Ellipsoidal Microhydrodynamics without Elliptic Integrals and How To Get There Using Linear Operator Theory: A Note on Weighted Inner Products
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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：10549-10551
- 全文大小：266K
- ISSN：1520-5045
文摘
In this research note we revisit the topic of microhydrodynamics of an ellipsoid in rigid body motion to arrive at the final resolution of a 140-year-old 鈥渕ystery鈥?that was featured in the dedication paper on the same topic in the Doraiswami Ramkrishna Festschrift. There, the initial focus was on the role of the theory of self-adjoint operators as the framework for proving that the surface tractions on a sphere had to be a constant multiple of the same rigid body motions of the boundary conditions. The ellipsoid was then considered as a simple example to illustrate the loss of this behavior for nonspherical particles. That goal was accomplished because for an ellipsoid, n路x, the dot product of the surface normal n and the point x on the ellipsoid surface, is the required nonconstant multiplier. The simplicity of this result is striking and has been noticed throughout its history with a number of authors remarking on the lengthy algebraic manipulations required to prove this simple result. In keeping with the theme of the Doraiswami Ramkrishna Festschrift, this note presents a short and simple proof that highlights the importance of the choice of the inner product, that is, the definition of the metric. The introduction of n路x = w(x) as a so-called weight function in the definition of the weighted inner product, as in 鉄?i>f, g鉄?sub>w = 鈭?f(s)g(s)w(s)ds over the appropriate metric space transforms the double layer operator (DLO) into a self-adjoint operator. From this it follows that the eigenfunctions of the adjoint with respect to the nonweighted inner product are w times the DLO eigenfunctions. Thus, the simplification noted in the companion paper is true for all eigenvalues and eigenfunctions of the double layer operator and not just the eigenvalue of 鈭? and its associated eigenfunction vRBM. These insights open the door to significant opportunities in the computational analysis of ellipsoidal particles in nanoparticle technology including topics such as perturbation methods for inertial and non-Newtonian effects, as we now have ready access to the spectral decomposition and biorthogonal expansions for the double layer operator.