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The topological aspects of fluid flows have long been fascinating subjects in the study of the physics of fluids. In this study, the topology of the second-order Boltzmann–Curtiss constitutive model beyond the conventional Navier–Stokes–Fourier equations and Stokes’s hypothesis was investigated. In the case of velocity shear, the topology of the second-order constitutive model was shown to be governed by a simple algebraic form. The bulk viscosity ratio in diatomic and polyatomic gases was found to play an essential role in determining the type of topology: from an ellipse to a circle, to a parabola, and then finally to a hyperbola. The topology identified in the model has also been echoed in other branches of science, notably in the orbits of planets and comets and Dirac cones found in electronic band structures of two-dimensional materials. The ultimate origin of the existence of the topology was traced to the coupling of viscous stress and velocity gradient and its subtle interplay with the bulk viscosity ratio. In the case of compression and expansion, the topology of the second-order constitutive model was also found to be governed by a hyperbola. The trajectories of solutions of two representative flow problems—a force-driven Poiseuille gas flow and the inner structure of shock waves—were then plotted on the topology of the constitutive model, demonstrating the indispensable role of the topology of the constitutive model in fluid dynamics.The topological aspects of fluid flows have long been fascinating subjects in the study of the physics of fluids. In this study, the topology of the second-order Boltzmann–Curtiss constitutive model beyond the conventional Navier–Stokes–Fourier equations and Stokes’s hypothesis was investigated. In the case of velocity shear, the topology of the second-order constitutive model was shown to be governed by a simple algebraic form. The bulk viscosity ratio in diatomic and polyatomic gases was found to play an essential role in determining the type of topology: from an ellipse to a circle, to a parabola, and then finally to a hyperbola. The topology identified in the model has also been echoed in other branches of science, notably in the orbits of planets and comets and Dirac cones found in electronic band structures of two-dimensional materials. The ultimate origin of the existence of the topology was traced to the coupling of viscous stress and velocity gradient and its subtle interplay with the bulk viscos...

Topology of the second-order constitutive model based on the Boltzmann–Curtiss kinetic equation for diatomic and polyatomic gases