A Systematic Literature Review of Fractional Differential Equations in Fluid Viscosity and Surface Tension Modeling

Fluids whose viscosity and surface tension depend on deformation history are poorly described by integer-order models, and fractional differential equations have been adopted for them across rheology and applied mathematics. Work on this subject remains scattered, and no synthesis has established which operators are in use, how the equations are solved, or what remains undone. This review supplies that synthesis and states the open problems that follow. Under the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) 2020 protocol, 682 records from Scopus, ScienceDirect, and SpringerLink were reduced to 23 eligible articles by deduplication of 61 records, title and abstract screening of 533, and full-text assessment of the 81 retrievable reports. A supplementary search on the term “fractional viscoelastic”, run as a sensitivity analysis, added five more, giving a final corpus of 28 studies from 1986 to 2026, classified by operator, model, solution method, fluid, property, and validation status. The springpot-based Fractional Maxwell Model and the Caputo-derivative dominate, no study applies the Atangana–Baleanu–Caputo or Caputo–Fabrizio operator constitutively, and joint modeling of the two properties is confined to three studies of one lubricating oil. Five open problems are stated, each with its supporting evidence and the direction that would resolve it.