The unsteady viscous flow induced by streamwise-travelling waves of spanwise wall velocity in an incompressible laminar channel flow is investigated. Wall waves belonging to this category have found important practical applications, such as microfluidic flow manipulation via electro-osmosis and surface acoustic forcing and reduction of wall friction in turbulent wall-bounded flows. An analytical solution composed of the classical streamwise Poiseuille flow and a spanwise velocity profile described by the parabolic cylinder function is found. The solution depends on the bulk Reynolds number R, the scaled streamwise wavelength $$\lambda $$λ, and the scaled wave phase speed U. Numerical solutions are discussed for various combinations of these parameters. The flow is studied by the boundary-layer theory, thereby revealing the dominant physical balances and quantifying the thickness of the near-wall spanwise flow. The Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) theory is also employed to obtain an analytical solution, which is valid across the whole channel. For positive wave speeds which are smaller than or equal to the maximum streamwise velocity, a turning-point behaviour emerges through the WKBJ analysis. Between the wall and the turning point, the wall-normal viscous effects are balanced solely by the convection driven by the wall forcing, while between the turning point and the centreline, the Poiseuille convection balances the wall-normal diffusion. At the turning point, the Poiseuille convection and the convection from the wall forcing cancel each other out, which leads to a constant viscous stress and to the break down of the WKBJ solution. This flow regime is analysed through a WKBJ composite expansion and the Langer method. The Langer solution is simpler and more accurate than the WKBJ composite solution, while the latter quantifies the thickness of the turning-point region. We also discuss how these waves can be generated via surface acoustic forcing and electro-osmosis and propose their use as microfluidic flow mixing devices. For the electro-osmosis case, the Helmholtz–Smoluchowski velocity at the edge of the Debye–Hückel layer, which drives the bulk electrically neutral flow, is obtained by matched asymptotic expansion.

中文翻译：

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通道流中沿流行进的粘性波

研究了不可压缩层流通道中由展向壁速度的流向行波引起的非定常粘性流动。属于这一类的壁波已经发现了重要的实际应用，例如通过电渗透和表面声学强迫进行微流体流动控制以及减少壁面湍流流动中的壁面摩擦。找到了由经典的流向泊肃叶流和由抛物线圆柱函数描述的展向速度分布组成的解析解。解决方案取决于体积雷诺数 R、按比例缩放的流向波长 $$\lambda $$λ 和按比例缩放的波相速度 U。讨论了这些参数的各种组合的数值解。流动由边界层理论研究，从而揭示主要的物理平衡并量化近壁展向流动的厚度。Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) 理论也被用来获得一个解析解，它在整个通道上都是有效的。对于小于或等于最大流向速度的正波速，通过 WKBJ 分析出现转折点行为。在壁面和转折点之间，壁面法向粘性效应仅由壁面力驱动的对流来平衡，而在转折点和中心线之间，泊肃叶对流平衡了壁面法向扩散。在转折点，泊肃叶对流和来自壁面力的对流相互抵消，导致恒定的粘性应力和 WKBJ 解的分解。该流态通过 WKBJ 复合膨胀和 Langer 方法进行分析。Langer 解比 WKBJ 复合解更简单、更准确，而后者量化了转折点区域的厚度。我们还讨论了如何通过表面声力和电渗透产生这些波，并建议将它们用作微流体流动混合装置。对于电渗透情况，Debye-Hückel 层边缘的 Helmholtz-Smoluchowski 速度驱动大量电中性流，通过匹配渐近膨胀获得。我们还讨论了如何通过表面声力和电渗透产生这些波，并建议将它们用作微流体流动混合装置。对于电渗透情况，Debye-Hückel 层边缘的 Helmholtz-Smoluchowski 速度驱动大量电中性流，通过匹配渐近膨胀获得。我们还讨论了如何通过表面声力和电渗透产生这些波，并建议将它们用作微流体流动混合装置。对于电渗透情况，Debye-Hückel 层边缘的 Helmholtz-Smoluchowski 速度驱动大量电中性流，通过匹配渐近膨胀获得。