# Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow

## TL;DR

This paper investigates the development of pinch-off singularities in axially symmetric Darcy flow, revealing new behaviors not seen in 2D Hele-Shaw flow, supported by a novel level set numerical scheme and similarity analysis.

## Contribution

It introduces a novel numerical method and demonstrates the occurrence of pinch-off singularities with a power-law behavior in axially symmetric porous media flow, extending understanding beyond 2D models.

## Key findings

- Pinch-off singularities occur in 3D axially symmetric Darcy flow.
- Minimum radius follows a power law with exponent 1/3 near pinch-off.
- Differences between 3D and 2D flow behaviors are highlighted.

## Abstract

We study a model for the evolution of an axially symmetric bubble of inviscid fluid in a homogeneous porous medium otherwise saturated with a viscous fluid. The model is a moving boundary problem that is a higher-dimensional analogue of Hele-Shaw flow. Here we are concerned with the development of pinch-off singularities characterised by a blow-up of the interface curvature and the bubble subsequently breaking up into two; these singularities do not occur in the corresponding two-dimensional Hele-Shaw problem. By applying a novel numerical scheme based on the level set method, we show that solutions to our problem can undergo pinch-off in various geometries. A similarity analysis suggests that the minimum radius behaves as a power law in time with exponent $\alpha = 1/3$ just before and after pinch-off has occurred, regardless of the initial conditions; our numerical results support this prediction. Further, we apply our numerical scheme to simulate the time-dependent development and translation of axially symmetric Saffman-Taylor fingers and Taylor-Saffman bubbles in a cylindrical tube, highlighting key similarities and differences with the well-studied two-dimensional cases.