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Complex fluid interfaces are commonplace in natural and engineered systems and a major topic in the fields of rheology and soft matter physics, providing boundary conditions for a system’s hydrodynamics. The relationship between structure and function dictates how constituents within complex fluids govern flow behavior via constituents changing conformation in response to the local microenvironment to minimize free energy. Both hydrodynamics, such as shear flow, and the presence of air–liquid interfaces are principal aspects of a complex fluid’s environment. The study of fluid interfaces coupled to bulk flows can be uniquely advanced through experimentation in microgravity, where surface tension containment can be achieved at relatively large length scales. This computational investigation assesses flow in the ring-sheared drop (RSD), a containerless biochemical reactor operating aboard the International Space Station for the study of complex fluids and soft matter physics. Specifically, the hydrodynamic effects of a generalized Boussinesq–Scriven interface with a shear-thinning surface shear viscosity are examined in flow regimes where the air–liquid interface remains coupled to the Newtonian bulk fluid. The results verify this interfacial model’s ability to affect system-wide hydrodynamics under specific parameter regimes, enabling future model validation with high-precision rheological measurements.