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
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.
For pressure driven pipe and channel flows, we have developed a
unified model which predicts the velocity
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.
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