Radial migration of suspended particles and its effect on multispecies flow inside a conduit

实验表明，当多物种悬浮液在管道中流动时，较大的物种向管道轴线迁移，而较小的物种向管道外围漂移。这种被称为血浆筛选的现象对于减少血流中的损失尤其重要，在血流中，较大的细胞在轴周围形成一个核心，而较小的颗粒则留在血管壁附近。尽管过去有几项研究，但尚不清楚等离子体筛选是由于多粒子流体动力学相互作用还是惯性动力学或细胞变形性而发生的。同样，对于上述因素对管道内粘性耗散的影响也没有准确的认识。在我们提出的研究中，我们将量化每个贡献因子对柱体约束多物种溶液中每个悬浮物种数量密度空间变化的个体影响。因此，我们将首先考虑压力驱动流动中具有不同尺寸和偏心距的刚性椭球的多物种系统，以解释流体动力学相互作用。然后，将惯性和颗粒变形能力分别考虑在内，以确定由于这些修改而导致的数密度的相对变化。对于每种情况，将计算通道内的压降，以描述不同流动条件下的粘性损失和流变性能。所提出的分析的复杂性是多方面的。首先，对于狭窄管道中的致密悬浮物，颗粒间和颗粒壁的粘性相互作用会导致流动应力的显著增加，并造成巨大的水动力摩擦的空间变化。因此，必须正确地解决数百个粒子之间以及粒子与围柱之间的相互作用。其次，如果考虑流体和溶质颗粒的惯性，则控制方程变得特别复杂。第三，如果认为悬浮体是可变形的，则必须在非预定义的表面上满足边界条件。幸运的是，我们最近开发的快速方法可以有效地求解这种情况下的流动方程。因此，我们将运用这一技术来克服预期的困难。智力优势：我们的关键数学创新是一种快速方案，该方案解决了代表管道和不同种类粒子的不连接的不同表面存在的流场。分子动力学、有限元和边界积分等传统方法难以考虑数百个悬浮体。相反，Stokesian动力学算法可以用于此目的。然而，尽管Stokesian动力学很有用，但它实际上仅限于无约束域的球形粒子，几次推广尝试都产生了不准确的或特定情况的模拟。因此，这种方法不能应用于圆柱形约束下的椭球粒子。而且，顾名思义，它只对不包含任何惯性项的Stokes方程有效。我们的广义方法解决了这些不足之处，因此我们可以解释惯性方程以及对应于管道束缚变形多物种系统的不同几何形状。更广泛的影响：本研究将解释在导管抛物面流中可变形颗粒径向迁移的不同原因的相对重要性。所得结果将有助于理解血浆在血管中筛选的原因及其对粘性耗散的影响。由于筛选过程取决于血液成分的基本特性，因此这种现象的任何差异都表明血液系统异常。因此，从长远来看，我们的分析将导致血栓形成、栓塞和异常出血等健康危害的定量预测。因此，通过注重及时预防而不是昂贵的治疗，可以减少医疗费用。我们的数学理论除了胶体体系的流动分析外，还有更广泛的科学意义。它也适用于其他方程或边界条件，如弹性或电动力学问题。如此广泛的应用范围将促进数学和生物流体两门具有启发性的课程，从而促进研究生和本科生的研究型教育。

When multispecies suspensions flow through conduits, experiments show that the larger species migrates towards the conduit axis whereas the smaller species drifts towards the periphery. The phenomenon known as plasma screening is especially crucial for loss reduction in blood flow where larger cells form a core around the axis leaving the smaller particles near the vessel walls. Despite several past studies, it is still not clear whether the plasma screening happens due to multiparticle hydrodynamic interactions or inertial dynamics or cell deformability. Similarly, there is still no accurate understanding on how the aforementioned factors affect the viscous dissipation inside the conduits. In our proposed study, we will quantify the individual effect of each contributing factor on the spatial variation of number density of each suspended species in a cylinder bound multispecies solution. Accordingly, we will first consider a multispecies system of rigid ellipsoids with different sizes and eccentricities in pressure driven flow to account for hydrodynamic interactions. Then, inertia and particle deformability will be included one by one to determine the relative changes in number density due to these modifications. For each case, the pressure drop inside the channel will be computed to describe the viscous loss and the rheological properties for different flow conditions. The complexity in the proposed analysis is manyfold. Firstly, for dense suspensions in narrow conduits, interparticle and particle wall viscous interactions cause major increase in flow stresses, and create huge spatial variation in hydrodynamic friction. Hence, the mutual interactions among hundreds of particles as well as between the particles and the confining cylinder have to be resolved properly. Secondly, if inertia of the fluid and the solute particles are taken into account, the governing equation becomes especially complicated. Thirdly, if the suspended bodies are considered deformable, the boundary conditions have to be satisfied on a surface which is not predefined. Fortunately, our recently developed fast methodology can efficiently solve flow equations in such situation. Thus, we will apply this technique to overcome the anticipated difficulties.Intellectual Merit: Our key mathematical innovation is a fast scheme which solves the flow field in presence of disconnected dissimilar surfaces representing the conduit and different species of particles. Conventional methods like molecular dynamics, finite element and boundary integrals encounter difficulties to take into account hundreds of suspended bodies. In contrast, Stokesian dynamics algorithm can be used for this purpose. However, despite its usefulness, Stokesian dynamics is actually restricted to spherical particles in unconfined domain several attempts for generalization yielded inaccurate or case specific simulations. So the method in the present form cannot be applied to ellipsoidal particles in cylindrical confinement. Moreover, as the name suggests, it is only valid for Stokes equation which does not involve any inertial term. Our generalized approach addresses these inadequacies so that we can account for inertial equations as well as different geometries corresponding to conduit bound deformable multispecies system.Broader Impact: This study will explain the relative importance of different causes contributing in radial migration of deformable particles in parabolic flow through a conduit. The resultant findings will be useful to understand the reason behind plasma screening in blood vessels and consequent effect on viscous dissipation. As the screening process depends on basic properties of blood components, any discrepancy in this phenomenon is indicative of hematological abnormality. Thus, in the long run, our analysis will lead to quantitative prediction of health hazards like thrombosis, embolism and abnormal hemorrhage. As a result, medical expenses can be reduced by focusing on timely prevention rather than expensive cure. Our mathematical theory has a wider scientific implication besides flow analysis in colloidal systems. It is applicable to other equations or boundary conditions as in elasticity or electrodynamics problems. Such broad scope of application will promote two stimulating courses on mathematics and biofluidics leading to research-based education of graduate and undergraduate students.