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Tham khảo tài liệu 'the mems handbook introduction & fundamentals (2nd ed) - m. gad el hak part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 10-26 MEMS Introduction and Fundamentals of the fluid outside of the EDL as a three-dimensional unsteady flow of a viscous fluid of zero net charge that is bounded by the following slip velocity condition -e Usip Esip 10.50 ụ where the subscript slip indicates a quantity evaluated at the slip surface at the top of the EDL in practice a few Debye lengths from the wall . The velocity along this slip surface is for thin EDLs similar to the electric field. This equation and the condition of similarity also hold for inlets and outlets of the flow domain that have zero imposed pressure-gradients. The complete velocity field of the flow bounded by the slip surface and inlets and outlets can be shown to be similar to the electric field Santiago 2001 . We nondimensionalize the Navier-Stokes equations by a characteristic velocity and length scale Us and Ls respectively. The pressure p is nondimensionalized by the viscous pressure pỉ.ỤẸ. The Reynolds and Strouhal numbers are Re pLsUs ụ and St Ls TUs respectively where T is the characteristic time scale of a forcing function. The equation of motion is ReSt Re u Vu Vp v2u 10.51 dt Note that the right-most term in Equation 10.51 can be expanded using a well-known vector identity V2u V V u V X V X u . 10.52 We can now propose a solution to Equation 10.52 that is proportional to the electric field and of the form u -Cy- E 10.53 Us where co is a proportionality constant and E is the electric field driving the fluid. Since we have assumed that the EDL is thin the electric field at the slip surface can be approximated by the electric field at the wall. The electric field bounded by the slip surface satisfies Faraday s and Gauss laws V E V X E 0 10.54 Substituting Equation 10.53 and Equation 10.54 into Equation 10.51 yields du ReSt dp Re u Vu Vp 10.55 This is the condition that must hold for Equation 10.53 to be a solution to Equation 10.51 . One limiting case where this holds is for very high Reynolds number flows where inertial and
