Predicting microscale gas flows and rarefaction effects through extended Navier–Stokes–Fourier equations from phoretic transport considerations

Abstract

We test an extended continuum-based approach for analyzing micro-scale gas flows over a wide range of Knudsen number and Mach number. In this approach, additional terms are invoked in the constitutive relations of Navier–Stokes–Fourier equations, which originate from the considerations of phoretic motion as triggered by strong local gradients of density and/or temperature. Such augmented considerations are shown to implicitly take care of the complexities in the flow physics in a thermo-physically consistent sense, so that no special boundary treatment becomes necessary to address phenomenon such as Knudsen paradox. The transition regime gas flows, which are otherwise to be addressed through computationally intensive molecular simulations, become well tractable within the extended quasi-continuum framework without necessitating the use of any fitting parameters. Rigorous comparisons with direct simulation Monte Carlo (DSMC) computations and experimental results support this conjecture for cases of isothermal pressure driven gas flows and high Mach number shock wave flows through rectangular microchannels.

Appendix: Phoretic transport of mass in the framework of extended Navier–Stokes equations

In the traditional derivations of the Navier–Stokes equations, the following assumption of no phoretic transport of mass is implicitly considered:

$$ u_{i} = 0 $$

(A1)

This suggests that the local density and temperature gradients (or corresponding pressure gradients) do not give rise to any additional transport of mass beyond the advective transport. However, this assumption contradicts the Fourier’s law of diffusive heat transport, given as

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

(A2)

This contradiction arises because local temperature gradients, acting as driving forces for heat diffusion, may also lead to phoretic (diffusive) mass flux.

Applying the kinetic theory of gases, the rate of molecular transport of mass may be given by (Bird et al. 1960): $\dot{m}_{i}^{ + } = 1/6\rho \bar{u}_{x}$ in the positive x _i -direction and $\dot{m}_{x}^{ - } = - {1 \mathord{\left/ {\vphantom {1 6}} \right. \kern-\nulldelimiterspace} 6}\rho \bar{u}_{x}$ in the negative x _i -direction, yielding a diffusive mean mass flux $\dot{m}_{i}^{D} = 0$ if no spatial density and temperature gradients exist in a flow. In the presence of thermodynamic property gradients of the fluid, a net phoretic mass flux results that can be expressed as

$$ \dot{m}_{i}^{p} = \frac{1}{6}\left[ {\rho \left( {x_{i} - \lambda } \right)\bar{u}_{i} \left( {x_{i} - \lambda } \right) - \rho \left( {x_{i} + \lambda } \right)\bar{u}_{i} \left( {x_{i} + \lambda } \right)} \right] $$

(A3)

where $\bar{u}_{i}$ is the molecular mean velocity in the i direction, which can be given as

$$ \bar{u}_{i} = \bar{u}_{M} = \sqrt {{\frac{8kT}{{\pi m_{\text{M}} }}}} $$

(A4)

where T is the absolute temperature, k is the Boltzmann constant, m _M is the molecular mass, and λ is the molecular mean free path of the considered ideal gas.

Series expansions of the ρ and $\bar{u}_{i}$ terms in the square brackets of Eq. A3, omitting the terms beyond first-order in λ, yield:

$$ \dot{m}_{i}^{p} \, = \frac{1}{6}\left[ {\left( {\rho \left( {x_{i} } \right) + {\frac{\partial \rho }{{\partial x_{i} }}}\left( { - \lambda } \right)} \right)\left( {\bar{u}_{i} \left( {x_{i} } \right) + {\frac{{\partial \bar{u}_{i} }}{{\partial x_{i} }}}\left( { - \lambda } \right)} \right) - \left( {\rho \left( {x_{i} } \right) + {\frac{\partial \rho }{{\partial x_{i} }}}\left( \lambda \right)} \right)\left( {\bar{u}_{i} \left( {x_{i} } \right) + {\frac{{\partial \bar{u}_{i} }}{{\partial x_{i} }}}\left( \lambda \right)} \right)} \right] $$

(A5)

The above equation can be rewritten to yield for the diffusive mass transport in the x _i -direction if only those product terms are considered that contain first-order derivatives, and also considering the isotropy of the molecular motion yielding $\bar{u}_{i} \left( {x_{i} } \right) = \bar{u}_{M} :$

$$ \dot{m}_{i}^{p} = - \frac{1}{3}\lambda \left[ {\bar{u}_{M} {\frac{\partial \rho }{{\partial x_{i} }}} + \rho {\frac{{\partial \bar{u}_{M} }}{{\partial x_{i} }}}} \right] $$

(A6)

Taking into account that the diffusion coefficient D can be written as $D = - {1 \mathord{\left/ {\vphantom {1 {3\left( {\lambda \bar{u}_{M} } \right)}}} \right. \kern-\nulldelimiterspace} {3\left( {\lambda \bar{u}_{M} } \right)}},$ it follows that:

$$ \dot{m}_{i}^{p} = - D\left[ {{\frac{\partial \rho }{{\partial x_{i} }}} + {\frac{\rho }{{\bar{u}_{M} }}}\,{\frac{{\partial \bar{u}_{M} }}{{\partial x_{i} }}}} \right] $$

(A7)

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

(A8)

With the above expressions for $\dot{m}_{i} ,$ the diffusive heat transport results in the following expression for the heat flux:

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

(A9)

The corresponding momentum flux, τ _ij ,  reads as follows:

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

(A10)

This expression may be rewritten to yield

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

(A11)

Considering the equation of state for ideal gases, one may write

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

(A12)

As an illustration, for the special cases of isothermal flows one may write

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

(A13)

In addition, one would also expect that the phoretic mass flux should be proportional to the gradient of the free energy, which is defined as $\psi = I + PV - TS;$ I and S being the internal energy and entropy of the system, respectively. Following this definition, one may write:

$$ {\frac{\partial \psi }{{\partial x_{i} }}} = {\frac{\partial U}{{\partial x_{i} }}} + {\frac{\partial PV}{{\partial x_{i} }}} - {\frac{\partial TS}{{\partial x_{i} }}} \Rightarrow {\frac{\partial U}{{\partial x_{i} }}} + \left( {V{\frac{\partial P}{{\partial x_{i} }}} + P{\frac{\partial V}{{\partial x_{i} }}}} \right) - \left( {T{\frac{\partial S}{{\partial x_{i} }}} + S{\frac{\partial T}{{\partial x_{i} }}}} \right) $$

(A14)

Using the Gibb’s relation (TdS = dU + PdV), one may express Eq. A14 as given below:

$$ {\frac{\partial \psi }{{\partial x_{i} }}} = \left( {V{\frac{\partial P}{{\partial x_{i} }}}} \right) - \left( {S{\frac{\partial T}{{\partial x_{i} }}}} \right) $$

(A15)

For isothermal flows of ideal gases, thus, one may have

$$ {\frac{\partial \psi }{{\partial x_{i} }}} \propto \frac{1}{P}{\frac{\partial P}{{\partial x_{i} }}} \propto {\frac{1}{\rho }}{\frac{\partial \rho }{{\partial x_{i} }}} \propto D{\frac{\partial \rho }{{\partial x_{i} }}} $$

(A16)