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1. Nonlinear Rayleigh–Taylor instability of inhomogeneous incompressible geophysical fluids with partial dissipation

   https://doi.org/10.1088/1361-6544/adca80  

   Fan, Yanlong; Han, Daozhi; Wang, Quan; Xing, Chao (April 2025, Nonlinearity)

   Abstract This article examines the Rayleigh–Taylor instability in an rotating inhomogeneous, incompressible fluid with partial viscosity. First, using the modified variational method, we demonstrate the existence of an exponentially growing normal mode in $H^k,k\text{⩾}0$and establish the instability of the linearised problem. Then, we obtain a nonlinear energy estimate for the problem with small initial data. In this process, we employ an innovative method to derive energy estimates for both density and velocity, effectively addressing the challenges posed by partial viscosity. Third, we prove the existence of a classical solution forH³initial data, provided it satisfies a compatibility condition. Finally, by integrating the results of the previous steps, we establish the nonlinear instability of the system in the Hadamard sense. 

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2. Second-gradient models for incompressible viscous fluids and associated cylindrical flows

   Balitactac, C; Rodriguez; C (May 2025, arxiv_25_b)

   We introduce second-gradient models for incompressible viscous fluids, building on the framework introduced by Fried and Gurtin. We propose a new and simple constitutive relation for the hyperpressure to ensure that the models are both physically meaningful and mathematically well-posed. The framework is further extended to incorporate pressure-dependent viscosities. We show that for the pressure-dependent viscosity model, the inclusion of second-gradient effects guarantees the ellipticity of the governing pressure equation, in contrast to previous models rooted in classical continuum mechanics. The constant viscosity model is applied to steady cylindrical flows, where explicit solutions are derived under both strong and weak adherence boundary conditions. In each case, we establish convergence of the velocity profiles to the classical Navier-Stokes solutions as the model's characteristic length scales tend to zero. 

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3. Energy growth for systems of coupled oscillators with partial damping

   https://doi.org/10.1088/1361-6544/adc8ee  

   Dolgopyat, Dmitry; Fayad, Bassam; Koralov, Leonid; Yan, Shuo (April 2025, Nonlinearity)

   Abstract We consider two interacting particles on the circle. The particles are subject to stochastic forcing, which is modeled by white noise. In addition, one of the particles is subject to friction, which models energy dissipation due to the interaction with the environment. We show that, in the diffusive limit, the absolute value of the velocity of the other particle converges to the reflected Brownian motion. In other words, the interaction between the particles is asymptotically negligible in the scaling limit. The proof combines averaging for large energies with large deviation estimates for small energies. 

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4. Magnetic Resonance Imaging of Multi‐Phase Lava Flow Analogs: Velocity and Rheology

   https://doi.org/10.1029/2023JB026464  

   Birnbaum, Janine; Zia, Wasif; Bordbar, Alireza; Lee, Ray F.; Boyce, Christopher M.; Lev, Einat (May 2023, Journal of Geophysical Research: Solid Earth)

   Abstract The rheology of lavas and magmas exerts a strong control on the dynamics and hazards posed by volcanic eruptions. Magmas and lavas are complex mixtures of silicate melt, suspended crystals, and gas bubbles. To improve the understanding of the dynamics and effective rheology of magmas and lavas, we performed dam‐break flow experiments using suspensions of silicone oil, sesame seeds, and N₂O bubbles. Experiments were run inside a magnetic resonance imaging (MRI) scanner to provide imaging of the flow interior. We varied the volume fraction of sesame seeds between 0 and 0.48, and of bubbles between 0 and 0.21. MRI phase‐contrast velocimetry was used to measure liquid velocity. We fit an effective viscosity to the velocity data by approximating the stress using lubrication theory and the imaged shape of the free surface. In experiments with both particles and bubbles (three‐phase suspensions), we observed shear banding in which particle‐poor regions deform with a lower effective viscosity and dominate flow propagation speed. Our observations demonstrate the importance of considering variations in phase distributions within magmatic fluids and their implications on the dynamics of volcanic eruptions. 

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5. Dispersion of solids in fracturing flows of yield stress fluids

   https://doi.org/10.1017/jfm.2017.465  

   Hormozi, S.; Frigaard, I. A. (November 2017, Journal of Fluid Mechanics)

   Solids dispersion is an important part of hydraulic fracturing, both in helping to understand phenomena such as tip screen-out and spreading of the pad, and in new process variations such as cyclic pumping of proppant. Whereas many frac fluids have low viscosity, e.g. slickwater, others transport proppant through increased viscosity. In this context, one method for influencing both dispersion and solids-carrying capacity is to use a yield stress fluid as the frac fluid. We propose a model framework for this scenario and analyse one of the simplifications. A key effect of including a yield stress is to focus high shear rates near the fracture walls. In typical fracturing flows this results in a large variation in shear rates across the fracture. In using shear-thinning viscous frac fluids, flows may vary significantly on the particle scale, from Stokesian behaviour to inertial behaviour across the width of the fracture. Equally, according to the flow rates, Hele-Shaw style models give way at higher Reynolds number to those in which inertia must be considered. We develop a model framework able to include this range of flows, while still representing a significant simplification over fully three-dimensional computations. In relatively straight fractures and for fluids of moderate rheology, this simplifies into a one-dimensional model that predicts the solids concentration along a streamline within the fracture. We use this model to make estimates of the streamwise dispersion in various relevant scenarios. This model framework also predicts the transverse distributions of the solid volume fraction and velocity profiles as well as their evolutions along the flow part. 

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