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We explore a class of splitting schemes employing implicit-explicit (IMEX) time-stepping to achieve accurate and energy-stable solutions for thin-film equations and Cahn-Hilliard models with variable mobility. This splitting method incorporates a linear, constant coefficient implicit step, facilitating efficient computational implementation. We investigate the influence of stabi- lizing splitting parameters on the numerical solution computationally, considering various initial conditions. Furthermore, we generate energy-stability plots for the proposed methods, examin- ing different choices of splitting parameter values and timestep sizes. These methods enhance the accuracy of the original bi-harmonic-modified (BHM) approach, while preserving its energy- decreasing property and achieving second-order accuracy. We present numerical experiments to illustrate the performance of the proposed methods. 

This paper investigates particle deposition driven by fluid evaporation in a single pore channel representative of those found in porous membranes. A moving boundary problem for the 2D heat equation is coupled with an evolution equation for the pore radius, and describes the physical processes of fluid evaporation, diffusion of the particle concentration, and deposition on the pore channel wall. Furthermore, a stochastic differential equation (SDE) approach based on a Brownian motion particle-level description of diffusion is used as a similar phenomenological representation to the partial differential equation (PDE) model. Sensitivity analysis reveals trends in dominant model parameters such as evaporation rate, deposition rate, the volume scaling coefficient, and investigates the monotonicity of concentration. Evaluations of the asymptotically reduced model and the SDE model against the 2D PDE model are done in terms of the pore radius and solute concentration over time. For further exploration, we apply the model to a 2D droplet as well with both deterministic and stochastic approaches. 

Abstract We study steady-state thin films on chemically heterogeneous substrates of finite size, subject to no-flux boundary conditions. Based on the structure of the bifurcation diagram, we classify the 1D steady-state solutions that exist on such substrates into six different branches and develop asymptotic estimates for the steady states on each branch. Using perturbation expansions, we show that leading-order solutions provide good predictions of the steady-state thin films on stepwise-patterned substrates. We show how the analysis in one dimension can be extended to axisymmetric solutions. We also examine the influence of the wettability contrast of the substrate pattern on the linear stability of droplets and the time evolution for dewetting on small domains. Results are also applied to describe 2D droplets on hydrophilic square patches and striped regions used in microfluidic applications.