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The MEMS Handbook Introduction & Fundamentals (2nd Ed) - M. Gad el Hak Part 3

Tham khảo tài liệu 'the mems handbook introduction & fundamentals (2nd ed) - m. gad el hak part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Flow Physics 4-5 Kn 0.0001 0.001 0.01 0.1 1 10 100 I I I Continuum flow ordinary density levels 1 1 Transition regime moderately rarefied 1 1 1 1 1 1 1 1 1 1 1 1 1 Slip-flow regime 1 Free-molecule flow slightly rarefied 1 highly rarefied I FIGURE 4.2 Knudsen number regimes. In boundary layers the relevant length scale is the shear-layer thickness s and for laminar flows s 1 T 4.9 L Re Ma Ma Kn 4.10 Res CRe where Res is the Reynolds number based on the freestream velocity vo and the boundary layer thickness s and Re is based on vo and the streamwise length scale L. Rarefied gas flows are in general encountered in flows in small geometries such as MEMS devices and in low-pressure applications such as high-altitude flying and high-vacuum gadgets. The local value of Knudsen number in a particular flow determines the degree of rarefaction and the degree of validity of the Navier-Stokes model. The different Knudsen number regimes are determined empirically and are therefore only approximate for a particular flow geometry. The pioneering experiments in rarefied gas dynamics were conducted by Knudsen in 1909. In the limit of zero Knudsen number the transport terms in the continuum momentum and energy equations are negligible and the Navier-Stokes equations then reduce to the inviscid Euler equations. Both heat conduction and viscous diffusion and dissipation are negligible and the flow is then approximately isentropic i.e. adiabatic and reversible from the continuum viewpoint while the equivalent molecular viewpoint is that the velocity distribution function is everywhere of the local equilibrium or Maxwellian form. As Kn increases rarefaction effects become more important and eventually the continuum approach breaks down altogether. The different Knudsen number regimes are depicted in Figure 4.2 and can be summarized as follows Euler equations neglect molecular diffusion Navier-Stokes equations with no-slip boundary conditions Navier-Stokes equations with slip boundary .