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An Analytical Investigation of Start-Up Couette Flow in a Flat Nano-Channel
Vladimir V. Kulish and Chan Weng Kong

The Newtonian constitutive relation between the shear stress and velocity gradient becomes invalid in the case of the fluid flow that happens in channels where the characteristic length is comparable to the molecular mean free path. Hence, the Navier-Stokes equation is not applicable to such flows. In order to avoid this difficulty, a new analytical approach is proposed in this paper. It is assumed that a finite time lag exists between the onset of the shear stress and velocity gradient. This leads to the appearance of the wave term in the Navier-Stokes equation.

The new model proposed in the present paper is validated by solving a simplified version of the Navier-Stokes equation with the wave term in the case of the Couette flow that occurs in a flat channel where the height is of the order of nanometers. The solution is written in terms of the integral Volterra equation with the kernel represented by the modified Bessel functions. The transient solution of this equation has then been found numerically and the velocity profile of the flow in question has been recovered from the solution. The results indicate that wave propagation within the channel causes a peak in the velocity profile during the first 20 pico-seconds. The velocity profile reaches the steady state condition after 30 pico-seconds. As expected, the wave effect is first observed near to the upper moving plate before propagating towards the lower plate. The effects of diffusion can be neglected during this time since the relaxation time is about 390 pico-seconds.

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