# Interdisciplinary Applied Mathematics

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where n is the number density of the gas, Ac is the collisional cross-sectional area, and c = a/3RT is the mean molecular speed. At standard conditions tr « 1 nanosecond, and the residence time from the throat to exit is about 2 microseconds for a typical micronozzle tested by (Bayt, 1999; Bayt and Breuer, 2000), so in this case the flow is in equilibrium. However, in the supersonic portion of the nozzle the relaxation time could be comparable to residence time for some conditions, and this may lead to strong nonequilibrium effects. This, in turn, translates into losses of thermal energy, which is converted into kinetic energy as the gas expands. Similar effects can be induced by rotational nonequilibrium associated with a second time scale, the rotational relaxation rate. This time scale may also be comparable to the residence time and is larger inside the boundary layer. A rotational frozen flow could have 15% less kinetic energy.

Finally, the effect of exit losses and of outflow boundary conditions may be quite pronounced. Ivanov et al. (1999) have considered the effect of extrapolated boundary conditions used at the outflow and noted a 5-10% loss in efficiency if the plume is not modeled explicitly; similar results were obtained for large nozzles where back-flow may occur (plume contamination) (Gatsonis et al., 2000). Clearly, standard extrapolation boundary conditions are questionable in micronozzles, with the core of inviscid flow relatively small compared to external aerodynamics applications, where such outflow boundary conditions are employed routinely.

We now turn to micronozzles with lower thrust level (< 1 mN). These nozzles correspond to lower pressures, and therefore rarefaction effects have to be accounted for in simulations. In general, in this regime, continuum approaches overpredict the specific impulse even for high Reynolds numbers of order 1000. We start by considering the Rothe nozzle, which is similar to    the    nozzle shown    in the sketch    of Figure    6.30, but    it    has a    throat

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