Ellipsoidal Microhydrodynamics without Elliptic Integrals and How To

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Ellipsoidal Microhydrodynamics without Elliptic Integrals and How to Get There Using Linear Operator Theory Sangtae Kim Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b01552 • Publication Date (Web): 12 Jun 2015 Downloaded from http://pubs.acs.org on June 18, 2015

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Industrial & Engineering Chemistry Research

Ellipsoidal Microhydrodynamics without Elliptic Integrals and How to Get There Using Linear Operator Theory Sangtae Kim ∗ School of Chemical Engineering, Purdue University, West Lafayette, IN 47906 E-mail: [email protected]

Abstract In this special dedication paper in the D. Ramkrishna Festschrift, several remarkably simple results in viscous uid 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 ow, 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. So as 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 ∗

To whom correspondence should be addressed 1

ACS Paragon Plus Environment


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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 inuence of Professor Ramkrishna over the past half-century.

Introduction Is is entirely appropriate that the special bond between chemical engineering and mathematics is featured as a primary theme in this dedication issue of IECR honoring Professor D. Ramkrishna.

Indeed, the chemical engineering community could do no better in 2004

than invite Professor Ramkrishna to pen (with his distinguished mentor N.R. Amundson as co-author) a special review article celebrating fty years of mathematical modeling in chemical engineering.

1

The present author was fortunate to participate in the second half

of this historical time frame, and as discussed in that review paper enjoyed the opportunity to explore the ramications of linear operator theory in viscous uid dynamics.

2

The foun-

dations of parallel computational microhydrodynamics, as described in Part IV of Kim and Karrila, show the profound inuence of Ramkrishna's classic book,

for Chemical Engineers , 3

Linear Operator Methods

and of his role in broadening the horizons of chemical engineers

beyond the world of dierential equations. On the occasion of this special issue, we revisit the linkage between linear operator theory and classical low Reynolds number microhydrodynamics (Stokes ow).

While the main

results concerning the surface tractions on a sphere in rigid body motion are not new (for

2

they are described in Part IV of Kim and Karrila ) these somewhat surprisingly simple results are not appreciated by broader chemical engineering community owing to our rational and well-founded adherence to pedagogical logic in undergraduate transport phenomena courses. This special celebration thus presents a compelling opportunity to share these wonders and their foundational connections to linear operator theory. So while the underlying theoretical foundations are not new, we will share new interpretations and insights that may be useful

2


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in the current research context. The organization of this paper is as follows. We start with the standard (spherical coordinates) discussion of the surface tractions on a translating sphere or a sphere in a uniform stream in Stokes ow. For those chemical enigneers whose careers did not continue as specialists in uid dynamics, this undoubtedly brings back bad memories of tedious coordinate tranformations of the equations of change and page-lling expressions for the components of the stress tensor of the Newtonian uid. It may come as a complete shock that when these classical results (in spherical coordinates) are transformed back to Cartesian coordinates, the surface traction simplies to a constant vector pointing in the direction of motion (or the direction of the uniform stream). And of course, that constant vector, upon multiplication with the surface area of the sphere, produces the correct result namely the famous

Stokes

Law for the hydrodynamic drag on a sphere. The initial surprise may turn into greater amazement and drift towards thoughts of some strange conspiracy, when the surface tractions on a sphere undergoing steady rigid body rotation in a viscous uid (or equivalently, a stationary sphere in a constant vorticity eld) turn out to be a constant multiple of that same rigid body rotation! The mystery of these amazing coincidences is solved by recourse to linear operator theory as applied to the integral representation for the velocity eld in Stokes ow. The six rigid body motions (translation along the three coordinate directions; rigid body rotations about the three coordinate axes) turn out to be eigenfunctions (of eigenvalue double layer operator

K

−1)

of the so-called

whereas the associated surface traction elds that result from the

solution of the six boundary value problems turn out to be eigenfunctions (for eigenvalue

−1)

of its adjoint

K∗ .

For the spherical particle (and only for the sphere) there is a special

geometric identity that forces

K = K∗ ,

i.e., the double layer operator is self-adjoint for a

sphere! The standard result in linear operator theory, e.g., as explained in

3

then leads to

the conclusion that the two sets of eigenfunctions are identical (to within a multiplicative constant).

3



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The unique situation of the sphere is further illustrated by considering a relatively simple generalization of the sphere to an ellipsoid with three unequal semi-axes. The expressions for the six surface tractions on an ellipsoid undergoing rigid body motion in Stokes ow are well known in the uid dynamics community, and thus we are in a position to demonstrate that as

K

is not self-adjoint for the ellipsoidal geometry, the resulting surface tractions are

not simply multiplicative constants of the associated rigid body motions. Moreover, we take the opportunity from our foray into linear operator theory and eigenspaces to arrive at more succinct expressions for surface tractions to highlight the fact that when the net force and torque are the known quantities (e.g., from a force and torque balance with external actions) the expression for the surface traction on an ellipsoid does not require ellipsoidal coordinates nor elliptic integrals!

This should make life easier for those involved in the creation of

mathematical models for colloidal suspensions, nanoparticles and other related research elds that may have steered clear of ellipsoidal shapes on the presumption of algebraic tedium.

Results and discussion Stokes Flow Revisited Undergraduate chemical engineering students are introduced to uniform Stokes ow past a sphere in the junior-level course on transport phenomena. For example, Chapter 2 of the classic text by Bird, Stewart and Lightfoot (BSL)

4

presents the solution for the velocity eld

followed by a detailed derivation of the Stokes drag on the sphere. To calculate the latter, expressions are presented for the pressure and stress tensor components which simplify as follows on the surface of the sphere:

4


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p|r=R = −

3 µv∞ cos θ 2 R

(1)

τrr |r=R = 0 τrθ |r=R =

Here

p denotes the dynamic pressure while τrr

(2)

3 µv∞ sin θ . 2 R and

τrθ

are the two relevant components of the

deviatoric stress tensor in an incompressible Newtonian uid of viscosity parameters are set by orienting the

v∞

and the spherical coordinates

to said radius

z -axis. R

r

z -axis and

θ

(3)

µ.

The geometric

in the direction of the uniform stream of speed

are dened by the usual convention with respect

The drag calculation involves only the values on the surface of the sphere of

(hence |r=R ) and BSL presents these results as

F

(n)

Z

2π

Z

= 0

F

(t)

Z

2π

Z

=

respectively.

F (n)

[−(p + τrr )] |r=R cos θ R2 sin θdθdφ = 2πµRv∞

(4)

[−τrθ ] |r=R (− sin θ) R2 sin θdθdφ = 4πµRv∞ ,

(5)

0

0

for the form drag

π

π

0

and friction drag

F (t)

arising from the normal and tangential forces

The pedagogical wisdom of employing

tive example of uniform ow past a sphere:

spherical coordinates

for this instruc-

is self-evident in an introductory course for

undergraduates; is convenient for deconstruction of the drag into the form drag and friction drag; and reinforces coordinate transformations and vector calculus reviews at the start of the semester! Nevertheless many students (and their instructors too) may be surprised by the hidden simplicity of the result when expressed in Cartesian coordinates. The pressure and stress components along with the corresponding unit vectors of the spherical coordinate system combine to produce:

t

= −p|r=R δr − τrθ |r=R δθ =

3 µv∞ 3 µv∞ (δr cos θ − δθ sin θ) = δz 2 R 2 R

5


(6)


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meaning that the surface traction

t

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(distribution of surface forces exerted by the uid on the

sphere) is a constant vector! Of course now that this result has been expressed in Cartesian coordinates, we see the immediate analogs of Equation (6) for uniform streams in the other two coordinate directions and the ultimate generalization (utilizing the linearity of the Stokes equations) for a sphere undergoing constant translation namely the surface traction

t

U

in a constant uniform stream,

v∞ ,

is a constant vector aligned in the net direction of the ambient

ow:

t

We note that the constancy of mechanics when the ansatz

t

=

3µ (v∞ − U ) . 2R

(7)

is often demonstrated in graduate level courses in uid

vi = (Ui − vi∞ )f (r) + xi xj (Uj − vj∞ )g(r)

and index notation are

used to solve for Stokes ow around a sphere. Because the surface traction is constant, a simple product with the surface area of the sphere gives the exact result for the classical Stokes drag on the sphere,

F

= 4πR2 t = 6πµR(v∞ − U ) .

(8)

As hinted in the introduction, this amazing result can be understood using the central concepts from linear operator theory. But rst we describe the corresponding results for a sphere undergoing steady rotation in a constant vorticity eld, so as to set the context for the general rigid body motion (RBM) of a sphere.

The Spinning Sphere in a Constant Vorticity Field We now consider a sphere of radius constant vorticity eld

Ω∞

× x.

R

rotating with a constant angular velocity

Ω

in a

For convenience, the sphere center is the origin and center

of rotation. As described in standard uid dynamics references

6


,2 4

the disturbance velocity

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eld is that of an anti-symmetric Stokes hydrodynamic dipole known as the rotlet,

3 R v = (Ω − Ω∞ ) × x r which decays as

r−2

and satises the appropriate RBM boundary condition at

again, the derivation in BSL's

Transport Phenomena

vφ

nonzero velocity component

4

(9)

r = R.

Here

is via spherical coordinates for the one

and we have omitted the straightforward steps in the gener-

alization of that result. The quantity of interest at this point is the hydrodynamic torque resisting the rotational motion and Chapter 3 of BSL

4

provides the standard derivation

(again in spherical coordinates) and the nal result is

T

= 8πµR3 (Ω∞ − Ω ) .

(10)

Here again, the following amazing and simple result for the surface traction is missed by the journey in spherical coordinates:

t

=

3 3µ (Ω∞ − Ω ) × x = T ×x , R 8πR4

(11)

that the surface traction is proportional to the rigid body rotation. For this simple expression for

t,

the BAC-CAB rule for the successive vector cross products in

x×t

=

3µ 3 x × [(Ω∞ − Ω ) × x] = x × (T × x ) , R 8πR4

(12)

leads to simplied surface integrals for the torque leading to the correct classical result. Alternatively, if we had known in advance that the surface traction on the sphere was proportional to the same rigid body rotation, the constant of proportionality on the RHS of

7



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Equation (11) could be determined by using the expression for the hydrodynamic torque:

Z

3 x×t dS = 8πR4 S

Z

3 x×(T ×x) dS = 8πR4 S

Z S

4π 3T 4π − dS = T . T R − x(T · x) dS = 8π 3

2

(13) We may now summarize the main point of this paper:

for rigid body motion (RBM) of

a spherical particle in Stokes ow, the surface traction is a vector proportional to that same RBM, with the constant of proportionality consistent with that required to produce the correct results for the Stokes drag and torque.

Integral Representations and Linear Operators We now shift gears to describe the connection between the previous discussion (concerning a sphere in Stokes ow) and linear operator theory.

The Stokes equations for ow past

submerged rigid particles have a formal solution in which the velocity in the uid domain (V ),

v (x),

is written as an integral representation that involve the Green's function and its

stress eld. So as to highlight the connection with the works of Professor Ramkrishna, we follow the notation as described in Kim and Karrila

2

  v (x) x ∈ V    Z  1 1 = − G(x − ξ) · t(ξ; v ) dS(ξ) v (x) x ∈ S 2  8πµ S    0 x∈P  Z 1 + K(x, ξ) · v (ξ) dS(ξ) , 2 S where

t(ξ ; v )

denotes the surface traction eld associated with

the particle surface

S.

v

(14)

evaluated at the point

ξ

on

The rst and second integrals on the RHS of this equation are known

as the single layer potential and double layer potential respectively, and it is the double layer potential that accounts for the jump discontinuity of domain (P ) to the uid domain ( V ).

v

in crossing

S

from the particle

The kernels in the single and double layer integral

operators correspond to the fundamental solution to the Stokes equation,

8


G(x)/(8πµ),

and

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its stress eld,

Σ (x),

i.e.,

−∇p + µ∇2 G = −F ext δ(x) ,

G(x)

=

1 ext F · G(x) , 8πµ

K(x, ξ) = −2n(ξ) · Σ (x − ξ) =

∇·G=0 ,

G(x) =

δ

|x|

+

xx

|x|3

(15)

.

(16)

(x − ξ)(x − ξ)(x − ξ) 3 n(ξ ) · . 2π |x − ξ|5

(17)

We now consider the six rigid body motions (RBMs) as solutions to the Stokes equations in the domain

P

(that is, temporarily reverse the role of particle and uid domains), and

apply the integral representation Equation (14), to those elds, so that

1 1 RBM v =− 2 2

Z

K(x, ξ) · v RBM dS(ξ) ,

x

∈S ,

(18)

S

where we used the fact that the RBM elds have zero tractions (to eliminate the single layer potential term) and introduced the minus sign on the RHS to account for the normal pointing into

P

when we applied the integral representation for the RBM elds. This then

leads to the rst major conclusion, namely,

the six rigid body motions are eigenfunctions of

K, the double layer operator, with eigenvalue −1,

K(v

RBM

Z )=

i.e., in operator notation,

K(x, ξ) · v RBM dS(ξ) = −v RBM .

(19)

S

It can be shown that for the single-particle geometry, this eigenspace for

λ = −1 has dimen2

sion six; the original proof due to Odqvist is reproduced in Kim and Karrila .

Surface Tractions and the Adjoint of the Double Layer Operator For the convenience of the reader, we summarize here the detailed derivations in Chapter

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Page 10 of 16

2

15 of Kim and Karrila . The integral representation Equation (14) of course holds for the special case of the disturbance velocity eld generated by a particle in rigid body motion in an otherwise quiescent uid of innite extent. When that result is added to the previously used representation for the six rigid body motions in the particle domain, we obtain three corresponding integral representations that are valid for the particle domain, particle surface and uid domain respectively. In each equation, the double layer potentials cancel because

v (ξ )

= v RBM

in all three cases and there was as before a minus sign to account for the normal

pointing into the particle when we applied the integral representation for the RBM elds in the particle domain. So we have simply:

v (x) v RBM v RBM

  x∈V    Z  1 G(x − ξ) · tRBM (ξ) dS(ξ) , =− x∈S  8πµ S    x∈P 

(20)

This formal procedure establishes for this special class of distubance velocity elds the reason why just the single layer potential alone can produce the required velocity representation A formal application of the Newtonian constitutive equation to Equation (20) leads to the corresponding equations for the surface traction; in particular the rst and third are with

x

approaching the surface from the uid side and the particle side:

  t (x) x ∈ V     Z 3 (x − ξ)(x − ξ)(x − ξ) RBM 1 RBM = n(x) · ·t (ξ) dS(ξ) , t (x) x ∈ S 2  |x − ξ|5 S 4π    0 x∈P  RBM

where

n(x)

appears in the integral equation because it appears on the LHS as part of

So ultimately for

x

∈ S , the middle case

(21)

tRBM .

of Equation (21), we arrive at a boundary integral

equation of the form:

1 RBM 1 t (x) = − 2 2

Z

K ∗ (x, ξ) · tRBM (ξ) dS(ξ) ,

S 10


(22)

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where the nal minus sign arises from the triadic in usual relationship between the kernels and

(x − ξ) because K ∗ (x, ξ) = K t (ξ, x) is the

K(x, ξ) is given in Equation ( ??).

is also the reason why for a given double layer density to uid domain is

2ψ

for the double layer operator

ψ,

This minus sign

the jump going from the particle

vs. −2ψ

for its adjoint operator, as can

be seen in Equations (14), (21) and (22). We have arrived at the second major conclusion of this section, namely,

the surface tractions produced by the six RBM boundary value problems

are eigenfunctions of K∗ , the adjoint operator of the double layer operator, with eigenvalue −1,

i.e., in operator notation,

∗

RBM

K (t

Z )=

K ∗ (x, ξ) · tRBM dS(ξ) = −tRBM .

(23)

S

This then sets the stage for the third and nal conclusion of this section. For the points and

ξ

x

on the surface of a sphere, there is a special geometrical identity involving the surface

normal at their locations,

n(ξ ) · (x − ξ )

= n(x) · (ξ − x)

as can be seen from the isoceles triangle with vertices at

(24)

x, ξ

and the sphere center (the

origin). When this result is inserted into the expressions for the kernels the conclusion is that

operator.

K = K∗ ,

i.e.,

K(x, ξ) and K ∗ (x, ξ)

for the sphere, the double layer operator is a self adjoint

The six-dimensional eigenspaces corresponding to

λ = −1

for

K

and

K∗

are thus

one and the same, and from the obvious geometric/symmetry connections associating the six basis functions we conclude that we must have six relations of the form

tRBM

= C v RBM .

Ellipsoids The geometric condition, Equation (24), required for the self-adjoint condition is not true in general for a nonspherical shape, so the eigenfunctions of

K

11


would not be eigenfunctions of


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K∗

Page 12 of 16

and vice versa. For example, for the ellipsoid

x2 y 2 z 2 + 2 + 2 =1 a2 b c with dierent values for the semi-axes

tRBM

are known explicitly

5

a > b > c,

(25)

the expressions for the surface tractions

from the (1876) velocity solution of Oberbeck

6

in ellipsoidal

coordinates, and thus we can show readily that the surface tractions are not RBM vector elds.

For example, for rigid body translation along a coordinate direction resulting in a

hydrodynamic force

F

in that direction, the traction eld

tRBM (x)

=

(n · x) F 4πabc

(26)

is the extension of Equation (8) to the ellipsoid where we have chosen instead of the net uniform stream

v∞

−U

F

as the known input

so as to highlight the analogy with the result for

the sphere. The integration of this traction eld on the surface of the ellipsoid involves the surface integral for

n·x

which equals

4πabc

(three times the volume of the ellipsoid from the diver-

gence theorem) and thus we do recover the hydrodynamic force, the factor

n·x

F,

as required. Furthermore,

is simply the perpendicular distance from the center of the ellipsoid to the

tangent plane at

x

and therefore varies in the range from

c

to

a

so as expected the traction

eld is not a constant vector, albeit it is always aligned in the direction of eld becomes a constant vector only for the special case when the sphere of radius

F.

The traction

a = b = c = R = n·x

(i.e.,

R).

In a similar fashion, the surface traction

tRBM (rot)

of the rotating ellipsoid is not a constant

multiple of that rotational velocity and again it is due to a factor of

12


n · x,

i.e., the analog of

Page 13 of 16

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Equation (11) is





2 −1

2

 (b + c )  P = 0   0

3(n · x) (P · T ) × x , tRBM (rot) (x) = 4πabc

0

0

(c2 + a2 )−1

0

0

(a2 + b2 )−1

   .  

(27)

This elliptic-integral-free result for the rotating ellipsoid is readily derived by combining

tRBM (rot)

as given in Brenner's (1964)

tRBM (rot) (x)



2

2

=

tour de force

3µ(n · x) (Q · (Ω ∞ − Ω )) × x , 4πabc

(28)



−1

 (b α2 + c α3 )  Q= 0   0

5

0

0

(c2 α3 + a2 α1 )−1

0

0

(a2 α1 + b2 α2 )−1

   ,  

with the mobility relations for ellipsoids in Table 3.3 of Kim and Karrila. expected elliptic integrals

Z αi = 0

∞

(a2i

αj , j = 1, 2, 3

a1 = a, a2 = b, a3 = c,

appear in Brenner's result for the surface tractions on the ellipsoid via his

erate limit of the sphere, our

P

Note that the

with

dλ p , 2 + λ) (a + λ)(b2 + λ)(c2 + λ)

his inputs are the rotational motions

2

(29)

(Ω ∞ − Ω )

(instead of the torque,

matrix reduces to

(2R2 )−1

Q

(30)

matrix, because

T ).

In the degen-

times the identity matrix and

Equation (27) simplies to Equation (11). The relatively simple results for the surface traction on an ellipsoid with the force or torque as the known inputs as given in Equation (26) and Equation (27) merit further discussion.

Even though the double layer operator is not self-adjoint for the ellipsoid, eigen-

functions of K∗ are n · x times the corresponding eigenfunctions of K for the eigenvalue −1. Note that for rigid body rotation, the rotational axis

P

·T

13


spans the same vector space


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as

T

becauses

Consequently,

P

Page 14 of 16

is a positive denite diagonal matrix and thus its inverse always exists.

(P · T ) × x

is a rigid body rotation and eigenfunction of

K.

In mathematical models of colloid science and nanotechnology, we often have a force and torque balance (e.g., the hydrodynamic drag equals all other external forces acting on the ellipsoid) at the microstructure level, and thus the hydrodynamic force and torque are known quantities or inputs into the model. In these so called

mobility

formulations of

particle science and technology, we now have simple (in the sense of elliptic-integral-free) expressions, Equations (26) and (27), for the surface traction distribution on the ellipsoid surface. This is in contrast to the historical solutions, the analogs of Equation (6), wherein the stress tensor components would be expressed in ellipsoidal coordinates and occupy entire pages of the original journal article. The elliptic integrals associated with the ratio of the semi-axes

a:b:c

are embedded within the input parameters

in lieu of the net uniform stream

v∞

−U

and net rotation

F

and

T

(Ω∞ − Ω ).

when we use them Finally, if the force

is not aligned with one of the principal axis of the ellipsoid (e.g., sedimentation of a tilted ellipsoid), the simple form shown here bypasses the classical decomposition/superposition procedure of solving translations in the three canonical directions and superposition of the three solutions weighted by the direction cosines.

Conclusions As part of the celebratory dedication to Professor Ramkrishna, this article drew attention to a remarkable property of the surface tractions on a sphere in Stokes ow with RBM boundary conditions, namely, that the surface tractions in the Stokes ow RBM boundary value problem are simply multiplicative constants of the associated RBM boundary condition. The claims can be veried readily by undergraduate students in the transport phenomena course, and yet the rational, mathematical explanation of these almost magical properties are beyond the scope of the undergraduate curriculum. But because they follow immediately

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from the theory of self-adjoint operators as covered in a graduate-level course on mathematical methods for chemical engineers, it would be appropriate to revisit them, along with the counter example described in this article for the ellipsoidal geometry, as illustrative examples of self-adjoint operators. For the microhydrodynamic community, this article also derives a previously unknown result for the ellipsoidal particle. eigenfunctions of

K∗

are simply

At the all important RBM eigenvalue at

n·x

−1,

times the six corresponding eigenfunctions of

the six

K (the six

RBMs). Given the importance of the ellipsoid in mathematical models of colloidal suspensions and nanoparticle technology, and despite the loss of the self-adjoint property, a more thorough investigation of the eigenspaces of

K∗

may bear useful fruit for the ellipsoid and

other nonspherical particle shapes that conform to classical curvilinear coordinate systems.

References 1. Ramkrishna, D.; Amundson, N.R. Mathematics in Chemical Engineering: Introspection.

2. Kim S.;

a 50 Year

AICHE J. 2004, 50, 7.

Karrila S. J.

Microhydrodynamics: Principles and Selected Applications ,

Butterworth-Heinemann: Stoneham, MA, 1991.

3. Ramkrishna D., Amundson, N.R.

Linear Operator Methods in Chemical Engineering with

Applications to Transport and Chemical Reaction Systems ,

Prentice-Hall:

Englewood

Clis, NJ, 1984.

4. Bird R. B.; Stewart W. E.; Lightfoot E. N.

Transport Phenomena , 2nd ed.; Wiley:

New

York, 2002.

5. Brenner, H., The Stokes Resistance of an Arbitrary Particle IV. Arbitrary Fields of Flow.

Chem. Eng. Sci. 1964, 19, 703.

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6. Oberbeck, A., On Steady-State Flow under Consideration of Inner Friction.

Angew. Math. 1876, 81, 62.

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J. Reine.

Ellipsoidal Microhydrodynamics without Elliptic Integrals and How To

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