A thermodynamically consistent variational framework for non-Newtonian fluids with evolving internal variables

Purpose This study presents a new variational model for non-Newtonian fluids based on Hamilton’s principle with an internal variable describing spatial and temporal variations of the viscosity. Design/methodology/approach The internal variable evolves in response to local flow conditions, enabling more refined and dynamic representation of complex non-Newtonian behaviors. The Type 1 model, originally introduced by Junker and Wick (2025) [P. Junker and T. Wick, “ Space-Time Modeling and Numerical Simulations of Non-Newtonian Fluids Using Internal Variables,” International Journal for Numerical Methods in Fluids 97, no. 12 (2025)] is revisited in the present work within an unified variational framework, and extended by introducing a new Type 2 model. Both models are derived from distinct free energy potentials: the Type 1 model describes viscosity evolution through the interaction between velocity and displacement gradients, while the Type 2 model captures viscosity variations driven by the magnitude of the strain. Findings Both formulations are capable of reproducing shear-thinning and shear-thickening behaviors within a unified and thermodynamically consistent setting. Unlike conventional constitutive laws expressed as nonlinear algebraic relations between stress and the rate of strain, the proposed framework naturally incorporates the evolution of the viscosity both in space and time. Simulations in two and three spatial dimensions demonstrate that the proposed models effectively capture a wide range of non-Newtonian flow characteristics. Originality/value The variational approach based on Hamilton’s principle, incorporating an internal variable, is novel. This allows for a wider range of non-Newtonian fluid flows models, which can be further studied in engineering and applied mathematics.