Interdisciplinary Applied Mathematics

Скачать в pdf «Interdisciplinary Applied Mathematics»

l2m1/2


i/u)



(3.15)


^V—


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,

Скачать в pdf «Interdisciplinary Applied Mathematics»

Метки