Ellipsoidal Microhydrodynamics without Elliptic Integrals and How To Get There Using Linear Operator Theory

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• 作者：Sangtae Kim
• 刊名：Industrial & Engineering Chemistry Research
• 出版年：2015
• 出版时间：October 28, 2015
• 年：2015
• 卷：54
• 期：42
• 页码：10497-10501
• 全文大小：244K
• ISSN：1520-5045

文摘

In this special dedication paper for the Doraiswami Ramkrishna Festschrift, several remarkably simple results in viscous fluid mechanics (also known as microhydrodynamics) are examined in the context of linear operator theory and the properties of self-adjoint operators. In particular, we highlight, for the broader chemical engineering community, that for a small sphere undergoing rigid body motion (RBM) in Stokes flow the surface tractions are simply a multiplicative constant times that same RBM and that this amazing and simple result is the consequence of linear operator theory applied to the relevant self-adjoint operator. Thus, to provide an illustrative example of the general theory that this can be true only for the sphere, we reexamine the corresponding classical (1876 and 1964) results for the surface tractions on an ellipsoid. The ellipsoid not only provides the example as expected, but produces a useful result that in the so-called mobility problem where the force and torque on the ellipsoid are the known inputs, the surface tractions can be cast in a very simple form that is independent of elliptic integrals and other complexities usually associated with ellipsoidal geometries. This connection between linear operators and transport phenomena highlights the power of mathematics in unifying the pedagogical framework for chemical engineers and the great influence of Professor Ramkrishna over the past half-century.