Abstract

Pressure-driven Newtonian fluid flow between grooved and flat surfaces is analysed with no-slip boundary conditions at walls. The effect of corrugation on the fluid flow is investigated using the mesh-free spectral method. The primary aim of the present work is to develop an asymptotic/semi-analytical theory for confined transverse flows to bridge the gap between the limits of thin and thick channels. The secondary aim is to calculate permeability with reference to the effect of wall corrugation (roughness) without the restriction of pattern amplitude. We performed mathematical modelling and evaluated the analytical solution for hydraulic permeability with respect to the flat channel. The Pad\(\acute{e}\) approximate is employed to improve the solution accuracy of an asymptotic model. The results elucidate that permeability always follows a decreasing trend with increasing pattern amplitude using the spectral approach at the long-wave and short-wave limits. The prediction of the spectral model is more accurate than the asymptotic-based model by Stroock et al. (Anal Chem 74(20):5306, 2002) and Pad\(\acute{e}\) approximate, regardless of the grooved depth and wavelength of the channel. The finite-element-based numerical simulation is also used to understand the usefulness of theoretical models. A very low computational time is required using the mesh-free spectral model as compared to the numerical study. The agreement between the present model and the fully resolved numerical results is gratifying. Regarding numerical values, we calculated the relative error for different theoretical models such as an asymptotic model, Pad\(\acute{e}\) approximate, and a mesh-free spectral model. The spectral model always predicts the maximum relative error as less than \(3 \%\), regardless of the large pattern amplitude and wavelength. In addition, the results of the molecular dynamic (MD) simulations by Guo et al. (Phys Rev Fluids 1(7):074102, 2016) and the theoretical model by Wang (Phys Fluids 15(5):1121, 2003) are found to be quantitatively compatible with the predictions of effective slip length from the spectral model in the thick channel limit.

Graphical abstract

中文翻译：

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使用谱法研究凹槽通道中的斯托克斯流

摘要

在壁面无滑移边界条件下，分析了凹槽表面和平坦表面之间的压力驱动牛顿流体流动。使用无网格谱法研究波纹对流体流动的影响。目前工作的主要目的是开发一种用于受限横向流的渐近/半解析理论，以弥合薄通道和厚通道极限之间的差距。第二个目标是参考壁波纹（粗糙度）的影响来计算渗透率，而不受图案幅度的限制。我们进行了数学建模并评估了平坦河道水力渗透率的解析解。采用Pad \(\acute{e}\)近似来提高渐近模型的求解精度。结果表明，使用长波和短波极限下的光谱方法，渗透率始终遵循随着模式幅度增加而下降的趋势。谱模型的预测比 Stroock 等人基于渐近的模型更准确。(Anal Chem 74(20):5306, 2002) 和 Pad \(\acute{e}\)近似，无论通道的凹槽深度和波长如何。基于有限元的数值模拟还用于了解理论模型的有用性。与数值研究相比，使用无网格光谱模型所需的计算时间非常短。目前的模型和完全解析的数值结果之间的一致性是令人满意的。关于数值，我们计算了不同理论模型的相对误差，例如渐近模型、Pad \(\acute{e}\)近似和无网格谱模型。无论图案幅度和波长如何，光谱模型始终预测最大相对误差小于\(3 \%\) 。此外，Guo等人的分子动力学（MD）模拟结果。(Phys Rev Fluids 1(7):074102, 2016) 和 Wang 的理论模型 (Phys Fluids 15(5):1121, 2003) 与来自谱模型的有效滑移长度的预测在数量上是一致的。粗通道限制。

图形概要