# Interdisciplinary Applied Mathematics

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In this case    the    molecules    are    reflected from    the    walls    with    zero    average

tangential velocity. Therefore, the diffuse reflection is an important case for tangential momentum exchange (and thus friction) of the gas with the walls.

The tangential momentum and thermal accommodation coefficients depend on the gas and surface temperatures, local pressure, and possibly the velocity and the mean direction of the local flow. They are usually tabulated for    some    common    gases and    surfaces; see    Table    2.1    and    for    details

(Seidl and Steinheil, 1974; Lord, 1976). Diffuse reflection is likely to occur for rough surfaces. The values of av and aT are not necessarily equal, as shown in Table 2.1. Typically, it takes a few surface collisions for a molecule to adopt the average tangential momentum of the surface, but it takes more surface collisions to obtain the energy level of the surface. Under laboratory conditions, values as low as 0.2 have been observed (Lord, 1976). Very low values of    av    will    increase the slip    on    the walls    considerably    even    for    small

Knudsen number flows due to the (2 — a)/a factor multiplying the velocity slip and temperature jump equations.

Measurements or direct computation of accommodation coefficients are very difficult to obtain. The accommodation coefficients for microchannel flows were measured indirectly using the first-order (Arkilic et al., 2001)

and second-order slip flow theories (Maurer et al., 2003; Colin et al., 2004). Measurements of accommodation coefficients in (Arkilic et al., 2001) were obtained in the microchannel described in Chapter 1 (see Figure 1.18). Using high-resolution measurements for the mass flowrate and plotting it against the inverse pressure, the slope was computed, and based on the slip theory equations (see Chapter 4), the tangential momentum accommodation coefficient (TMAC) was obtained. The results of such measurements for argon and nitrogen are plotted in Figure 2.3 as a function of the Knud-sen number.    The    measured    value    is    av    « 0.80    for    nitrogen    or    argon    or

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