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1. A diffuse domain method for two-phase flows with large density ratio in complex geometries

   https://doi.org/10.1017/jfm.2020.790  

   Guo, Zhenlin; Yu, Fei; Lin, Ping; Wise, Steven; Lowengrub, John (January 2021, Journal of Fluid Mechanics)

   null (Ed.)

   We present a quasi-incompressible Navier–Stokes–Cahn–Hilliard (q-NSCH) diffuse interface model for two-phase fluid flows with variable physical properties that maintains thermodynamic consistency. Then, we couple the diffuse domain method with this two-phase fluid model – yielding a new q-NSCH-DD model – to simulate the two-phase flows with moving contact lines in complex geometries. The original complex domain is extended to a larger regular domain, usually a cuboid, and the complex domain boundary is replaced by an interfacial region with finite thickness. A phase-field function is introduced to approximate the characteristic function of the original domain of interest. The original fluid model, q-NSCH, is reformulated on the larger domain with additional source terms that approximate the boundary conditions on the solid surface. We show that the q-NSCH-DD system converges to the q-NSCH system asymptotically as the thickness of the diffuse domain interface introduced by the phase-field function shrinks to zero ( $$\epsilon \rightarrow 0$$ ) with $$\mathcal {O}(\epsilon )$$ . Our analytic results are confirmed numerically by measuring the errors in both $$L^{2}$$ and $$L^{\infty }$$ norms. In addition, we show that the q-NSCH-DD system not only allows the contact line to move on curved boundaries, but also makes the fluid–fluid interface intersect the solid object at an angle that is consistent with the prescribed contact angle. 

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2. Analysis and finite element approximation of a diffuse interface approach to the Stokes–Biot coupling

   https://doi.org/10.1093/imanum/draf002  

   Aznaran, Francis_R_A; Bukač, Martina; Muha, Boris; Salgado, Abner_J (March 2025, IMA Journal of Numerical Analysis)

   Abstract We consider the interaction between a poroelastic structure, described using the Biot model in primal form, and a free-flowing fluid, modelled with the time-dependent incompressible Stokes equations. We propose a diffuse interface model in which a phase field function is used to write each integral in the weak formulation of the coupled problem on the entire domain containing both the Stokes and Biot regions. The phase field function continuously transitions from one to zero over a diffuse region of width $$\mathcal{O}(\varepsilon)$$ around the interface; this allows the weak forms to be integrated uniformly across the domain, and obviates tracking the subdomains or the interface between them. We prove convergence in weighted norms of a finite element discretization of the diffuse interface model to the continuous diffuse model; here the weight is a power of the distance to the diffuse interface. We, in turn, prove convergence of the continuous diffuse model to the standard, sharp interface, model. Numerical examples verify the proven error estimates, and illustrate application of the method to fluid flow through a complex network, describing blood circulation in the circle of Willis. 

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3. Chemotaxis in shear flow: Similarity solutions of the steady‐state chemoattractant and bacterial distributions

   https://doi.org/10.1002/aic.16713  

   Shim, Suin; Stone, Howard_A; Ford, Roseanne_M (July 2019, AIChE Journal)

   Abstract When chemotactic bacteria are exposed to a concentration gradient of chemoattractant while flowing along a channel, the bacteria accumulate at the interface between the chemoattractant source and bacterial suspension. Assuming that the interface is no‐slip, we can apply the shear flow approximation near the no‐slip boundary and solve a steady‐state convection‐diffusion model for both chemoattractant and bacterial concentrations. We suggest similarity solutions for the two‐dimensional problem and identify a critical length scaleη_cfor bacteria chemotaxis in a given concentration gradient. The analysis identifies three dimensionless groups representing, respectively, chemotactic sensitivity, the chemotaxis receptor constant, and the bacteria diffusion coefficient, which typically show coupled effects in experimental systems. We study the effect of the dimensionless groups separately and provide understanding of the system involving shear flow and chemotaxis. 

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4. Simulating hindered grain boundary diffusion using the smoothed boundary method

   https://doi.org/10.1088/1361-651X/ad4d0b  

   Hanson, Erik; Andrews, W_Beck; Powers, Max; Jenkins, Kaila_G; Thornton, Katsuyo (June 2024, Modelling and Simulation in Materials Science and Engineering)

   Abstract Grain boundaries can greatly affect the transport properties of polycrystalline materials, particularly when the grain size approaches the nanoscale. While grain boundaries often enhance diffusion by providing a fast pathway for chemical transport, some material systems, such as those of solid oxide fuel cells and battery cathode particles, exhibit the opposite behavior, where grain boundaries act to hinder diffusion. To facilitate the study of systems with hindered grain boundary diffusion, we propose a model that utilizes the smoothed boundary method to simulate the dynamic concentration evolution in polycrystalline systems. The model employs domain parameters with diffuse interfaces to describe the grains, thereby enabling solutions with explicit consideration of their complex geometries. The intrinsic error arising from the diffuse interface approach employed in our proposed model is explored by comparing the results against a sharp interface model for a variety of parameter sets. Finally, two case studies are considered to demonstrate potential applications of the model. First, a nanocrystalline yttria-stabilized zirconia solid oxide fuel cell system is investigated, and the effective diffusivities are extracted from the simulation results and are compared to the values obtained through mean-field approximations. Second, the concentration evolution during lithiation of a polycrystalline battery cathode particle is simulated to demonstrate the method’s capability. 

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5. Fast and Accuracy-Preserving Domain Decomposition Methods for Reduced Fracture Models with Nonconforming Time Grids

   https://doi.org/10.1007/s10915-023-02251-0  

   Huynh, Phuoc-Toan; Cao, Yanzhao; Hoang, Thi-Thao-Phuong (July 2023, Journal of Scientific Computing)

   This paper is concerned with the numerical solution of compressible fluid flow in a fractured porous medium. The fracture represents a fast pathway (i.e., with high permeability) and is modeled as a hypersurface embedded in the porous medium. We aim to develop fast-convergent and accurate global-in-time domain decomposition (DD) methods for such a reduced fracture model, in which smaller time step sizes in the fracture can be coupled with larger time step sizes in the subdomains. Using the pressure continuity equation and the tangential PDEs in the fracture-interface as transmission conditions, three different DD formulations are derived; each method leads to a space-time interface problem which is solved iteratively and globally in time. Efficient preconditioners are designed to accelerate the convergence of the iterative methods while preserving the accuracy in time with nonconforming grids. Numerical results for two-dimensional problems with non-immersed and partially immersed fractures are presented to show the improved performance of the proposed methods. 

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