Detailed analysis under fully constrained and relaxed boundary conditions of linear fields in the vicinity of a corner

Abstract

This work examines the behavior of fields near corners under various boundary conditions (BCs), focusing on singularities arising from fully constrained and relaxed BCs. We analyze this behavior across diverse physical systems governed by similar equations, including electromagnetism, superconductivity, and two-phase fluid flow. The corner geometry presents a challenge due to potentially diverging field solutions as the corner is approached (r$\to$ 0). This motivates the investigation of relaxed BCs, which regularize the field by introducing a characteristic length (Ls) that relates the field’s value to its normal derivative at the boundary.

We explore both single-medium (single-phase) and double-medium (two-phase) systems. While prior research has addressed relaxed BCs in specific contexts, their application to corners, particularly in diverse physical systems, remains under-explored. We develop a series solution method to analyze the field behavior near the corner under different BCs. Concrete examples illustrate the theoretical framework, examining both fully constrained and relaxed scenarios. The implications of this work extend to fields such as fluid mechanics, electromagnetism, and heat transfer.