# Interdisciplinary Applied Mathematics

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(2.23)

dEi — dEr dEidEw

where dEi and dEr denote the energy fluxes of incoming and reflected molecules per unit time, respectively, and dEw denotes the energy flux if all the incoming molecules had been reemitted with the energy flux corresponding to the surface temperature Tw. The perfect energy exchange case corresponds to aT = 1. A separate thermal accommodation coefficient can be defined for the effects of gas-surface interactions on transitional, rotational, and vibrational energies of the molecules. Experimental evidence indicates that under such interactions the transitional and rotational energy components are more affected compared to the vibrational energy of the molecules (Schaaf and Chambre, 1961). However, such refinements cannot be applied to macroscopic models, since the rarefaction effects are treated by solving the continuum energy equation with the temperature jump boundary condition. DSMC models (see Section 15.1) can be more flexible in employing various molecule-wall collision models for different modes of energy transfer, as we show in Section 15.4.

The tangential momentum accommodation coefficient (av) can be defined for tangential momentum exchange of gas molecules with surfaces

^    (2.24)

Ti    Tw

where Ti and Tr show the tangential momentum of incoming and reflected molecules, respectively, and tw is the tangential momentum of reemitted molecules, corresponding to that of the surface (tw = 0 for stationary surfaces).

The case of av =0 is called specular reflection,

where the    tangential    velocity    of    the    molecules    reflected from    the    walls    is

unchanged, but the normal velocity of the molecules is reversed due to the normal momentum transfer to the wall. In this case there is no tangential momentum exchange of fluid with the wall, resulting in zero skin friction. This is a limit of inviscid flow, where viscous stresses are zero. Hence

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