PDEs on deformable domains: Boundary Arbitrary Lagrangian-Eulerian (BALE) and Deformable Boundary Perturbation (DBP) methods

Rivero-Rodriguez, Javier; Perez-Saborid, Miguel; Scheid, Benoit

doi:10.48550/arxiv.1810.10001

Cited by 3 publications

(3 citation statements)

References 22 publications

(45 reference statements)

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“…In addition, the mesh was symmetrical with respect to x = 0 and the skewness of the mesh was always around 0.93 to ensure a good mesh quality. The steady results were verified by comparing the numerical solution with the theoretical unidirectional flow solution given above in (2.13) and (2.14) and with the results given by Rivero-Rodriguez, Perez-Saborid & Scheid (2018).…”

Section: Numerical Methods and Stability Of The Interfacementioning

confidence: 53%

Solid particles moving parallel to a deformable liquid–liquid interface in a micro-channel: migration forces

2022

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This work focuses on the dynamics of a train of solid particles, separated by a distance $L$ , flowing near a deformable interface formed by two co-flowing immiscible fluids in a microchannel of height $h$ . Our study includes a systematic analysis of the influence of the governing parameters (fluids viscosity ratio, interface and particle positions, Reynolds $Re$ and capillary $Ca$ numbers and the inter-particle distance $L$ ) on the hydrodynamic force $f$ exerted on the particle. In the pure inertial regime with non-deformable interfaces $Ca=0$ , the particle is driven towards the wall (interface) when the particle is close to the interface (wall). Up to three neutral equilibrium positions $f=0$ , two of them stable, are found in this limit. The contrary is obtained in the pure capillary regime $Re=0$ . In this limit, we also carried out an asymptotic analysis in the distinguished limits of very large and very small surface tension. In the latter case, the amplitude of the interface deformation induced by the particle is large, comparable to its diameter, but its influence is limited to a small region upstream and downstream of the particle. In the limit of very large surface tension, the amplitude of the interface deformation is small but the presence of the particle modifies the shape of the interface in a region of length $2\lambda$ , much larger than the particle diameter $d$ . The parameter $\lambda$ , introduces an additional characteristic length that determines the asymptotic behaviour of the flow properties in the limit of large surface tension.

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Section: Numerical Methods and Stability Of The Interfacementioning

confidence: 53%

Solid particles moving parallel to a deformable liquid–liquid interface in a micro-channel: migration forces

2022

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“…Consequently, the boundary conditions (2.3), (2.4a) and (2.4b) cannot be imposed explicitly. To circumvent this difficulty, we followed the methodology developed by Rivero-Rodriguez, Perez-Saborid & Scheid (2018) to write these boundary conditions referred to the unperturbed particle's position x p,0 . This process resulted in (B3), ( B5) and (B6), respectively, as outlined in detail in Appendix B.…”

Section: Boundary Conditions Of the Stability Problemmentioning

confidence: 99%

Influence of immersed particles on the stability of the liquid–liquid interface in a two-layer channel flow

Ruiz-Martín,

Rivero-Rodríguez,

Sánchez-Sanz

2024

J. Fluid Mech.

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The current study investigates the global linear stability of a two-layer channel flow with a train of solid particles flowing near the liquid–liquid interface. Three different mechanisms of instability (shear, interfacial and migration modes) are identified, and their interactions are examined. The interfacial instability, associated with the viscosity jump at the liquid–liquid interface, is found to be coupled to the migration of the particle. The stability of the flow configuration is evaluated for various governing parameters, including fluid viscosities and flow rate ratios, particle position, inter-particle distance, and Reynolds and capillary numbers. Our numerical results are compared with the particle-free flow configuration, indicating that the presence of the particle in the more viscous fluid promotes the destabilization of the interface. Remarkably, under certain flow parameters, the presence of the particle stabilizes the interface when flowing in the less viscous liquid. The impact of particles is more significant as the capillary number increases or the Reynolds number decreases.

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“…Note that the jet breaking is intrinsically transient as it results from the Rayleigh-Plateau instability but the droplet generation, as mentioned above, can be quasi-static in the limit of low dispersed flow rate. Van Brummelen et al [42] having shown that solving the transient Navier-Stokes equation to obtain the steady solution is often inefficient, Rivero-Rodriguez et al [43] have developed the Boundary Arbitrary Lagrangian-Eulerian (BALE) method to facilitate the solution of steady Navier-Stokes equations with free surfaces. Rivero-Rodriguez and Scheid [44,45] have then applied this method to study the dynamics of deformable and off-centered bubbles in microchannels, allowing for exhaustive parametric analysis, taking the advantage of continuation methods suitable for tracking stationary solutions.…”

Section: Introductionmentioning

confidence: 99%

Raydrop : a universal droplet generator based on a non-embedded co-flow-focusing

Dewandre,

Rivero-Rodriguez,

Vitry

et al. 2020

Preprint

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Most commercial microfluidic droplet generators rely on the planar flow-focusing configuration implemented in polymer or glass chips. The planar geometry, however, suffers from many limitations and drawbacks, such as the need of specific coatings or the use of dedicated surfactants, depending on the fluids in play. On the contrary, and thanks to their axisymmetric geometry, glass capillary-based droplet generators are a priori not fluid-dependent. Nevertheless, they have never reached the market because their assembly requires art-dependent and not scalable fabrication techniques. Here we present a new device, called Raydrop, based on the alignment of two capillaries immersed in a pressurized chamber containing the continuous phase. The dispersed phase exits one of the capillaries through a 3D-printed nozzle, placed in front of the extraction capillary for collecting the droplets. This non-embedded implementation of an axisymmetric flow-focusing is referred to co-flow-focusing. Experimental results demonstrate the universality of the device in terms of the variety of fluids that can be emulsified, as well as the range of droplet radii that can be obtained, without neither the need of surfactant nor coating. Additionally, numerical computations of the Navier-Stokes equations based on the quasi-steadiness assumption are shown to correctly predict the droplet radius in the dripping regime and the dripping-jetting transition when varying the geometrical and fluid parameters. The monodispersity ensured by the dripping regime, the robustness of the fabrication technique, the optimization capabilities from the numerical modeling and the universality of the configuration confer to the Raydrop technology a very high potential in the race towards high-throughput droplet generation processes.

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PDEs on deformable domains: Boundary Arbitrary Lagrangian-Eulerian (BALE) and Deformable Boundary Perturbation (DBP) methods

Cited by 3 publications

References 22 publications

Solid particles moving parallel to a deformable liquid–liquid interface in a micro-channel: migration forces

Solid particles moving parallel to a deformable liquid–liquid interface in a micro-channel: migration forces

Influence of immersed particles on the stability of the liquid–liquid interface in a two-layer channel flow

Raydrop : a universal droplet generator based on a non-embedded co-flow-focusing

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