Micro- and nanodevices are often operated in gaseous environments and thus their performances are affected by the gas around them. Since the smallest characteristic length of typical micro-electro-mechanical systems (MEMS) devices is comparable with (or smaller than) the mean free path of the gas molecules, the traditional computational fluid dynamics methods fail in predicting the flows related to these devices. Therefore, an accurate analysis of such microfluidic systems requires the solution of the Boltzmann equation. To extend the applicability of the continuum description, which is significantly more efficient compared to molecular-based approaches, several authors suggested to use slip boundary conditions in conjunction with the Navier-Stokes equations to extend their applicability to the slip and transitional regimes. Unfortunately, as yet, no consensus has been reached on the correct form of second-order formulation, with recent experimental studies revealing large discrepancies between the experimentally determined values of the second-order slip coefficient and the theoretical values proposed in literature. In the current paper, we provide an analytical expression for the first- and second-order velocity slip coefficients by means of a variational technique which applies to the integrodifferential form of the Boltzmann equation based on the true linearized collision operator and the Maxwell scattering kernel of the gas-surface interaction. Our findings concerning the slip coefficients compare favorably with recent experimental results.

Variational derivation of second-order slip coefficients on the basis of the Boltzmann equation for hard-sphere molecules

CERCIGNANI, CARLO;LORENZANI, SILVIA
2010

Abstract

Micro- and nanodevices are often operated in gaseous environments and thus their performances are affected by the gas around them. Since the smallest characteristic length of typical micro-electro-mechanical systems (MEMS) devices is comparable with (or smaller than) the mean free path of the gas molecules, the traditional computational fluid dynamics methods fail in predicting the flows related to these devices. Therefore, an accurate analysis of such microfluidic systems requires the solution of the Boltzmann equation. To extend the applicability of the continuum description, which is significantly more efficient compared to molecular-based approaches, several authors suggested to use slip boundary conditions in conjunction with the Navier-Stokes equations to extend their applicability to the slip and transitional regimes. Unfortunately, as yet, no consensus has been reached on the correct form of second-order formulation, with recent experimental studies revealing large discrepancies between the experimentally determined values of the second-order slip coefficient and the theoretical values proposed in literature. In the current paper, we provide an analytical expression for the first- and second-order velocity slip coefficients by means of a variational technique which applies to the integrodifferential form of the Boltzmann equation based on the true linearized collision operator and the Maxwell scattering kernel of the gas-surface interaction. Our findings concerning the slip coefficients compare favorably with recent experimental results.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11311/574536
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