We consider two-dimensional mass transport to a finite absorbing strip in a uniform shear flow as a model of surface-based biosensors. The quantity of interest is the Sherwood number Sh, namely the dimensionless net flux onto the strip. Considering early-time absorption, it is a function of the Péclet number Pe and the Damköhler number Da, which, respectively, represent the characteristic magnitude of advection and reaction relative to diffusion. With a view towards modelling nanoscale biosensors, we consider the limit Pe«1. This singular limit is handled using matched asymptotic expansions, with an inner region on the scale of the strip, where mass transport is diffusively dominated, and an outer region at distances that scale as Pe-1/2, where advection enters the dominant balance. At the inner region, the mass concentration possesses a point-sink behaviour at large distances, proportional to Sh. A rescaled concentration, normalised using that number, thus possesses a universal logarithmic divergence; its leading-order correction represents a uniform background concentration. At the outer region, where advection by the shear flow enters the leading-order balance, the strip appears as a point singularity. Asymptotic matching with the concentration field in that region provides the Sherwood number as $${\rm{Sh}} = {\pi \over {\ln (2/{\rm{P}}{{\rm{e}}^{1/2}}) + 1.0559 + \beta }},$$ wherein β is the background concentration. The latter is determined by the solution of the canonical problem governing the rescaled inner concentration, and is accordingly a function of Da alone. Using elliptic-cylinder coordinates, we have obtained an exact solution of the canonical problem, valid for arbitrary values of Da. It is supplemented by approximate solutions for both small and large Da values, representing the respective limits of reaction- and transport-limited conditions.

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小 Péclet 数质量传输到有限条：基于表面的生物传感器的平流-扩散-反应模型

我们考虑将二维质量传输到均匀剪切流中的有限吸收条作为基于表面的生物传感器的模型。感兴趣的量是舍伍德数 Sh，即带材上的无量纲净通量。考虑到早期吸收，它是 Péclet 数 Pe 和 Damköhler 数 Da 的函数，它们分别表示平流和反应相对于扩散的特征幅度。为了对纳米级生物传感器进行建模，我们考虑了极限 Pe«1。这个奇异极限是使用匹配的渐近扩展来处理的，内部区域在条带的尺度上，其中质量传输占主导地位，而外部区域在距离上的尺度为 Pe-1/2，平流进入主导平衡。在内部区域，质量浓度在远距离处具有点汇行为，与 Sh 成正比。使用该数字归一化的重新调整的浓度因此具有普遍的对数散度；它的前导校正表示均匀的背景浓度。在外部区域，剪切流的平流进入前序平衡，条带表现为点奇点。与该区域的浓度场渐近匹配提供舍伍德数为$${\rm{Sh}} = {\pi \over {\ln (2/{\rm{P}}{{\rm{e}}^{1/2}}) + 1.0559 + \beta } },$$其中β是背景浓度。后者由控制重新调整的内部浓度的规范问题的解决方案决定，因此是 Da 单独的函数。使用椭圆柱坐标，我们得到了典型问题的精确解，对任意 Da 值有效。它还辅以小 Da 值和大 Da 值的近似解，分别代表反应限制和传输限制条件的限制。