[Paper Review] A thermo-diffusion system with Smoluchowski interactions: well-posedness and homogenization

Oleh Krehel, Adrian Muntean+1 more|arXiv (Cornell University)|Apr 16, 2014

Advanced Mathematical Modeling in Engineering20 references3 citations

TL;DR

This paper establishes the well-posedness and homogenization of a thermo-diffusion system with Smoluchowski-type interactions in a periodically perforated domain, modeling colloidal particle transport and deposition under thermal gradients. Using two-scale convergence, the authors derive a macroscopic upscaled system that captures effective diffusion, heat transfer, and particle deposition in porous media.

ABSTRACT

We study the solvability and homogenization of a thermal-diffusion reaction problem posed in a periodically perforated domain. The system describes the motion of populations of hot colloidal particles interacting together via Smoluchowski production terms. The upscaled system, obtained via two-scale convergence techniques, allows the investigation of deposition effects in porous materials in the presence of thermal gradients.

Motivation & Objective

• To establish the mathematical well-posedness of a microscale thermo-diffusion system with Smoluchowski-type particle interactions in a periodically perforated domain.
• To derive a macroscopic upscaled model via homogenization techniques, enabling efficient simulation of transport and deposition in porous materials.
• To incorporate thermal gradients and their effects on colloidal particle motion, including Dufour and Soret-like couplings.
• To model particle deposition on solid grain boundaries via surface mass evolution equations.
• To provide a rigorous asymptotic limit using two-scale convergence, yielding effective coefficients for the macroscopic system.

Proposed method

• Formulate a microscale system of PDEs for temperature $\theta^\varepsilon$, particle concentrations $u_i^\varepsilon$ in the pore space $\Omega^\varepsilon$, and deposited masses $v_i^\varepsilon$ on the boundary $\Gamma^\varepsilon$.
• Apply two-scale convergence to pass from the microscale to the macroscale, using periodic unfolding and weak convergence in $L^2$-spaces.
• Solve auxiliary cell problems in the unit cell $Y_1$ to compute effective coefficients such as $\mathbb{K}$, $\mathbb{D}^i$, and $\mathbb{F}^i$.
• Derive the strong formulation of the upscaled system by integrating over the fast $y$-variables and identifying effective fluxes.
• Use mollifiers $J_\delta$ to regularize data and ensure convergence in the homogenization process.
• Establish existence and uniqueness of weak solutions to the microscopic problem using Galerkin approximation and energy estimates.

Experimental results

Research questions

• RQ1 How can the well-posedness of a coupled thermo-diffusion system with Smoluchowski interactions be rigorously established in a periodically perforated domain?
• RQ2 What is the effective macroscopic behavior of colloidal particles under thermal gradients when diffusion, convection, and deposition are coupled?
• RQ3 How do the Dufour and Soret effects manifest in the homogenized system, and what are their contributions to the effective fluxes?
• RQ4 What are the effective transport coefficients (e.g., $\mathbb{K}$, $\mathbb{D}^i$, $\mathbb{F}^i$) in the upscaled model, and how are they computed from cell problems?
• RQ5 How does the deposition of particles on the solid grain boundary $\Gamma^\varepsilon$ influence the macroscopic evolution of particle mass?

Key findings

• The microscopic system admits a unique weak solution under standard assumptions on data, established via Galerkin approximation and energy estimates.
• The two-scale convergence method successfully yields a macroscopic system with effective coefficients derived from cell problems in the unit cell $Y_1$.
• The upscaled system features a macroscopic heat equation with effective thermal conductivity $\mathbb{K}$, incorporating both diffusion and cross-coupling with concentration gradients via $\mathbb{T}$.
• The effective diffusion tensor $\mathbb{D}^i$ includes contributions from both self-diffusion and cross-diffusion terms due to the $\bar{u}_i^j$ correctors.
• The deposition rate is modeled via effective coefficients $A_i$ and $B_i$, computed as averages over the boundary $\Gamma$, capturing the loss of particles from the pore space.
• The homogenized system preserves the physical structure of the original problem, including boundary conditions on $\Gamma_R^\varepsilon$ and $\Gamma_N^\varepsilon$, and correctly accounts for fluxes at the macroscopic scale.