Multi-species viscous models for tissue growth: incompressible limit and qualitative behaviour

TL;DR

This paper develops two 2D viscous tissue models that simulate cell swirling and segregation during embryo elongation, analyzes their incompressible limits, and compares their behaviors, revealing a pressure jump and ghost effects relevant to biology.

Contribution

It introduces two novel viscous tissue models with segregation mechanisms, analyzes their incompressible limits, and compares their qualitative behaviors, including pressure effects.

Findings

Models reproduce swirling cell motions and segregation.

Incompressible limits produce strictly segregated solutions.

Active segregation model shows persistent pressure and ghost effects.

Abstract

We introduce two 2D mechanical models reproducing the evolution of two viscous tissues in contact. Their main property is to model the swirling cell motions while keeping the tissues segregated, as observed during vertebrate embryo elongation. Segregation is encoded differently in the two models: by passive or active segregation (based on a mechanical repulsion pressure). We compute the incompressible limits of the two models, and obtain strictly segregated solutions. The two models thus obtained are compared. A striking feature in the active segregation model is the persistence of the repulsion pressure at the limit: a ghost effect is discussed and confronted to the biological data. Thanks to a transmission problem formulation at the incompressible limit, we show a pressure jump at the tissues' boundaries.