具壁面吸附層的同心圓管內冪律流體擴散滲透流探究

Abstract

本文旨在藉由研究具壁面吸附層的同心圓管內冪律流體擴散滲透流現象，了解吸附層對於管內非牛頓流體流況的影響，並冀望將其應用於驅動微流道中的流體以及控制管內的流向。本理論模型將同心圓管分為三個流體層，由內而外依次為內管壁面吸附層、核心流體層以及外管壁面吸附層。此外，流道上下游存在一固定的濃度梯度，因此會產生擴散滲透流動。工作流體方面，在內外吸附層中本文以牛頓流體作模擬，而核心流體層則使用冪律流體模型。再者，內外管壁面上的帶電高分子吸附層假設為存在平均分佈的固定電荷、電解液離子可自由通透且流體流動會受阻的多孔性介質層。基於上述物理模型，再藉由線性化的帕松-波茲曼方程組、布林克曼方程式以及柯西動量方程式可解得同心圓管內的電位、速度分佈及體積流率，並且以無因次內管半徑、無因次內外吸附層厚度、外管半徑與德拜長度比值、無因次內外吸附層阻力係數、無因次離子擴散係數差參數、無因次內外管壁面電位、無因次吸附層特徵電位、流動特性指數及稠度指數表達。研究結果顯示：具吸附層的同心圓管模型可近似為平行板、圓管及無吸附層同心圓管模型，故具通用性。再者，我們發現變動無因次離子擴散係數差參數與無因次吸附層特徵電位能改變管內的流率方向。不僅如此，藉由降低流動特性指數、降低稠度指數或者提高無因次吸附層阻力係數皆會使零體積流率線圖中無因次離子擴散係數差參數與無因次內外吸附層特徵電位同號區域 (第一與三象限) 內的正流率區面積變大。最後，變動外吸附層的帶電與阻力特性 (無因次的外吸附層阻力係數與特徵電位) 造成管內體積流率的最大變動範圍相較於變動內吸附層特性 (無因次的內吸附層阻力係數與特徵電位) 大。

Our research aims to theoretically analyze the diffusioosmotic flow of power-law fluids in an annular channel coated with polyelectrolyte layers (PELs). Through the investigations, we can realize the role of polyelectrolyte layers in non-Newtonian fluid transport and look forward to applying our work on fluid pumping or flow direction control in microfluidic devices. In the theoretical model, the annular channel was assumed to be composed by three different fluid layers, which are the inner PEL, the core flow region, and the outer PEL. Apart from that, a constant concentration gradient of the electrolyte was conducted downstream, so diffusioosmotic flows would be generated. Furthermore, we implemented the Newtonian fluid model in the inner and outer PELs and the power-law fluid model in the central flow region respectively. The polyelectrolyte layers mentioned above were defined as ion-penetrable and solute-permeable porous media where fluid would be subjected to frictional forces. On the basis of the physical model, we introduced modified Poisson-Boltzmann equations, Brink-man equations, and Cauchy momentum equation to obtain the analytical solutions to electric potentials and the semi-analytical solutions to velocity field and volume flow rate in terms of dimensionless inner radius, dimensionless thickness of the inner and outer PELs, ratio of the outer radius to Debye length, dimensionless drag coefficients of the inner and outer PELs, diffusivity difference parameter, dimensionless inner and outer wall zeta potentials, dimensionless characteristic potentials in the inner and outer PELs, flow behavior index, and flow consistency index. The results showed that our model was proved to be general that it could approach several simple models. Besides, we found that the choice of the diffusivity difference parameter and the dimensionless characteristic potentials in the inner and outer PELs could lead to different flow directions in the channel. Furthermore, by decreasing the flow behavior index, decreasing the flow consistency index or increasing the dimensionless drag coefficients of the inner and outer PELs would enlarge the 'toward downstream' area in the zero flow rate border curve plot where the diffusivity difference parameter and the dimensionless characteristic potentials in the inner and outer PELs are of the same sign. At last, the change of outer PEL properties (the dimensionless drag coefficient and characteristic potential in the outer PEL) causes a wider range of flow rate values than the change of inner PEL properties (the dimensionless drag coefficient and characteristic potential in the inner PEL).