This study presents a variational modeling framework for non-Newtonian fluid flow with viscosity evolv- ing in both space and time, derived from Hamilton’s principle. The key idea is the introduction of an internal variable that evolves in response to local flow conditions, allowing spatially and temporally vary- ing viscosity to be described in a natural and thermodynamically consistent manner. This approach en- ables a unified representation of complex non-Newtonian behaviors, including shear-thinning and shear- thickening, within a single variational setting. Within this framework, the Type 1 model originally proposed by Junker and Wick (2025) is revisited and reformulated from a unified variational perspective, and a new Type 2 model is introduced as an extension. The two models are derived from distinct free energy potentials and reflect different physical mechanisms governing viscosity evolution. While the Type 1 model describes viscosity changes driven by the interaction between velocity and displacement gradients, the Type 2 model captures viscosity variations governed by the magnitude of the strain. This formulation allows a systematic comparison of different flow-dependent mechanisms within a common theoretical structure. In contrast to conventional non-Newtonian constitutive laws based on nonlinear algebraic relations be- tween stress and strain rate, the proposed framework treats viscosity as an independent internal state variable whose evolution is governed by a differential equation in space and time. The numerical formu- lation is based on a space–time variational structure with a semi-discrete finite element implementation: velocity, displacement, and pressure fields are discretized globally, while the internal variable is evalu- ated locally at each integration point using a global–local solution strategy. Numerical simulations of benchmark channel flow and flow around a circular cylinder demonstrate that the proposed models successfully reproduce a wide range of non-Newtonian flow characteristics. In particular, the results show that different evolution mechanisms of the internal variable lead to funda- mentally distinct viscosity distributions, flow stability properties, and vortex formation patterns. These findings indicate that the proposed framework provides a flexible and extensible basis for the simulation and analysis of time-dependent non-Newtonian fluid flows.