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The ultrarelativistic shock layer around the triangle prism is numerically analyzed using the relativistic Boltzmann equation to investigate the dissipation process under two types of ultrarelativistic limits: namely, the Lorentz contraction limit, in which the uniform flow velocity approximates to the speed of light, and the thermally relativistic limit, in which the temperature of the uniform flow approximates to infinity. The relativistic Boltzmann equation is numerically solved using the direct simulation Monte Carlo method. We discuss dissipation process in the flow field by focusing on profiles of the dynamic pressure and heat flux along the stagnation streamline under the Lorentz contraction limit or the thermally relativistic limit. Our numerical results confirm that profiles of the dynamic pressure and heat flux along the stagnation streamline strongly depend on the Lorentz contraction and thermally relativistic effects under their ultrarelativistic limits, as predicted by Chapman-Enskog expansion on the basis of the generic Knudsen number.

The transport coefficients of the inelastic variable hard sphere are calculated using infinite Maxwellian iterations of Grad’s 14 moment equations and dimensional analysis on the basis of the Chapman–Enskog expansion. The viscosity coefficient (μ) obtained using infinite Maxwellian iterations coincides with that obtained using dimensional analysis on the basis of the Chapman–Enskog expansion. Two transport coefficients (κ and η) calculated using infinite Maxwellian iterations, which define the heat flux, coincide with those calculated using dimensional analysis on the basis of the Chapman–Enskog expansion, only when κ and η obtained using infinite Maxwellian iterations converge. Divergences of κ and η occur whenever the dissipating rate of the heat flux, via the inelastic collisional term, is equal to or smaller than the characteristic increasing or decreasing rate of the heat flux via the decrease in the temperature.

This paper investigates the effect of the form of an obstacle on the time that a crowd takes to evacuate a room, using a toy model. Pedestrians are modeled as active soft matter moving toward a point with intended velocities. An obstacle is placed in front of the exit, and it has one of four shapes: a cylindrical column, a triangular prism, a quadratic prism, or a diamond prism. Numerical results indicate that the evacuation-completion time depends on the shape of the obstacle. Obstacles with a circular cylinder (C.C.) shape yield the shortest evacuation-completion time in the proposed model.