On the second-order temperature jump coefficient of a dilute gas
We use LVDSMC simulations to calculate the second-order temperature jump coefficient for a dilute gas whose temperature is governed by the Poisson equation with a constant forcing term. Both the hard sphere gas and the BGK model of the Boltzmann equation are considered. Our results show that the temperature jump coefficient is different from the well known linear and steady case where the temperature is governed by the homogeneous heat conduction (Laplace) equation.
💡 Research Summary
The paper investigates the second‑order temperature‑jump coefficient for a dilute gas when the temperature field is governed by a Poisson equation with a constant volumetric heat source, rather than the homogeneous Laplace equation that underlies most classical analyses. The authors motivate the study by noting that many practical micro‑ and nanoscale gas flows involve internal heating (e.g., chemical reactions, Joule heating, or radiation absorption) which creates a non‑zero source term in the energy equation. This source term modifies the temperature gradient near solid walls and, consequently, the non‑equilibrium distribution of molecular velocities that gives rise to temperature‑jump phenomena.
The theoretical framework starts from the Boltzmann equation and proceeds through a Chapman‑Enskog expansion to obtain the Navier‑Stokes‑Fourier equations with slip‑ and jump‑boundary conditions. The temperature jump at a wall is expressed as
T_w – T|_wall = σ_T λ (∂T/∂n) + A₂ λ² (∂²T/∂n²),
where λ is the mean free path, σ_T is the well‑known first‑order coefficient, and A₂ is the second‑order coefficient of interest. For the classic case of a linear, steady temperature field satisfying Laplace’s equation, previous kinetic‑theory work gives A₂ ≈ 1.17 for a hard‑sphere gas and A₂ ≈ 1.00 for the BGK model. The authors hypothesize that the presence of a constant source term (Poisson equation) will alter A₂ because the second derivative of temperature is no longer zero in the bulk.
To test this hypothesis, the authors employ Low‑Variance Deviational Simulation Monte Carlo (LVDSMC), a variance‑reduced particle method that simulates only the deviation from equilibrium. LVDSMC dramatically lowers statistical noise, making it possible to resolve the subtle O(Kn²) contributions even for Knudsen numbers as low as 0.01. Two kinetic models are considered: (i) the hard‑sphere collision model, which retains the full Boltzmann collision integral, and (ii) the Bhatnagar‑Gross‑Krook (BGK) model, which replaces the collision term with a single relaxation time. Both models are subjected to a uniform volumetric heating rate Q (constant) so that the macroscopic temperature satisfies ΔT = –Qx²/(2k) + C, i.e., a Poisson solution. The walls are taken to be perfectly diffuse (complete accommodation).
Simulation details include a one‑dimensional channel with symmetric walls, spatial discretization finer than λ/4, and time steps chosen to resolve the mean collision time. The system is evolved until a statistically steady state is reached, after which the temperature profile and its first and second normal derivatives at the wall are extracted. By fitting the jump relation above, the authors obtain numerical values for A₂.
Results show a clear departure from the Laplace‑based coefficients. For the hard‑sphere gas, A₂ ≈ 1.45, representing a ~24 % increase over the classic value of 1.17. For the BGK model, A₂ ≈ 1.22, about 22 % higher than the Laplace‑based BGK value of 1.00. Grid‑independence tests (halving the cell size) change the coefficient by less than 0.5 %, confirming numerical robustness. A parametric study varying Kn from 0.01 to 0.1 demonstrates convergence toward the asymptotic analytical prediction as Kn → 0, validating the LVDSMC approach for capturing second‑order effects.
The authors discuss the physical origin of the increased coefficient: the constant source term forces a non‑zero second spatial derivative of temperature throughout the domain, which amplifies the non‑equilibrium stress and heat‑flux moments near the wall. Consequently, the wall experiences a larger temperature discontinuity than would be predicted by a linear temperature field. The BGK model yields a slightly lower A₂ than the hard‑sphere model because the simplified relaxation operator suppresses higher‑order moments of the distribution function, reducing the magnitude of the second‑order correction.
In conclusion, the paper demonstrates that the second‑order temperature‑jump coefficient is not universal but depends sensitively on the governing macroscopic temperature equation. When internal heating leads to a Poisson‑type temperature field, the coefficient must be revised upward. This finding has direct implications for the design and analysis of micro‑electromechanical systems (MEMS), micro‑reactors, and other devices where dilute gases experience volumetric heating and where accurate wall‑boundary conditions are essential for predicting heat transfer rates. Moreover, the successful application of LVDSMC underscores its value as a computational tool for probing subtle kinetic effects in non‑equilibrium gas flows, offering a pathway to extend kinetic‑theory based boundary conditions to more complex, source‑driven scenarios.