Interdisciplinary Applied Mathematics

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3.1 Couette Flow: Slip Flow Regime


Shear-driven flows are encountered in micromotors, comb mechanisms, and microbearings. In the simplest form, the linear Couette flow can be used as a prototype flow to model such flows driven by a moving plate. Since the Couette flow is shear-driven, the pressure does not change in the stream-wise direction. Hence, the compressibility effects become important for large temperature fluctuations or at high speeds. Considering this, we first present an analysis of incompressible Couette flow with slip.


The velocity profile for incompressible Couette flows with slip can be obtained by considering a two-dimensional channel extending between y = 0 to y = h, with the top surface moving with a prescribed velocity UTO. By integrating the momentum equation (2.10) assuming no dependence on the streamwise direction, the following velocity profile is obtained in nondimensional form,


U _y/h+ Kn 3 (7 — 1) Kn2 Re 8TS t/oo 1 + 22~<t« Kn 2w 7 Ec dx ’


where dTs/dx is the tangential temperature gradient along the surfaces due to the thermal creep effects, and Ec is the Eckert number defined as (Ec = MCAT). The linear velocity profile of the Couette flow makes it impossible to incorporate high-order slip effects using equation (2.42), since d2U/dn? = 0. Therefore, high-order rarefaction effects cannot be captured unless the first-order slip coefficient Ci is modified as a function of Kn. In Section 3.2, we present a high-order slip model for Couette flows that is valid in the transition and free molecular flow regimes. In deriving the velocity profile in equation (3.1), we utilized Ci = 1.0. However, more accurate values for    the    slip    coefficient    have    been    determined    using    the    Boltzmann

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