Purdue Professor Solves 140-Year Fluid
Mechanics Enigma
Perdue University
WEST LAFAYETTE , Ind.
Anderson
School of Chemical
Engineering , Purdue
University
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 "mystery" 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·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·x = w(x) as a so-called weight function in the definition of the weighted inner product, as in ⟨ f, g⟩ w = ∫ f(s) g(s) w(s)d s over the appropriate metric space transforms the double layer operator Κ 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 eigenfunctions of Κ. 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 −1 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.
http://www.purdue.edu/newsroom/releases/2015/Q4/purdue-professor-solves-140-year-fluid-mechanics-enigma.html
The findings have potential implications for research and industry because
ellipsoid nanoparticles are encountered in various applications including those
involving pharmaceuticals, foods and cosmetics.
Like a sphere, the oblong ellipsoids undergo "rigid body motion"
when submerged in a fluid, meaning they do not deform while moving from side to
side and rotating. However, because an ellipsoid is not perfectly spherical, it
is counterintuitive that its rigid-body motion in a fluid could be described
using the same simple mathematical expression as spheres.
"In general, you would expect a very complicated expression because an
ellipsoid is not a perfect sphere," said Sangtae Kim, a distinguished professor in
Purdue's School of
Chemical Engineering.
Yet, that is not the case, presenting a quandary that Kim has been
pondering since his days as an undergraduate in the 1970s.
"It's been gnawing at me since then," he said.
MIT chemical engineer Howard Brenner wrote a paper in 1964 showing the
mathematics behind the simple formula, but it took pages of complex
calculations to arrive at the simple result.
"Dr. Brenner highlighted this simplicity," Kim said. "But
the simplicity of the result could only be shown by going through five to 10
pages of very messy algebra and calculations. In the end, everything cancels
and you get this final very simple result. It's almost like a miracle, which
has bothered me for a long time."
He has solved the enigma in a new paper appearing in November in a special
issue of the American Chemical Society (ACS) journal Industrial & Engineering
Chemistry Research. The special issue honors the 50th anniversary of Purdue's Doraiswami
Ramkrishna's doctoral dissertation, which is considered a landmark
in the history of chemical engineering. Ramkrishna is Purdue's Harry Creighton
Peffer Distinguished Professor of Chemical Engineering. Kim's paper was
highlighted as an ACS "editor's choice."
The ellipsoid enigma begins in 1876 and 1892, when scientists described how
an ellipsoid moves through surrounding fluid while traveling side-to-side and
rotating, respectively, causing pressure and stress on the object's skin
referred to as surface traction. Brenner later unified the mathematics for both
the side-to-side motion and rotation.
The new research demonstrated how an ellipsoid's interaction with fluid can
be described using the same type of simple mathematical pattern that applies to
spheres.
"The pattern has been known for 140 years, and the fundamental
underlying reason for why this simple pattern has to be true is now apparent
because of this new work being published," Kim said.
John Anderson, president emeritus of the Illinois Institute of Technology
and a professor of chemical engineering, said, "Dr. Kim's paper
definitively finalizes the 140-year development of intriguing relationships
among hydrodynamic properties of ellipsoids, relationships that have proven
invaluable to theorists trying to model the motion of particles in flowing
liquids and electric fields. A fascinating backstory is that Professor Kim
maintained his interest in proving the exactness of these useful relationships
even during his years in executive management in the pharmaceutical industry
and the National Science Foundation. The crucial spark was reignited last fall
when he was invited to speak in memory of his late colleague Howard
Brenner."
Professor Henry Power of the University
of Nottingham
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
=
ABSTRACT
Ellipsoidal Microhydrodynamics without Elliptic Integrals
and How To Get There Using Linear Operator Theory: A Note on Weighted Inner
Products
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 "mystery" 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·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·x = w(x) as a so-called weight function in the definition of the weighted inner product, as in ⟨ f, g⟩ w = ∫ f(s) g(s) w(s)d s over the appropriate metric space transforms the double layer operator Κ 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 eigenfunctions of Κ. 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 −1 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.
http://www.purdue.edu/newsroom/releases/2015/Q4/purdue-professor-solves-140-year-fluid-mechanics-enigma.html
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