Interdisciplinary Applied Mathematics

Скачать в pdf «Interdisciplinary Applied Mathematics»


3 (y — 1) Kn2 Re dTs    , N


(57)


since |)|^| =    . The leading-order variation in the volumetric flowrate


under fixed is linear in Kn due to velocity slip, and quadratic in Kn due to thermal creep effects (for fixed Eckert number). However, since Kn ж M/Re and Ec ж M2, then we see that the thermal creep term is linear in Kn, i.e., proportional to Kn /M.


In order to maintain zero average flowrate in a channel under a prescribed pressure gradient for an incompressible flow, the following condition should be maintained:


In this case, if



dp


dx



9 (7-1)


2tt 7



Kn



2 этL


dx



Ec(l + 3(^)



_Kn_i


1+i Kn’



(5.8)




the flow creeps from cold to hot along the channel surface, where a positive pressure gradient creates back-flow in the middle of the channel (Kennard, 1938; Loeb, 1961). With regard to the effects of thermal creep on Fanning friction coefficient    of    the    flow    for    a fixed    volumetric flowrate,    the    ratio    of


the friction coefficient of a slip surface Cf to the friction coefficient Cf0 of a no-slip surface is given by









It is seen that for fixed flowrate Q, Eckert number Ec, and Reynolds number Re, the ratio of Fanning friction coefficients of slip flow to the no-slip flow changes significantly by varying the Knudsen number Kn. For flows without thermal creep effects (i.e.,    = 0.0), the extra terms in

Скачать в pdf «Interdisciplinary Applied Mathematics»

Метки