In micro-scales the characteristic dimension of the flow conduits are comparable to the mean-free-path of the gas media they operate in. Under such conditions the "continuum hypothesis" break-down. Constitutive laws that determine stress tensor and heat flux vector for continuum flows have to be modified in order to incorporate the rarefaction effects. The very well known "no-slip" boundary conditions for velocity and tempearture of the fluid on the walls are subject to modifications in order to incorporate the reduction of momentum and energy exchange of the molecules with the surroundings.
Deviation from continuum hypothesis is identified with Knudsen number, which is the ratio of the mean-free-path of the molecules to a characteristic length scale. This length scale should be choosen in order to include the gradients of density, velocity and temperature within the flow domain. For example for external flows boundary layer thickness, and for internal fully developed flows channel half thickness should be used as the characteristic length scale.
According to the Knudsen number the flow regimes can be divided into various regimes. These are: continuum, slip, transition and free-molecular flow regimes. In the figure below we show the governing equations for these flow regimes. Discrete particle or molecular based model is the Boltzmann equation. Boltzmann equation is an integro differential equation and solution of this equation is limited to few cases. The continuum based models are the Navier-Stokes equations. Euler equations corresponds to inviscid continuum limit which shows a singular limit sice the fluid is assumed to be inviscid and non-conducting. Euler flow correspods to Kn=0.0. The Navier-Stokes equations can be derived from the Boltzmann equation using the Chapman-Enskog expansion. As Knudsen number is larger than 0.1 the Navier-Stokes equations break-down and a higher level of approximation is obtained by carrying second order terms (in Kn) in the Chapman-Enskog expansion. A special form of such an equation is called the Burnett equations, for which the solution requires second-order accurate slip boundary conditions in Kn. The Burnett equations and consistant second-order slip boundary conditions is subject to some contraversy and a better way of solving high Knudsen number flow is through molecular based direct simulation techniques such as the Direct Simulation Monte Carlo method (DSMC)

## Flow regimes and governing equations:

Microflows are typically in the slip and early transitional flow regimes. Therefore, Navier-Stokes equations with appropriate slip boundary conditions govern these flows.

In our analysis the main assumption is the dynamic similarity of microflows to low pressure (rarefied) gas flows. Therefore, rarefied gas flows in low pressure environments and micro flows are dynamically similar if two flow conditions have identical Knudsen number, Reynolds number and Mach number, with similar geometries.

In the coming pages we present the main findings of our research in micro-flows in the slip flow regime. The details of our findings will be refered to the corresponding publications where the postcript files for these publications can be down loaded.

We mainly concantrate on the pressure-driven channel flows including separation effects and shear-driven channel flows as well as external flows around a micro-probe.
In our studies we have identified four main effects in micro-flows. These are:

• ## Thermal Creep

For pressure driven pipe and channel flows, we have developed a unified model which predicts the velocity distribution, volumetric and mass flow rate as well as pressure distribution for the entire Knudsen regime (0.0 < Kn < oo) . Please visit our Rarefied gas dynamics site.

### Acknowledgements

Back to Ali Beskok's web pages.