Cassie状态下内流的渐近Nusselt数
Asymptotic Nusselt numbers for internal flow in the Cassie state
摘要：
我们认为层流，充分发展，Poiffille流动的液体在卡西态通过非绝热，平行板微通道对称纹理与等通量脊。通过使用匹配的渐近展开，我们解析地开发（表观流体动力学）滑移长度和各种定义的努塞尔数的表达式。我们的小参数（$ε$）是脊的间距除以微通道的高度。当脊的方向平行于流动，我们量化的努塞尔数表达式在文献中的错误，并提供了一个新的封闭形式的结果。后者精确到$O\left(ε^2\right)$，对任何固体（脊）分数都有效，而目前文献中的那些精确到$O\left(ε^1\right)$，当固体分数接近零时，在重要极限处失效。当脊的方向横向于（周期性充分发展的）流动时，忽略滑移长度中惯性效应的相关误差为$O\left(ε^3\mathrm{Re}\right)$，其中$\mathrm{Re}$是基于其水力直径的通道尺度雷诺数。相应的努塞尔数表达式是新的，其精度被证明是依赖于雷诺数，Peclet数和普朗特数除了$ε$。操纵的解决方案遇到的内部温度的问题在附近的脊表明，热扩散阻力的经典结果更好地表示在polylogarithm函数。
英文摘要：
We consider laminar, fully-developed, Poiseuille flows of liquid in the Cassie state through diabatic, parallel-plate microchannels symmetrically textured with isoflux ridges. Through the use of matched asymptotic expansions we analytically develop expressions for (apparent hydrodynamic) slip lengths and variously-defined Nusselt numbers. Our small parameter ($ε$) is the pitch of the ridges divided by the height of the microchannel. When the ridges are oriented parallel to the flow, we quantify the error in the Nusselt number expressions in the literature and provide a new closed-form result. The latter is accurate to $O\left(ε^2\right)$ and valid for any solid (ridge) fraction, whereas those in the current literature are accurate to $O\left(ε^1\right)$ and breakdown in the important limit when solid fraction approaches zero. When the ridges are oriented transverse to the (periodically fully-developed) flow, the error associated with neglecting inertial effects in the slip length is shown to be $O\left(ε^3\mathrm{Re}\right)$, where $\mathrm{Re}$ is the channel-scale Reynolds number based on its hydraulic diameter. The corresponding Nusselt number expressions are new and their accuracy is shown to be dependent on Reynolds number, Peclet number and Prandtl number in addition to $ε$. Manipulating the solution to the inner temperature problem encountered in the vicinity of the ridges shows that classic results for thermal spreading resistance are better expressed in terms of polylogarithm functions.
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