Reconsideration of the implicit boundary conditions in pressure driven rarefied gas flows through capillaries

The pressure driven rarefied gas flow through plane channels and tubes of various lengths is solved in a wide range of the gas rarefaction parameter by implementing the implicit boundary conditions for the incoming distributions at the capillary ends. The advantage of this formulation is the small size of the computational domain which includes only the capillary, while the typical formulation requires the inclusion of adequately large inlet and outlet regions to properly impose the boundary conditions far from the capillary ends. However, by comparing the corresponding flow rates of the two formulations it has been found that the error introduced by the implicit boundary conditions formulation may be significant and depends mostly on the dimensionless length. The relative error for capillaries with dimensionless length equal to 10 reaches up to 20%, while for shorter capillaries becomes unacceptably large. The range of validity of the implicit boundary conditions is extended by introducing in the formulation the end effect theory. This way the computational cost remains small, while the introduced error is drastically decreased. Overall, the implementation of the implicit boundary conditions formulation must be always performed with caution, carefully considering its effect on the accuracy of the results.