Pressure-driven diffusive gas flows in micro-channels: from the Knudsen to the continuum regimes

Abstract

Despite the enormous scientific and technological importance of micro-channel gas flows, the understanding of these flows, by classical fluid mechanics, remains incomplete including the prediction of flow rates. In this paper, we revisit the problem of micro-channel compressible gas flows and show that the axial diffusion of mass engendered by the density (pressure) gradient becomes increasingly significant with increased Knudsen number compared to the pressure driven convection. The present theoretical treatment is based on a recently proposed modification (Durst et al. in Proceeding of the international symposium on turbulence, heat and mass transfer, Dubrovnik, 3–18 September, pp 25–29, 2006) of the Navier–Stokes equations that include the diffusion of mass caused by the density and temperature gradients. The theoretical predictions using the modified Navier–Stokes equations are found to be in good agreement with the available experimental data spanning the continuum, transition and free-molecular (Knudsen) flow regimes, without invoking the concept of Maxwellian wall-slip boundary condition. The simple theory also results in excellent agreement with the results of linearized Boltzmann equations and Direct Simulation Monte Carlo (DSMC) method. Finally, the theory explains the Knudsen minimum and suggests the design of future micro-channel flow experiments and their employment to complete the present days understanding of micro-channel flows.

Appendix: Treatments of molecular diffusion in ideal gas flows

When diffusion in ideal gas flows is treated in such a way that the conventional Navier–Stokes equations are derived, the following is assumed:

$$ \dot{m}_{i}^{D} = 0\quad \left( {\text{no mass diffusion}} \right) $$

(30)

This readily suggests that no density and temperature gradients (or corresponding pressure gradients) are present in the flow field. Hence this assumption contradicts Fourier’s law of diffusive heat transport, usually given as

$$ \dot{q}_{i} = - \lambda \frac{{\partial {\rm T}}}{{\partial x_{i} }} $$

(31)

The contradiction arises because every temperature gradient is related to mass diffusion. The derivations for $\dot{m}_{i}^{D}$ based on self-diffusion yield, see Durst et al. (2006):

$$ \dot{m}_{i}^{D} = - \rho D\left( {\frac{1}{\rho }\frac{\partial \rho }{{\partial x_{i} }} + \frac{1}{{2{\rm T}}}\frac{{\partial {\rm T}}}{{\partial x_{i} }}} \right) $$

(32)

With this expression for $\dot{m}_{i}^{D},$ the diffusive heat transport results as

$$ \dot{q}_{i} = - \lambda \left( {\frac{\partial T}{{\partial x_{i} }}} \right) + \dot{m}_{i}^{D} c_{p} T $$

(33)

The corresponding momentum transport for τ _ij , the molecular momentum transport, reads as follows:

$$ \tau_{ij} = - \mu \left( {\frac{{\partial U_{j} }}{{\partial x_{i} }} + \frac{{\partial U_{i} }}{{\partial x_{j} }}} \right) + \frac{2}{3}\mu \delta_{ij} \frac{{\partial U_{k} }}{{\partial x_{k} }} + \dot{m}_{i}^{D} U_{j} + \dot{m}_{i}^{D} U_{i} - \frac{2}{3}\delta_{ij} \dot{m}_{i}^{D} U_{k} $$

(34)

This expression can be rewritten to yield

$$ \tau_{ij} = - \nu \left[ {\frac{{\partial (\rho U_{j} )}}{{\partial x_{i} }} + \frac{{\partial (\rho U_{i} )}}{{\partial x_{j} }}} \right] + \frac{2}{3}\nu \delta_{ij} \frac{{\partial (\rho U_{k} )}}{{\partial x_{k} }} - \frac{\mu }{2T}\left( {U_{j} \frac{\partial T}{{\partial x_{i} }} + U_{i} \frac{\partial T}{{\partial x_{j} }} - \delta_{ij} U_{k} \frac{\partial T}{{\partial x_{k} }}} \right) $$

(35)

For the considerations in this paper, the above diffusive transport terms are of importance for the special case of T = constant. Because of the small Mach number, flows treated in micro-channel fluid mechanics are isothermal. Hence we can write, using the equation of state for ideal gases:

$$ \frac{1}{P}\frac{\partial P}{{\partial x_{i} }} = \frac{1}{\rho }\frac{\partial \rho }{{\partial x_{i} }} + \frac{1}{T}\frac{\partial T}{{\partial x_{i} }} $$

(36)

Hence, one can derive for isothermal flow:

$$ \dot{m}_{i}^{D} = - \rho D\left( {\frac{1}{\rho }\frac{\partial \rho }{{\partial x_{i} }}} \right) = - \rho D\left( {\frac{1}{P}\frac{\partial P}{{\partial x_{i} }}} \right) $$

(37)

It is important to note that in $\dot{m}_{i}^{D}$ derived by Brenner (2005) is identical with $\dot{m}_{i}^{D}$ derived by Durst et al. (2006) for T =  constant:

$$ {\text{Brenner}}:\quad \dot{m}_{i}^{D} = \alpha \cdot \frac{\partial }{{\partial x_{i} }}\left( {\ln \rho } \right) $$

(38)

$$ {\text{Durst}}:\quad \dot{m}_{i}^{D} = - \rho D\frac{\partial }{{\partial x_{i} }}\left[ {\ln \left( {\rho \sqrt T } \right)} \right] $$

(39)

It can be shown that $\alpha = \frac{\lambda }{{c_{p} }} = \left( {\rho D} \right)$ by Brenner (2005, 2006) is identical with (−ρD) by Durst et al. (2006), where $D = \frac{1}{3}\bar{u}_{M} \lambda$ and hence, for T = constant. $\dot{m}_{\text{Brenner}}^{D} = \dot{m}_{\text{Durst}}^{D} .$