一阶速度滑移边界条件下的Stokes第二问题

Stokes' Second Problem Under First-Order Velocity Slip Boundary Condition

本文基于分离变量和积分变换方法,针对在微尺度滑移区的Stokes第二问题进行了求解,得到了速度分布的解析解,并进一步得到了穿透深度的表达式.通过分析,发现由于微尺度速度滑移边界条件的影响,速度幅值和穿透深度皆有所减小.在此基础上,对平板受到的切应力、阻尼力和弹性力等进行了分析,结果表明弹性系数和阻尼系数均随振动频率的增大而增大.最后,通过与已有实验结果进行比较,发现基于本文的计算公式得到的器件谐振频率和品质因子与实验测量结果吻合很好.

Based on the methods of separation of variables and integral transform, Stokes＇ second problem was solved in the microscale slip region and the analytical solution to velocity distribution was obtained. Then the expression of penetration depth was derived. It is found through analysis that due to the effect of microscale velocity slip boundary condition, both velocity oscillation amplitude and penetration depth are reduced. On this basis, shear stress, damping force and elastic force on the plate are investigated. It is shown that both elastic coefficient and damping coefficient increase with the increase of oscillation fre- quency. Finally, compared with the available experimental resuks, the resonance frequency and quality factor of the device calculated by those computational formulae which are derived in this paper agree well with those measured results.