A bulk-surface continuum theory for fluid flows and phase segregation with finite surface thickness

Anne Boschman, Luis Espath, Kristoffer G. van der Zee

Research output: Contribution to journalArticleResearchpeer-review


In this continuum theory, we propose a mathematical framework to study the mechanical interplay of bulk-surface materials undergoing deformation and phase segregation. To this end, we devise a principle of virtual powers with a bulk-surface dynamics, which is postulated on a material body P where the boundary ∂P may lose smoothness, that is, the normal field may be discontinuous on an edge ∂2P. The final set of equations somewhat resemble the Navier–Stokes–Cahn–Hilliard equation for the bulk and the surface. Aside from the systematical treatment based on a specialized version of the virtual power principle and free-energy imbalances for bulk-surface theories, we consider two additional ingredients: an explicit dependency of the apparent surface density on the surface thickness and mixed boundary conditions for the velocity, chemical potential, and microstructure.

Original languageEnglish
Article number134055
Number of pages12
JournalPhysica D: Nonlinear Phenomena
Publication statusPublished - Apr 2024
Externally publishedYes


  • Bulk-surface partial differential equations
  • Continuum mechanics
  • Fluid Mechanics

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