Laminar Drag Reduction in Micro-Channels Using Ultrahydrophobic Surfaces

4 Laminar Drag Reduction in Micro-Channels Using Ultrahydrophobic Surfaces 

The question of the conditions to be satisfied by a moving fluid in contact with a solid body was one of considerable difficulty for quite some time, as pointed out by Goldstein (1965), and the assumption of no-slip is now generally accepted for practical purposes. On the other hand, if we can make an artificial solid surface where there is very little interaction between the surface and the liquid in contact with it, slip would be appreciable for liquid flow. The analysis of the phenomenon was presented by Watanabe et al. (1999). 

Because fluid slip occurs at highly water-repellent walls when the contact angle is about 150°, Watanabe et al. (1999) analyzed the friction factor of slip flow in a circular pipe. For a fully developed steady flow in a pipe, the Navier-Stokes equation can be written as 

By integrating this equation, and owing to the physical consideration that the velocity must be finite at r = 0, 

The constant Q is evaluated under the boundary conditions at the pipe wall r = a, u = Us. Then, the slip velocity Ug is determined (Goldstein 1965) from a macroscopic point of view  

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