Let us consider a fully-developed laminar flow between infinitely-large parallel plates with the stream wise pressure gradient (laminar Poiseuille flow). The pressure gradient in the stream wise (x) direction is given by dp/dx and constant. The gap between the top and bottom walls is H. The bottom and top walls are fixed. For simplicity, the momentum accommodation factor can be assumed to be v1.0. Choose the origin of the coordinate system at the bottom wall (y-H corresponds to the top wall). Answer the following questions. 1) Derive the velocity profile u-u(y) for the wall boundary condition with the non-slip flow. 2) Compute the volume flow rate for the non-slip flow QNs. 3) Determine the optimum parameter b in the second-order slip flow boundary condition proposed by Karniadakis & Beskok. 4) Derive the velocity profile u-u(y) for the second-order slip flow boundary condition proposed by Karniadakis & Beskok with the optimum parameter b determined in 3). 5) For large Knudsen numbers, the fluid viscosity u can be modeled as μ = μ o/(1+9.658Kn1-159), where pois the viscosity in the ordinary scale. Under this assumption, plot a graph for QSL2/QNs at 0<Kn<10, where QSL2 is the volume flow rate with the 2nd-order slip flow boundary condition. Ref.: A. Beskok, and G. Karniadakis, "REPORT: A MODEL FOR FLOWS IN CHANNELS, PIPES, AND DUCTS AT MICRO AND NANO SCALES," Microscale Thermophysical Engineering, Vol. 3, pp. 43-77 (1999).

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Let us consider a fully-developed laminar flow between infinitely-large parallel plates with the stream wise pressure gradient (laminar Poiseuille flow). The pressure gradient in the stream wise (x) direction is given by dp/dx and constant. The gap between the top and bottom walls is H. The bottom and top walls are fix