arXiv subjects

Piotr Skrzypacz

Publications and source records attributed to Piotr Skrzypacz.

4 recordsLinked to original sources

arXiv2026

Geometry-induced formation and merging of multiple dead zones in diffusion-reaction problems

This study focuses on dead-core solutions for a two-dimensional isothermal reaction-diffusion problem involving a single chemical reaction described by power-law kinetics within a bounded reactor domain. The steady-state nonlinear diffusion-reaction boundary value problem is solved numerically by applying a suitable time-stepping technique. The lumped finite element technique with piecewise linear shape functions is utilized to discretize the spatial domain. Dead zones, defined as regions inside a reactor where no reaction occurs because the reactant has been completely consumed, present a major difficulty in the design and operation of chemical reactors. The impact of the reaction exponent and the Thiele modulus on the concentration fields and on the size of such inactive regions is numerically investigated. The findings indicate that particular reactor shapes can lead to the formation of several separate dead zones.

math.NA↗

arXiv2026

Physics-Informed Neural Network for Diffusion-Reaction Problems with Dead-Core Formation in Catalyst Slabs

This work investigates a nonlinear two-point boundary value problem arising in diffusion--reaction processes in catalyst slabs with power-law kinetics and fractional reaction order. For sufficiently large Thiele modulus, the solution develops a dead-core region separated from the active region by an unknown free boundary. We propose a structured Physics-Informed Neural Network (PINN) framework that incorporates the asymptotic behavior at the dead-core interface into a hard-constrained trial solution and treats the interface location as a trainable parameter. The concentration profile and free boundary are therefore approximated simultaneously without explicit interface tracking or penalty-based enforcement of the interface conditions. The method is validated against the exact solution for power-law kinetics and a high-precision numerical shooting method. Numerical experiments covering near-critical and strongly supercritical regimes demonstrate accurate recovery of both the concentration profile and dead-core location, together with robustness to random initialization and collocation sampling. While classical shooting is more efficient for the present one-dimensional benchmark, the proposed formulation provides a flexible framework for extensions to multidimensional geometries and problems for which analytical solutions are unavailable.

math.NA↗

arXiv2020

Periodic solutions and the avoidance of pull--in instability in non--autonomous micro--electro--mechanical systems

We study periodic solutions of a one-degree of freedom micro-electro-mechanical system (MEMS) with a parallel-plate capacitor under $T$--periodic electrostatic forcing. We obtain analytical results concerning the existence of $T-$ periodic solutions of the problem in the case of arbitrary nonlinear restoring force, as well as when the moving plate is attached to a spring fabricated using graphene. We then demonstrate numerically on a $T-$ periodic Poincar{é} map of the flow that these solutions are generally locally stable with large "islands" of initial conditions around them, within which the pull-in stability is completely avoided. We also demonstrate graphically on the Poincar{é} map that stable periodic solutions with higher period $nT, n>1$ also exist, for wide parameter ranges, with large "islands" of bounded motion around them, within which all initial conditions avoid the pull--in instability, thus helping us significantly increase the domain of safe operation of these MEMS models.

math.DS↗

arXiv2016

On the solvability of the Brinkman-Forchheimer-extended Darcy equation

The nonlinear Brinkman-Forchheimer-extended Darcy equation is used to model some porous medium flow in chemical reactors of packed bed type. The results concerning the existence and uniqueness of a weak solution are presented for nonlinear convective flows in medium with nonconstant porosity and for small data. Furthermore, the finite element approximations to the flow profiles in the fixed bed reactor are presented for several Reynolds numbers at the non-Darcy's range.

math.AP↗