A new finite-volume-based finite-element method using the quadratic elements is developed in the present study, to analyze the flow in micro and nano sizes with higher-order slip boundary conditions. The method is applied to gaseous flow in micro and nanoscale-channels. The developed method is carried out over a wide range of Knudsen numbers, which cover not only the continuum slip flow regime with 0≤Kn≤0.1 but also it entire the range of transient regime with 0.1<Kn≤10. To make the present computational model capable of simulating micro and nano sizes with the help of the Navier-Stokes equations, the modified high-order slip boundary conditions are applied which need utilizing the advantages of general quadratic second order elements in the computational domain. In other words, this paper introduces a new developed method, which is applied on higher-order elements, and employing reliable boundary conditions that all of these issues are used for the first time in the Micro/Nano study as well. The results reveal excellent agreement with those represented by analytical, DSMC, and Boltzmann calculations. The proposed method (using finite-volume-element strategy which benefits from the advantages of general quadratic second-order elements) is proved to be an efficient, practical, and accurate tool, which robustly extends the capability of our primitive large scale Navier-Stokes solver to micro and nano-scale flow predictions in slip and transient regimes. It can be regarded as a super alternative to classical molecular dynamics-based methods.

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