Mathematical Modeling of Non-Newtonian Fluid Dynamics

Mathematical modelling of non-Newtonian fluid dynamics seeks to characterise and predict the flow behaviour of materials whose stress–strain relationship departs from the linearity inherent to Newtonian fluids. Such models incorporate constitutive relations that capture shear-thinning or shear-thickening viscosities, yield stresses and time-dependent elastic responses. Classical frameworks include power-law and Herschel–Bulkley descriptions for rate-dependent viscosity, as well as integral and differential viscoelastic models such as Oldroyd-B, Giesekus and FENE (finitely extensible nonlinear elastic) formulations. Governing equations extend the Navier–Stokes system by embedding these constitutive laws, leading to coupled partial differential equations that often defy analytic solution and demand robust numerical schemes. Advances in stability analysis, regularity theory and efficient discretisation have enabled detailed studies of flow instabilities, transient phenomena and complex boundary interactions. These developments underpin applications ranging from polymer processing and food science to physiological flows and geophysical suspensions, where accurate prediction of stress fields, free surfaces and yield zones is crucial for design and control.

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Mathematical Modeling of Non-Newtonian Fluid Dynamics publication trend

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Technical terms

Non-Newtonian fluid: A fluid whose viscosity depends on shear rate or history, exhibiting shear-thinning, shear-thickening or yield stress behaviour rather than a constant viscosity.

Constitutive relation: A mathematical expression linking stress and strain (or strain rate), reflecting the material’s rheological properties.

Power-law model: A rate-dependent viscosity model in which shear stress scales as a power of shear rate, accommodating shear-thinning or shear-thickening responses.

Viscoelastic model: A formulation that accounts for both viscous dissipation and elastic stress storage, often represented by combinations of springs and dashpots or by integral/differential equations.

Spectral collocation method: A numerical technique using global basis functions (such as Chebyshev polynomials) to discretise differential equations, yielding high accuracy for smooth solutions.

Finite element method: A spatial discretisation approach that subdivides a domain into elements and employs variational formulations to approximate solutions of partial differential equations.

Maximal monotone operator: A mathematical object defining an implicit relation between variables (e.g., stress and strain rate) that is both single-valued in a monotonic sense and admits a maximal graph over a function space.

References

1. Linear stability of a Couette flow for non-monotone stress-power law models. The European Physical Journal Plus (2023).
2. A review of implicit algebraic constitutive relations for describing the response of nonlinear fluids. Comptes Rendus Mécanique (2024).
3. Fully discrete finite element approximation of unsteady flows of implicitly constituted incompressible fluids. IMA Journal of Numerical Analysis (2019).