ Measuring the viscoelastic response of complex fluids to a shear stress is often challenging, because the resulting shear strain can be very weak or shows only at short relaxation times. Light-scattering techniques such as Dynamic Light Scattering (DLS) and Diffusing-Wave Spectroscopy (DWS) overcome these difficulties and allow us to extract the Mean-Square Displacement and the elastic and loss moduli of complex liquids as function of frequency reaching from a few up to MHz (1). I will show how DNA nanostars can undergo a transition from a Newtonian liquid to an equilibrium gel as function of temperature (2,3). Moreover, using DWS the formation of inter-percolating colloidal networks and their viscoelasticity can be studied.
1. R. Liu, A. Caciagli, J. Yu, X. Tang, R. Ghosh, E. Eiser ‘Dynamic Light Scattering based microrheology of End-functionalised triblock copolymer solutions’ Polymers 15, 481 (2023).
2. I.D. Stoev, T. Cao, A. Caciagli, J. Yu, R. Liu, R. Ghosh, T. O’Neill, D. Liu, E. Eiser “On the Role of Flexibility in Linker-Mediated DNA Hydrogels” Soft Matter 16, 990 (2020).
3. Z. Xing, A.Caciagli, T. Cao, I. Stoev, M. Zupkauskas, T. O'Neill, T. Wenzel, R. Lamboll, D. Liu, E. Eiser “Microrheology of DNA-Hydrogels” PNAS 115, 8137 (2018). (TCPL 201) |

14:30 - 15:00 |
Mahdi Izadi: Dynamics and Stoppage Mechanisms in Viscoplastic Suspensions: Modelling Capillary Forces, Pore Blockage, and Yield Stress Effects ↓ Motivated by the problem of sealing fractured rocks with a cement slurry, we study the flow of a viscoplastic suspension (representing the cement slurry) into a randomized Hele-Shaw geometry (representing a varying aperture fracture or microannulus), which is initially filled with Newtonian fluid. The aim is to investigate the flow dynamics and, more specifically, the stoppage of the invading fluid as it enters further into the fracture. Stoppage mechanisms for such flows include yield stress, pore blockage and capillary forces. To address these effects, we have created a 2D model based on continuum mechanics to solve the flow issue. Our model uses the augmented Lagrangian approach to locate the yield surface. We then account for pore blockage by adjusting the fluid's rheology, according to the size of particles in the suspension and the size of the fracture. Next we have incorporated capillary pressure into our model. (TCPL 201) |

15:00 - 15:30 |
Parisa Mirbod: Viscoelastic Flow Stability overlying Porous Media: Insights from Linear Stability Analysis ↓ This talk presents the complex dynamics of Oldroyd-B fluid within a plane Poiseuille flow confined by porous layers, aiming to unravel the intricate interplay of elasticity, inertia, and flow over the porous region. We consider rigid, homogeneous, isotropic porous layers and explore porous materials of small permeability, where inertial effects can be neglected. The flow stability is governed by key non-dimensional parameters including Reynolds number, permeability parameter, ratio of porous layer thickness to channel width, Weissenberg number, and ratio of solvent to the solution viscosity. Our investigation, maintaining fixed values for porosity, employs a modified Darcy-Brinkman-Oldroyd-B model to establish a comprehensive set of equations and boundary conditions tailored to our specific case.
We validate our results by recovering linear stability analysis outcomes of both viscoelastic and Newtonian flows in relevant scenarios. Through analysis of eigenspectra across varied dimensionless parameters, we observe significant impacts of small porous layer permeability on the elasto-inertial eigenspectrum, leading to instability in viscoelastic channel flow. Also, the emergence of new damped modes, attributed to the presence of porous layers with polymer additives, is noted. Moreover, incorporation of porous walls within the channel initiates the unstable mode from the wall mode, diverging from the behavior observed in viscoelastic flows constrained by smooth walls. Further investigation into neutral stability curves unveils a non-monotonic effect of Weissenberg number on flow stability, with higher permeabilities shifting the Wi threshold to lower values. Particularly, for higher Wi values and high permeabilities, the flow is stabilized compared to Newtonian fluid. This talk provides valuable insights into the complex dynamics of porous media flows, with implications for various engineering applications involving viscoelastic fluids and porous structures. (TCPL 201) |

15:30 - 16:00 | Coffee Break (TCPL Foyer) |

16:00 - 16:30 |
Gwynn Elfring: Viscoelastic propulsion due to rotating bodies ↓ An arbitrary axisymmetric body generates purely circular streamlines when rotating about its axis of symmetry in a Newtonian fluid at low Reynolds numbers. In viscoelastic fluids that symmetry is broken due to elastic hoop stresses that generate axial flow as is exemplified in the well-known rod-climbing effect. In this talk, we discuss the effect of shape on the purely viscoelastic axial flows generated by rotating bodies, and the propulsive force that is generated if the body is not front-back symmetric. In particular we investigate optimal shapes that generate the strongest viscoelastic propulsion, and reveal flow reversal and non-symmetric stationary states. (TCPL 201) |

16:30 - 17:00 |
Christopher Browne: Harnessing elastic instabilities for enhanced mixing and reaction kinetics in porous media ↓ A wide range of environmental, industrial, and energy processes rely on reactive transport in disordered 3D porous media, but laminar flow under strong geometric confinement (Re<<1) typically imposes a fundamental limit on transport. We report a novel technique to mimic turbulent-enhanced reactivity by the addition of dilute high molecular weight polymers, which induce an elastic flow instability. By directly visualizing the flow in transparent 3D porous media, we demonstrate that the flow exhibits chaotic spatiotemporal fluctuations reminiscent of inertial turbulence, despite the vanishingly small Reynolds number. These pore-scale velocity fluctuations can enhance mixing by stretching and folding solute gradients exponentially in time-analogous to turbulent Batchelor mixing. We observe a dramatic 3 x reduction in the required mixing length and 6 x improvement in the dispersion of the concentration gradient, suggesting a cooperation between the elastic instability and the laminar chaotic advection inherent to disordered 3D porous media. We show these two mixing mechanisms can be modeled with additive independent mixing rates, representing a dramatic conceptual simplification. We then extend these results to reactive mixing, accelerating a model reaction by 5 x while simultaneously increasing throughput by 20 x -circumventing the traditional trade-off between throughput and reactor length. Our results thus provide the first demonstration, to our knowledge, that elastic flow instabilities can provide turbulent-like enhancements in chemical reaction rates, which can operate cooperatively with laminar chaotic advection in industrially-relevant geometries. (TCPL 201) |

18:00 - 19:30 |
Dinner ↓ |

Friday, July 19 | |
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07:00 - 08:30 |
Breakfast ↓ |

08:30 - 09:00 |
José Soares de Andrade Jr.: Flow through three-dimensional self-affine fractures ↓ We investigate through numerical simulations of the Navier-Stokes equations the influence of the surface roughness on the fluid flow through fracture joints. Using the Hurst exponent H to characterize the roughness of the self-affine surfaces that constitute the fracture, our analysis reveals the important interplay between geometry and inertia on the flow. Precisely, for low values of Reynolds numbers Re, we use Darcy’s law to quantify the hydraulic resistance G of the fracture and show that its dependence
on H can be explained in terms of a simple geometrical model for the tortuosity τ of the channel. At sufficiently high values of Re, when inertial effects become relevant, our results show that nonlinear corrections up to third-order to Darcy’s law are
approximately proportional to H. These results imply that the resistance G to the flow follows a universal behavior by simply rescaling it in terms of the fracture resistivity and using an effective Reynolds number, namely, Re/H. We also show the presence of quasi one-dimensional channeling, even considering the absence of shear displacement between upper and lower surfaces of the self-affine fracture. (Online) |

09:00 - 09:30 |
J. Esteban López-Aguilar: Non-Newtonian Flow Modelling in Fractal Porous Media ↓ In this work, we propose a new fractal toy-model dealing with the flow of Newtonian and non-Newtonian fluids in porous media. The porous medium is hypothesized as a bundle of tortuous capillaries, i.e. tortuous tubes grouped in the sample cross-sectional surface distributed according to a fractal scaling. Two fractal scalings are applied: one to the length of the capillaries (making them tortuous) and another to the pore distribution in the sample. Hence, to calculate the flow through the samples promoted by constant pressure-gradients, the sum of all individual-pore flow-contributions is taken into account. Such flow problem is solved using a numerical algorithm constructed in the Generalized Newtonian Fluid constitutive-approach rationale. This predictive tool, using a fourth-order Runge-Kutta ordinary differential-equation (ODE) solver, provides solutions to the momentum balance-equation modified by a fractal scaling, applied to a single pore that accounts for its length, and subsequently integrated numerically over the contribution of each pore, to obtain the total volumetric flow in the porous medium. The algorithm is validated and contrasted against analytically-solvable Newtonian, power-law and Bingham base cases, whilst flow predictions for poly (n-methyl cellulose) (PMC)-aqueous solutions in a porous medium are extracted using the Herschel-Bulkley and a hybrid Carreau-Yasuda-Bingham equations-of-state. The new implementation of these fractal scalings generates solutions that concur with predictions published elsewhere, and, in some cases, improve over some previously-reported predictions for flow of visco-plastic PMC aqueous solutions reported by Sochi (2010) and Park (1972), for which the minimum pressure-drop for flow-inception is estimated accurately, in contrast to other approaches. (TCPL 201) |

09:30 - 10:00 |
Quirine Krol: Microscale fluid fluctuations during drainage and imbibition in porous media measured with rapid NMR profiling: The non-linear relation between macroscopic flow rate and local flow dynamics ↓ Drainage, imbibition, and steady-state two-phase flow in porous media have a wide range of applications in physics, chemistry, and engineering. Experimental methods to capture these phenomena are often limited to two-dimensional setups, rely on refractive index matching, and/or are too slow to capture the dynamics at a resolution relevant to the physics at the pore scale. We utilize nuclear magnetic resonance methods to retrieve 1D spin-echo intensity and phase-angle profiles with a temporal and spatial resolution of 17 ms and 70 μm, respectively.
During a 34-second-long drainage and imbibition of a cylindrical tank filled with 1 mm and 3 mm beads, we capture saturation and flow dynamics associated with velocity fluctuation-induced signal attenuation. These signal attenuations show typical time and length scales during a single experiment. We demonstrate that when flow rates increase, the spacing between fluctuating dynamics events decreases, but the duration remains relatively constant. When the flow rates decrease over time, we observe less attenuation from low saturation regions, giving more detail on individual events, but also showing less attenuation between events suggesting that the fluid relaxes at lower flow rates between bursts. We observe that during drainage, the extent and the amplitude of the fluctuations are much higher than during imbibition, which might indicate that the measured fluctuation events are related to Haines jumps. We discuss the results in relation to flow rate, variable pore size, and viscosity of the invading fluid. These measurements can thus help with the qualitative interpretation of the physical origins associated with various types of local and long-range correlated flow dynamics in two-phase flow in porous media. (TCPL 201) |

10:00 - 11:00 |
Coffee and checkout by 11AM ↓ 5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM. (TCPL Foyer) |

11:15 - 12:00 | Ian Frigaard: Workshop wrapping up, plans for future, etc. (TCPL 201) |

12:00 - 13:30 | Lunch and leave! (Vistas Dining Room) |