Evaporation Boundary Conditions for the Linear R13 Equations Based on the Onsager Theory

Abstract

Due to failure of the continuum hypothesis for higher Knudsen numbers, rarefied gases and microflows of gases are particularly difficult to model. Macroscopic transport equations compete with particle methods, such as DSMC to find accurate solutions in the rarefied gas regime. Due to growing interest in micro flow applications, such as micro fuel cells, it is important to model and understand evaporation in this flow r egime. H ere, e vaporation b oundary c onditions f or t he R 13 equations, which are macroscopic transport equations with applicability in the rarefied gas regime, are derived. The new equations utilize Onsager relations, linear relations between thermodynamic fluxes and forces, with constant coefficients, that need to be d etermined. For this, the boundary conditions are fitted to DSMC data and compared to other R13 boundary conditions from kinetic theory and Navier-Stokes-Fourier (NSF) solutions for two one-dimensional steady-state problems. Overall, the suggested fittings of the new phenomenological boundary conditions show better agreement to DSMC than the alternative kinetic theory evaporation boundary conditions for R13. Furthermore, the new evaporation boundary conditions for R13 are implemented in a code for the numerical solution of complex, two-dimensional geometries and compared to NSF solutions. Different flow patterns between R13 and NSF for higher Knudsen numbers are observed.

New exact solutions for microscale gas flows

“…Similar computations confirm that the fields due to a point heat source placed in a isobaric stationary-state rarefied gas flow close to a no-temperature jump surface (Eqs. (27), solutions (28) and boundary condition (18)) are equivalent to the ones obtained in (43a) and (43b).…”

“…We start with the case of a point heat source (thermal Gradlet) placed in a heat flow close to a temperature jump surface (Eqs. (26), solutions (28) and boundary condition (30)). The Fourier transformed temperature jump boundary condition is now quite complicated:…”

“…In order to simplify the equations far enough that they can be solved in this setting, we will have to assume one dimension: this is a standard assumption found elsewhere in the literature on the unsteady Grad and R13 equations [40]. We start with the linearised Grad equations from [28] in one dimension with a time-dependent point heat source applied to the conservation of energy equation. This gives us the linearised dimensionless conservation equations…”

“…where δ is the Dirac delta function and g is the strength of the point heat source. The other coefficients are defined in [28] and depend on the collision model which is being used. The Prandtl number is Pr = μc p /k, where μ is the shear viscosity, c p is the isobaric specific heat and k is the thermal conductivity.…”

“…Our assumption of one dimension seems prohibitive, but is quite typical in the existing literature [28]. Many of the solutions which we derive refer to a flow with constant velocity and pressure.…”