Interdisciplinary Applied Mathematics

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where v is the kinematic viscosity.

3.3.1 Quasi-Steady Flows

In this Section, we present extensions of the steady flow velocity and shear stress models in Section 3.2 to include oscillatory Couette flows. This particular approach will be valid for

any Stokes number flow in the slip flow regime (Kn < 0.1),

   quasi-steady flows for Kn < 12.

We define the quasi-steady conditions as the flow, where the velocity amplitude distribution always passes through (y/L,u/u0) = (0.5,0.5), resulting in a linear velocity distribution with equal amounts of slip on the oscillating and stationary walls. Our observations have shown that such conditions are typically achieved when в < 0.25.

Velocity Model

For oscillatory Couette flows, the momentum equation reduces to the following form:

du(y,t)    d2u(y,t)

= (31e)

An analytical solution of the above equation can be obtained for oscillatory flows with a specified frequency ш and amplitude U0, as shown in (Sherman, 1990). For a sinusoidal velocity excitation, a velocity response of the form u(y,t) = A{V(y) exp(jwt)} is expected, where the symbol A denotes the imaginary part of a complex expression, and V(y) is the amplitude governed by

Alternatively, we can write this as

where ф = у ^ is the complex frequency variable, and generalized solution of the above equation can be written form,

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