Gas mixing at Martian atmospheric conditions through a Smoothed Particle Hydrodynamics approach

Introduction

Many fluid dynamics problems in planetary science involve multiple phases and require advanced numerical methods. While Eulerian approaches are difficult to adapt to complex multi-phase systems, Lagrangian methods like Smoothed Particle Hydrodynamics (SPH) [1] offer greater flexibility, making them well-suited for multi-component flows. In SPH, the fluid is represented by particles, and each particle’s properties are determined by neighboring particles within the range of an interpolation function. Although SPH has been widely used for gas-dust systems [2,3], relatively little work has been done on binary gas mixtures. While [2] suggests that gas-dust models can be extended to gas-gas mixtures, an added challenge arises: molecular collision timescales are much shorter than hydrodynamic timescales, complicating diffusion modeling, especially in multi-phase systems with solid components like dust and ice.

In this work, we present an SPH model for binary gas mixtures using a multi-fluid approach, where each species follows its own set of equations. To handle the short diffusion timescales efficiently, a splitting scheme is implemented, allowing hydrodynamic and diffusive processes to be treated separately, significantly reducing the computational cost. The method applies to any binary mixture of monatomic gases and is designed for future extension to more realistic scenarios, such as the interaction between water vapor and the CO₂-dominated Martian atmosphere, where contributions from rotational and vibrational modes must be included.

Gas mixing models are relevant in a wide range of astrophysical applications, including planetary missions like ExoMars and, in particular, the ESA Rosalind Franklin rover. Scheduled for launch in 2028, the rover will drill up to 2 meters below the surface at Oxia Planum [4]. In an ongoing study [5], we are exploring how drilling operations may influence the presence and stability of possible subsurface volatiles, such as water ice. The gas mixing framework developed here provides a key foundation for incorporating atmospheric effects into such models.

Model 

In this two-fluid model formulation, the density, momentum, and energy of each gas are treated separately. The standard Euler equations for an inviscid, ideal gas with zero thermal conductivity are modified to include collisional terms that account for momentum and energy exchange between the species. These terms are derived from a kinetic relaxation model based on the Boltzmann equation [6,7]. A first-order operator splitting [8] separates the hydrodynamic evolution from the collisional terms. Hydrodynamics are advanced using a two-step Euler scheme, while collisional terms are applied as a corrective step within each timestep. Standard hydrodynamic terms, such as pressure gradients in the momentum and energy equations, follow the usual SPH formulation [1]. The exponentially relaxing collisional system is then solved, and the original hydrodynamic velocities and energies are updated with the new collisional terms before advancing to the next integration timestep. 

Results 

The proposed approach is validated through numerical tests, starting with a comparison of Trotter splitting against the standard SPH method. Additionally, tests on Xe-Ne mixture in a controlled environment verify that the simulations align with expected hydrodynamic and thermodynamic behavior. The tests consider two monatomic gases in a closed cylindrical domain, initially set at Martian atmospheric pressure. The heavier gas occupies the upper half, and the lighter gas the lower half. Finally, no external forces are considered. Fig.1 illustrates the initial setup for the SPH simulation.

Fig.1: Illustration of the initial system setup. 

Firstly, we fix the temperature at 300 K and compare results from the formal SPH formulation, using the smaller collisional timestep, with those from Trotter splitting. Fig.2 shows that both approaches yield consistent results, with a maximum discrepancy of 6%. Using 50000 particles on an 18-core workstation, the formal SPH simulation took 6 hours, while the Trotter splitting method finished in about 40 minutes.

Fig.2: Time evolution of mean densities for neon (top) and xenon (bottom). Solid lines show results with the smaller collisional timestep, while dotted lines represent the Trotter splitting approach.

In a second test, xenon is set to an initial temperature of 500 K, and neon is set to 300 K. The simulation shows that the collisional energy exchange causes the gases' temperatures to relax around 366 K  (Fig.3), close to the expected equilibrium temperature of 375 K. Fig.4, instead, demonstrates that both gases reach an equilibrium density slightly below half their initial values. Although the final mean density is underestimated by about 8–9%, the simulation correctly reflects the expected behavior. Indeed, since mass is conserved and the volume occupied by each gas doubles during mixing, the final equilibrium densities are expected to be half of their initial values. The small underestimation is primarily attributed to the influence of boundary conditions in the SPH framework.

Fig.3: Time evolution of the mean temperatures of neon (blue) and xenon (red).

Fig.4: Time evolution of the mean densities of neon (blue) and xenon (red).

Conclusions

We developed a novel SPH model for simulating binary gas mixing, with explicit treatment of interspecies collisional momentum and energy exchange based on a kinetic relaxation model. Our results show that even a first-order scheme can yield physically consistent outcomes, capturing realistic thermal equilibration, density relaxation, and energy conservation across various test cases. The proposed framework is well-suited for multi-component systems, as it allows each gas to be modeled with its own set of Euler equations and interspecies interaction terms. This structure provides a natural pathway for extending the model to include additional physical processes, such as drag interactions with ice and dust [3]. Therefore, it offers a robust foundation for modeling gas dynamics in complex planetary science scenarios.

Acknowledgments 

Work supported by the ASI-INAF grant "Attività scientifica di preparazione all'esplorazione marziana 2023-3-HH.0".

References

[1] Monaghan (2005), Rep. Prog. Phys. 68, 1703.

[2] Monaghan & Kocharyan (1995), Comput. Phys. Commun., 87, 225.

[3] Laibe & Price (2012a), MNRAS, 420, 2345

[4] Vago et al. (2017), Astrobiology, 17, 471.

[5] Maggioni et al., in preparation

[6] Gross & Krook (1956), Phys. Rev., 102, 593

[7] Vega Reyes et al. (2007), Phys. Rev.

[8] Trotter, H. F. (1959), Proc. Am. Math. Soc., 10, 545