Analytical and numerical description for isothermal gas flows in microchannels

Abstract

Analytical solutions for the pressure and the velocity profiles in a microchannel are derived from the quasi gasdynamic equations (QGD). An expansion method according to a small geometric parameter ɛ is undertaken to obtain the isothermal flow parameters. The deduced expression of the mass flow rate is similar to the analytical expression obtained from the Navier-Stokes equations with a second order slip boundary condition and gives results in agreement with the measurements. The analytical expression of the pressure predicts accurately the measured pressure distribution. The effects of the rarefaction and of the compressibility on pressure distributions are discussed. The numerical calculations based on the full system of the QGD equations were carried out for different sizes of the microchannels and for different gases. The numerical results confirm the validity of the analytical approach.

Appendix A

If we take into account the second order terms in the logarithmic function expansion in the Taylor series (33), we obtain the second order approximation of the pressure distribution:

$$\begin{aligned}\frac{{\tilde{p}}({\tilde{x}})}{{\tilde{p}}_{o}} =&{\sqrt{\frac{\left(\left({\mathcal{P}} + 6K_{{\rm slip}} Kn_{o} \right)^{2} - {\tilde{x}} F_{1} ({\mathcal{P}}) + 24\frac{Kn_{o}^{2}}{k_{\lambda}^{2}}-(6K_{{\rm slip}}Kn_{o})^{2} + \frac{F_{2}({\mathcal{P}})^{2}}{F_{3} ({\mathcal{P}})}\right)}{F_{3}({\mathcal{P}})}}- \frac{F_{2}({\mathcal{P}})}{F_{3}({\mathcal{P}})}}, \\F_{1}({\mathcal{P}}) =& {\mathcal{P}}^{2} - 1 + 12K_{{\rm slip}} Kn_{o} ({\mathcal{P}} - 1) + \frac{24}{{\mathcal{P}}}\frac{Kn_{o}^{2}}{k_{\lambda}^{2}} \left(\frac{3}{2}{\mathcal{P}} - 2 + \frac{1}{{\mathcal{P}}}\right), \\F_{2}({\mathcal{P}}) =& 6 K_{{\rm slip}} Kn_{o} + \frac{12}{{\mathcal{P}}} \frac{Kn_{o}^{2}}{k_{\lambda}^{2}}, \\F_{3}({\mathcal{P}}) =& 1 - \frac{12}{{\mathcal{P}}} \frac{Kn_{o}^{2}}{k_{\lambda}^{2}}. \\\end{aligned} $$

(37)

The corresponding pressure gradient for the second order approximation gives:

$$ \frac{{\text{d}}{\tilde{p}}({\tilde{x}})}{{\text{d}}{\tilde{x}}} = - \frac{{\tilde{p}}_{o} F_{1}({\mathcal{P}})/F_{3} ({\mathcal{P}})} {2\sqrt{\left(\left({\mathcal{P}} + 6K_{{\rm slip}} Kn_{o} \right)^{2} - {\tilde{x}} F_{1} ({\mathcal{P}}) + 36\frac{Kn_{o}^{2}}{k_{\lambda}^{2}} - (6K_{{\rm slip}} Kn_{o})^{2} + \frac{F_{2} ({\mathcal{P}})^{2}}{F_{3} ({\mathcal{P}})}\right)/F_{3} ({\mathcal{P}})}}. $$