Microfluidic systems are usually fabricated with soft materials that deform due to the fluid stresses. Recent experimental and theoretical studies on the steady flow in shallow deformable microchannels have shown that the flow rate is a nonlinear function of the pressure drop due to the deformation of the upper soft wall. Here, we extend the steady theory of Christov et al. (2018) by considering the start-up flow from rest, both in pressure-controlled and in flow-rate-controlled configurations. The characteristic scales and relevant parameters governing the transient flow are first identified, followed by the development of an unsteady lubrication theory assuming that the inertia of the fluid is negligible, and that the upper wall can be modeled as an elastic plate under pure bending satisfying the Kirchhoff-Love equation. The model is governed by two non-geometrical dimensionless numbers: a compliance parameter $\beta$, which compares the characteristic displacement of the upper wall with the undeformed channel height, and a parameter $\gamma$ that compares the inertia of the solid with its flexural rigidity. In the limit of negligible solid inertia, $\gamma \to 0$, a quasi-steady model is developed, whereby the fluid pressure satisfies a nonlinear diffusion equation, with $\beta$ as the only parameter, which admits a self-similar solution under pressure-controlled conditions. This simplified lubrication description is validated with coupled three-dimensional numerical simulations of the Navier equations for the elastic solid and the Navier-Stokes equations for the fluid. The agreement is very good when the hypotheses behind the model are satisfied. Unexpectedly, we find fair agreement even in cases where the solid and liquid inertia cannot be neglected.

中文翻译：

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浅可变形微通道中的启动流

微流体系统通常由软材料制成，这些材料会因流体应力而变形。最近对浅层可变形微通道中稳定流动的实验和理论研究表明，由于上部软壁变形，流速是压降的非线性函数。在这里，我们扩展了 Christov 等人的稳态理论。(2018) 通过考虑压力控制和流速控制配置中的静止启动流量。首先确定控制瞬态流动的特征尺度和相关参数，然后发展非定常润滑理论，假设流体的惯性可以忽略不计，并且上壁可以模拟为纯弯曲下的弹性板，满足基尔霍夫-洛夫方程。该模型由两个非几何无量纲数控制：柔量参数 $\beta$，它将上壁的特征位移与未变形的通道高度进行比较，以及一个参数 $\gamma$，将固体的惯性与其抗弯刚度。在固体惯性可忽略的极限$\gamma\to 0$下，建立了一个拟稳态模型，其中流体压力满足非线性扩散方程，$\beta$为唯一参数，允许自相似压力控制条件下的溶液。这种简化的润滑描述通过弹性固体的 Navier 方程和流体的 Navier-Stokes 方程的耦合三维数值模拟得到验证。当模型背后的假设得到满足时，一致性非常好。不料，