Second-order slip

Velocity profiles of Plane Poiseuille Flow Continuum-------First-order slip Second-order slip - -... 

We can see that the velocity profiles of these models become more and more different from each other with the decrease in the inverse Knudsen number. The difference between the velocity profiles of the first-order and second-order slip models is in the magnitude of the slip velocity, but the shapes of the velocity profiles are almost same. The second-order slip model predicts a larger and larger slip velocity than the first-order slip model when the inverse Knudsen number decreases. The Fukui-Kaneko model gives a medium slip velocity and velocity profile. 

Based on the nanoscale effect function, Peng et al. further derived the modihed Re5molds equations of the first-order slip, the second-order slip and the Fukui-Kaneko models [19]. The flow rate coefficients become the following expressions. 

Deissler, R.G., An analysis of second-order slip flow and temperature-jump boundary conditions for rarefied gases, Int. J. Heat Mass Transfer, Vol. 7, pp. 681-694, (1964). 

More reeently, [26] has eonfirmed the need to include the second order slip condition at higher Kn number values. Their work was both theoretical and experimental using nitrogen and helium in a silicon channels. They used the second order slip approximation to obtain the equation for the volumetric flow rate and related it to the ratio of inlet to outlet pressure. It was shown that when using the Navier-Stokes equation, the boundary conditions must be modified to include second order slip terms as the Knudsen number increases. They also studied in depth the accommodation coefficient Fv and verified the need for further study. It was shown that as the Knudsen number increases, the momentum accommodation value deviates further and further from unity for instance Kn -0.5 yields Fv 0.8 for helium. The values found for nitrogen were quite similar. The measurements agreed with past studies such as [11] for lower Kn. 

For a comparison we also included similar predictions by the second-order slip boundary condition of Hsia and Domoto (large dashed line). The form of their boundary conditions is similar to Cercignani s, Deissler s, and Schamberg s, and they all become invalid at around Kn = 0.1. This boundary condition performs worse than even the first-order Maxwell s boundary condition for large Kn values. Only the general slip boundary condition predicts the scaling of the velocity profiles accurately. 

Figure 2. Velocity scaling at wall and centerline of the channels for slip and transition flows. The linearized Boltzmann solution of Aoki is shown by triangles, and the DSMC simulations are shown by points. Theoretical predictions of velocity scaling for different values of b, and Hsia and Domoto s second-order slip boundary condition are also shown.

If the above equation (IP-based viscosity coefficient model) was further incorporated with the second-order slip velocity expression, i.e., Eq. 13... 

Maurer J, Tabelin P, Joseph P, Willaime H (2003) Second-order slip laws in microchannels for helium and nitrogen. Phys Fluids 15 2613-2621... 

Colin S, Lalonde P, Caen R (2004) Validation of a second-order slip flow model in rectangular microchannels. Heat Transfer Eng 25(3) 23-30... 

Hadjiconstantinou NG (2003) Comment on Cercignani s second-order slip coefficient. Phys Fluids 15(8) 2352-2354... 

Zhang HW, Zhang ZQ, Zheng YG, Ye HF (2010) Corrected second order slip boundary condition for fluid flows in nanochannels. Phys Rev E 81 066303... 

The slip flow boundary condition trend follows the experimental data. Equation (3.77) shows that the contribution of second-order term in the slip flow boundary condition is contrary to the first-order term. The effect of second-order slip correction is to reduce the increase in mass flow rate due to first-order slip leading to a closer comparison with the experiment. 

The coefficients tu, depend on the gas model and R is the specified gas constant. Since the Burnett equations are obtained by a second-order Chapman-Enskog expansion in Kn, they require second-order slip boundary condition. However, it may be noted that it has been observed that the second-order slip b.c. are inaccurate for Kn > 0.2. The Burnett equation can be used to obtain analytical/numerical solutions for at least a portion of the transition regime for a monoatomic gas. 

The implementation of second-order slip boundary condition requires obtaining the second derivative of the tangential velocity in the normal direction to the surface which may lead to computational difficulties. To circumvent this problem, the following general velocity slip boundary condition in nondimensional form is used. 

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