# Interdisciplinary Applied Mathematics

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FIGURE 4.6. Mach number (left) and density (right) variation along the channel as a function of ^ for inlet conditions of Table 4.2 (the legend refers to the inlet Mach number).

aged over the channel of length L. The Fanning friction factor at any point in the channel can be evaluated as a function of local conditions at station

x,

f (x)

Ts(x)

±p(x)£/(x)2

where rs (rs = p^j) is the shear stress on the wall. Since the mass flux (pU) in the channel is constant, the Fanning friction factor can be assumed to be a function of dynamic viscosity only, i.e., p(T), assuming ЩT-/IJ is approximately a constant. Since the dynamic viscosity variations can be obtained from Sutherland’s law, the Fanning friction factor f essentially becomes a function of temperature. In particular,

ii =(t3/2T0 + 110

Po To) T +100

where T and To are the local and reference temperatures, respectively, and p and po are the local and reference dynamic viscosities, respectively. Therefore, for small temperature changes reported in the experiments (see Table 4.2) the friction factor is approximately constant. Similarly, the Reynolds number in the channel (Re = £^^) is a function of temperature, and becomes approximately constant for small temperature variations.

The Mach number variation in the channel for Fanno flow can be obtained for specified friction factor as see (Thompson, 1988), equation (6.34):

dM2    7M4(1 + ^M2)4/

1 -M2

dx

D

(4.10)

This equation,    integrated    from    inlet    state    (Min) to    any    station    x    down

stream of the channel, gives

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