Boundary Conditions and Polymeric Drag Reduction for the Navier–Stokes Equations
Boundary Conditions and Polymeric Drag Reduction for the Navier–Stokes Equations
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DOI：
10.1007/s00205-021-01689-6
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发表时间：
2019-04
影响因子：
2.5
通讯作者：
Theodore D. Drivas;J. La
中科院分区：
文献类型：
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作者：
Theodore D. Drivas;J. La
减少湍流管道和通道流中的壁阻力是一个具有重要实际意义的问题。在工程应用中，末端官能化的聚合物链通常被用作减阻剂。这些聚合物漂浮在溶剂中，并在其链端之一附着（通过吸附或通过不可逆的化学结合）到基材（壁）。我们提出了一个偏微分方程模型来在简单的设置中研究这种设置，其中溶剂是粘性不可压缩的纳维-斯托克斯流体，占据光滑域的大部分，并且壁接枝聚合物处于所谓的蘑菇状态（聚合物间间距约为典型聚合物长度）。聚合物的微观描述通过流体壁切向应力的动态边界条件进入流体运动的宏观描述，类似于（但不同于）历史相关的滑移长度。我们建立了两个空间维度上强解的全局适定性，并证明了强欧拉解的无粘极限在一定速率下成立。此外，壁面摩擦系数和全局能量耗散与雷诺数成反比。这种缩放比例对应于层流摩擦系数的泊肃叶定律，从而量化了我们设置中的阻力减少。这些结果与没有聚合物添加剂的物理边界形成鲜明对比，这些物理边界通过例如无滑移条件建模，即使在二维中，通常也没有这样的结果。
Reducing wall drag in turbulent pipe and channel flows is an issue of great practical importance. In engineering applications, end-functionalized polymer chains are often employed as agents to reduce drag. These are polymers which are floating in the solvent and attach (either by adsorption or through irreversible chemical binding) at one of their chain ends to the substrate (wall). We propose a PDE model to study this setup in the simple setting where the solvent is a viscous incompressible Navier–Stokes fluid occupying the bulk of a smooth domain, and the wall-grafted polymer is in the so-called mushroom regime (inter-polymer spacing on the order of the typical polymer length). The microscopic description of the polymer enters into the macroscopic description of the fluid motion through a dynamical boundary condition on the wall-tangential stress of the fluid, something akin to (but distinct from) a history-dependent slip-length. We establish the global well-posedness of strong solutions in two-spatial dimensions and prove that the inviscid limit to the strong Euler solution holds with a rate. Moreover, the wall-friction factorand the global energy dissipationvanish inversely proportional to the Reynolds number. This scaling corresponds to Poiseuille’s law for the friction factorfor laminar flow and thereby quantifies drag reduction in our setting. These results are in stark contrast to those available for physical boundaries without polymer additives modeled by, for example, no-slip conditions, where no such results are generally known even in two-dimensions.