Ricco Pierre, Hicks Peter D
1Department of Mechanical Engineering, University of Sheffield, Sheffield, S1 3JD UK.
2School of Engineering, Fraser Noble Building, King's College, University of Aberdeen, Aberdeen, AB24 3UE UK.
J Eng Math. 2018;111(1):23-49. doi: 10.1007/s10665-018-9953-y. Epub 2018 Feb 23.
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 , the scaled streamwise wavelength , and the scaled wave phase speed . 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.
研究了不可压缩层流通道中展向壁面速度的流向行波所诱导的非定常粘性流动。属于此类的壁面波已发现重要的实际应用，例如通过电渗和表面声激励进行微流体流动操纵以及减少湍流壁面边界层流动中的壁面摩擦。找到了一个由经典的流向泊肃叶流动和由抛物柱面函数描述的展向速度剖面组成的解析解。该解取决于整体雷诺数 、尺度化的流向波长 以及尺度化的波相速度 。讨论了这些参数各种组合的数值解。通过边界层理论研究该流动，从而揭示主要的物理平衡并量化近壁展向流动的厚度。还采用温策尔 - 克拉默斯 - 布里渊 - 杰弗里斯（WKBJ）理论来获得一个在整个通道内有效的解析解。对于小于或等于最大流向速度的正波速，通过WKBJ分析会出现转折点行为。在壁面和转折点之间，壁面法向粘性效应仅由壁面强迫驱动的对流平衡，而在转折点和中心线之间，泊肃叶对流平衡壁面法向扩散。在转折点处，泊肃叶对流和来自壁面强迫的对流相互抵消，这导致粘性应力恒定并导致WKBJ解失效。通过WKBJ复合展开和兰格方法分析此流动区域。兰格解比WKBJ复合解更简单、更精确，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激励和电渗产生这些波，并提出将它们用作微流体流动混合装置。对于电渗情况，通过匹配渐近展开获得德拜 - 休克尔层边缘处驱动整体电中性流动的亥姆霍兹 - 斯莫卢霍夫斯基速度。
