Deep learning-based algorithms have recently been utilized to solve several nonlinear partial differential equations (PDEs) on curved surfaces. The mesh-free nature of these algorithms reduces computational complexity, especially for curved surfaces, where generating meshes is significantly challenging. The purpose of this paper is to generate an accurate solution of the highly nonlinear Allen-Cahn equation on various curved surfaces using a deep learning-based algorithm. However, generating a convergent solution of the Allen-Cahn equation for sharp interphase is significantly challenging, especially for curved surfaces, where the curvature of the geometry plays a crucial role in the solution accuracy. Due to the sharp interphase between two layers, the equation becomes very stiff; as a result, instability is a common practice. To mitigate this, we employ a transfer learning-based approach with PINNs to solve the Allen-Cahn equation on curved surfaces. A thorough numerical study is carried out to show the effectiveness of the numerical results presented for various complex closed curved surfaces with several initial conditions. Benchmark examples are provided for some cases to compare against the previous literature, along with a comparison with the analytic solution for the sphere case to test the convergence and accuracy of the present solution.

The primary goal of this paper is to model and study the behavior of ellipsoidal microplastic particles in a 3D lid-driven cavity. An Eulerian equation for the hydrodynamics is solved toward stationarity under the Reynolds number 1000 and coupled with a Lagrangian system governing the particle dynamics. The key points of departure in the modeling are laminar flow and small particle Reynolds number under which the drag force has extensively been studied. We then address the question, what would be the behavior of prolate and oblate particles under extreme conditions, where the aspect ratio tends to either ∞ (thin needle) or 0 (thin disk). The corresponding tool, Singular Perturbation Theory, not only settles the analysis of the critical manifold but also underlies a quasi-steady state approximation (QSSA) of the particle velocity around the manifold. We show that in a certain range of particle aspect ratio, the QSSA gives quite good approximations of the particle positions yet more computational efficiency. Sedimentation is shown to highly be dependent on the particle's initial position, aspect ratio, and size. Meanwhile initial position determines to which direction the fluid stream drifts the particles, larger size and aspect ratio closer to 1 increase the likelihood of sedimenting. We also show that neutrally buoyant particles prefer to deposit on the cavity base as well as on vorticity-dominating regions. Generally, buoyant particles with aspect ratios between 1/20 and 20 spin faster than tumble, and they spin even faster as the aspect ratio gets smaller.

In this paper, a rigorous computational study is carried out on the three numerical methods: cell average technique (CAT), weighted finite volume scheme (WFVS), and quadrature method of moments (QMOM). Each method is analyzed to solve the population balance equation coupled with hydrodynamics. Different test cases have been considered for aggregation, breakage, simultaneous aggregation and breakage processes with four different aggregation kernels and uniform breakage function. Both the advantages and disadvantages of those methods are thoroughly investigated for the PBE coupled with hydrodynamics. Based on accuracy and efficiency, it is recommended that the WFVS is a smart choice for computing number density and moments in the case of inhomogeneous PBE.

Microplastic in freshwater has been known to absorb, adsorb, and later desorb persistent organic pollutants as well as in its tiny size acts as an infiltrator to vital tissues; it may therefore corrupt physiological processes of organic lives. The fate of microplastic particles can be understood by revealing to what extent certain material properties (e.g., size and density) determine local behavior such as sedimentation and interaction with biofilm. This work seeks to gain an understanding of the short-range transport of microplastic particles in freshwater through devising a lid–driven cavity with a biofilm-covering obstacle as the medium. A stationary Navier–Stokes equation for an incompressible fluid at a moderate Reynolds number provides the background flow field. Microbeads are injected into the flow field, where their motion is governed by a Lagrangian system of equations. Advanced features such as dry particle–particle and particle–wall collisions as well as adhesion between particles and biofilm portraying particle entrapment are presented. Various simulations and parameterization studies are carried out to determine the impact of material properties, obstacle geometry, and adhesion force on the deposition profiles. In most cases, particles are trapped in the biofilm and in regions around the cavity with negative Okubo–Weiss numbers whereby the relative vorticity is dominating against the local strains.