Department of Mathematics
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Item Thermophoresis and uniform flow in rarefied polyatomic gases: The role of constitutive relations and boundary conditions(AIP, 2023) Rana, Anirudh SinghRecently, Rana and Barve [“A second-order constitutive theory for polyatomic gases: Theory and applications,” J. Fluid Mech. 958, A23 (2023)] developed a second-order coupled constitutive relations (CCR) for polyatomic gases that include quadratic nonlinearities in the entropy flux and apply the second law. However, in that work, the boundary conditions were heuristically obtained to match the drag coefficient on a sphere and may not be accurate in situations where thermal transpiration and thermal stress are significant factors, as indicated by their asymptotic analysis. This article presents a systematic approach for deriving thermodynamically admissible boundary conditions for the CCR model. We also propose a set of higher-order boundary conditions based on an asymptotic analysis of the solutions for drag on flow past a sphere and thermophoretic drag. The goal of deriving these boundary conditions is to improve the accuracy of the CCR model when applied to external flows, such as slow flow past particles and thermophoretic motion of a spherical particle and doublet. The results of the study demonstrate that the combination of the newly derived boundary conditions in conjunction with the CCR equations shows excellent agreement with both theoretical predictions and experimental data over a wide range of Knudsen numbers. The study suggests that the approach presented in this article can be used to improve the accuracy of the CCR model in a variety of external flow applications.Item H-theorem and boundary conditions for two-temperature model: Application to wave propagation and heat transfer in polyatomic gases(AIP, 2023-12) Rana, Anirudh SinghPolyatomic gases find numerous applications across various scientific and technological fields, necessitating a quantitative understanding of their behavior in nonequilibrium conditions. In this study, we investigate the behavior of rarefied polyatomic gases, particularly focusing on heat transfer and sound propagation phenomena. By utilizing a two-temperature model, we establish constitutive equations for internal and translational heat fluxes based on the second law of thermodynamics. A novel reduced two-temperature model is proposed, which accurately describes the system's behavior while reducing computational complexity. Additionally, we develop phenomenological boundary conditions adhering to the second law, enabling the simulation of gas-surface interactions. The phenomenological coefficients in the constitutive equations and boundary conditions are determined by comparison with relevant literature. Our computational analysis includes conductive heat transfer between parallel plates, examination of sound wave behavior, and exploration of spontaneous Rayleigh-Brillouin scattering. The results provide valuable insights into the dynamics of polyatomic gases, contributing to various technological applications involving heat transfer and sound propagation.Item Stokes' paradox in rarefied gases: A perspective through the method of fundamental solutions(2024-06) Rana, Anirudh SinghIn the realm of fluid dynamics, a curious and counterintuitive phenomenon is Stokes' paradox. While Stokes equations -- used for modeling slow and steady flows -- lead to a meaningful solution to the problem of slow and steady flow past a sphere, they fail to yield a non-trivial solution to the problem of slow and steady flow past an infinitely long cylinder (a two-dimensional problem essentially); this is referred to as Stokes' paradox. We revisit this paradox in the context of rarefied gas flows by means of the method of fundamental solutions (MFS). To this end, we adopt an extended hydrodynamic model, referred to as the CCR model, consisting of the balance equations for the mass, momentum and energy and closed with the coupled constitutive relations. We determine an analytic solution of the CCR model for the problem and compare it with the MFS-based numerical solution. Apart from addressing flow past a circular cylinder, we aim to showcase the capability of the MFS to predict the flow past other objects in two dimensions for which the analytic solutions do not exist. For that, we investigate the problem of rarefied gas flow past an infinitely long semicircular cylinder.Item Capturing non-equilibrium in hypersonic flows: Insights from a two-temperature model in polyatomic rarefied gases(AIP, 2024) Rana, Anirudh SinghThe study utilizes a two-temperature model to analyze non-equilibrium in normal shocks within hypersonic flows in polyatomic rarefied gases. Derived from the extended second law of thermodynamics, this model separates translational and internal temperatures in polyatomic gases, providing a more accurate depiction of non-equilibrium gas flow compared to classical theories like the Navier–Stokes and Fourier (NSF) system. Notably, the analysis reveals that the two-temperature model incorporates an additional contribution to the heat flux due to the gradient of the dynamic temperature, resulting in improved accuracy, especially for high Mach numbers. Results show that the model gives satisfactory shock density and temperature profiles up to Mach 10, with very good agreement observed up to Mach 6.1 compared to the classical NSF model. We conduct an order of magnitude analysis on the dynamic temperature and heat flux gradients appearing in the new constitutive equation using the Mott-Smith method. This analysis highlights the impact of these terms on accurately modeling polyatomic gas behavior in high-speed flows. The effects of bulk viscosity and incoming temperature on shock profiles are also investigated, contributing to a better understanding of shock wave structures in polyatomic gases and their implications for hypersonic flow dynamics.