Interdisciplinary Applied Mathematics

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equation, DSMC, and molecular dynamics simulations. For example, the theoretical model derived in (Ohwada et al., 1989b), using linearized Boltzmann equation for hard-sphere molecules predicts C1 = 1.111, for Kn < 0.1 for linear Couette flows.


The volumetric flowrate per channel width can be found by integrating equation (3.1)


Q 1    3 (7 — 1) Kn2 Re dTs


Uooh 2 2tt 7 Ec dx


(3.2)


The first term is independent of the Knudsen number, while the second term is due to the thermal creep effects, which can result in change of the flowrate in    the    channel.    In    Section    5.1    we    will present    heat    transfer analysis    of


shear-driven microflows, including the thermal creep effects corresponding to various prescribed heat flux conditions.


Once the analytical solution for the velocity distribution is known, the ratio of the skin friction coefficient for shear-driven slip flows and no-slip flows is given by


(3.3)


Cf    1


Cf0 1 + 2^-Kn’


where Cf = tw/(^pUwith tw the wall shear stress. The above equation is    obtained    for    constant    mass flow    rate    in    the    channel;    hence the    con


tribution of thermal creep is neglected. If the thermal creep effects were considered, the driving velocity Ux should have been modified in order to keep the volumetric flowrate of slip and no-slip flows the same.


For compressible shear-driven flows, an analytical solution can also be obtained given the simplicity of the geometry. In order to include the compressibility effects we must also specify the thermal boundary conditions. For the following analysis, we assumed that the upper plate temperature is specified to be TTO, while the bottom plate is adiabatic. Also for simplification, viscosity and thermal conductivity are assumed to vary linearly with temperature (i.e., к ~ p ~ T), and the Prandtl number is fixed. In this case it is possible to obtain the friction coefficient (see page 313 in Liepmann and Roshko, 1957):

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