pith. sign in

arxiv: 1907.07444 · v1 · pith:4PFVMDLQnew · submitted 2019-07-17 · ❄️ cond-mat.soft

Adapting the Teubner reciprocal relations for stokeslet objects

Thomas A. Witten , Aaron Mowitz This is my paper

Pith reviewed 2026-05-24 20:12 UTC · model grok-4.3

classification ❄️ cond-mat.soft
keywords stokesletreciprocal theoremcolloidal swimmerselectrophoretic mobilityOseen matrixsurface velocity profilediscrete sourcesTeubner method
0
0 comments X

The pith

The net velocity of a body represented by discrete stokeslets equals a sum of shape-dependent contributions from each surface point, obtained by linear operations on the Oseen interaction matrix.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a method to predict the translation and rotation of an asymmetric colloidal body driven by any prescribed surface velocity profile. It does so by representing the body with a discrete set of stokeslet sources and adapting the Lorentz reciprocal theorem to those sources. The resulting contributions from each surface point depend only on the fixed shape and can be computed once via operations on the Oseen matrix between stokeslet pairs. The approach is then used to recover Teubner's method for electrophoretic mobilities of nonuniformly charged bodies. A reader would care because it turns the fluid problem into a set of precomputable linear combinations instead of repeated full solutions for every velocity profile.

Core claim

Self-propelled colloidal swimmers move by pushing the adjacent fluid backwards. The resulting motion of an asymmetric body depends on the profile of pushing velocity over its surface. We describe a method of predicting the motion arising from arbitrary velocity profiles over a given body shape, using a discrete-source stokeslet representation. The net velocity and angular velocity is a sum of contributions from each point on the surface. The contributions from a given point depend only on the shape. We give a numerical method to find these contributions in terms of the stokeslet positions defining the shape. Each contribution is determined by linear operations on the Oseen interaction matrix

What carries the argument

Adapted Lorentz reciprocal theorem for discrete stokeslet sources, which converts surface velocity profiles into net body motion via linear operations on the Oseen matrix between stokeslet pairs.

If this is right

  • Motion for any surface velocity profile on a given shape reduces to a linear combination of precomputed point contributions.
  • The shape-dependent contributions can be obtained numerically from the Oseen matrix without resolving the full flow field for each new profile.
  • The same construction recovers Teubner's method for finding mobilities of bodies with nonuniform surface charge.
  • The discrete-source representation directly supplies both translational and rotational velocities as sums over the same set of contributions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be applied to other surface-driven propulsion problems once the velocity profile is known, such as phoretic or Marangoni swimmers.
  • Precomputing the contribution vectors for a fixed shape would allow rapid evaluation of many different driving profiles, useful for design or optimization tasks.
  • The linear structure suggests a route to systematic coarse-graining or reduced-order models of micro-swimmers.

Load-bearing premise

The Lorentz reciprocal theorem can be adapted to discrete stokeslet sources so that the discrete representation accurately reproduces the continuous body's surface velocity and resulting motion.

What would settle it

Compute the electrophoretic mobility for a uniformly charged sphere using the stokeslet method and check whether it matches the known analytical result within the discretization error.

Figures

Figures reproduced from arXiv: 1907.07444 by Aaron Mowitz, Thomas A. Witten.

Figure 1
Figure 1. Figure 1: FIG. 1. Electrophoresis of a partially charged stokeslet ob ~ [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
read the original abstract

Self-propelled colloidal swimmers move by pushing the adjacent fluid backwards. The resulting motion of an asymmetric body depends on the profile of pushing velocity over its surface. We describe a method of predicting the motion arising from arbitrary velocity profiles over a given body shape, using a discrete-source "stokeslet" representation. The net velocity and angular velocity is a sum of contributions from each point on the surface. The contributions from a given point depend only on the shape. We give a numerical method to find these contributions in terms of the stokeslet positions defining the shape. Each contribution is determined by linear operations on the Oseen interaction matrix between pairs of stokeslets. We first adapt the Lorentz Reciprocal Theorem to discrete sources. We then use the theorem to implement the method of Teubner[1] to determine electrophoretic mobilities of nonuniformly charged bodies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to adapt the Lorentz reciprocal theorem to discrete stokeslet representations of colloidal bodies, enabling computation of net translational and angular velocities arising from arbitrary surface velocity profiles as a sum of shape-dependent contributions. Each contribution is obtained via linear operations on the Oseen interaction matrix between stokeslet pairs. The method is then used to implement Teubner's approach for electrophoretic mobilities of nonuniformly charged bodies.

Significance. If the discrete adaptation is shown to be exact or to converge controllably to the continuous case, the approach would allow precomputation of mobility contributions that depend only on shape, providing an efficient, linear-algebra-based route to mobilities for arbitrary profiles without refitting parameters. This would be a useful extension of reciprocal-theorem methods to stokeslet objects in soft-matter hydrodynamics.

major comments (2)
  1. [Adaptation of the Lorentz Reciprocal Theorem (abstract and main text)] The central claim that net velocity/angular velocity is exactly a sum of per-point contributions determined by linear operations on the Oseen matrix rests on the adaptation of the Lorentz reciprocal theorem to discrete singular sources. No derivation is supplied showing how the theorem is modified for the discrete case, how the singularity on the diagonal is handled, or that the resulting mobility exactly enforces the prescribed velocity at each stokeslet (as opposed to a weak/integrated sense). This is load-bearing for the assertion that the discrete representation faithfully reproduces continuous surface profiles.
  2. [Numerical method and Teubner implementation] No error analysis, continuum-limit proof, or numerical validation (e.g., recovery of known mobilities for a uniformly charged sphere or comparison against boundary-integral solutions for nonuniform profiles) is presented. Without such tests the mapping from continuous velocity profile to discrete sum remains unquantified, especially for nonuniform profiles where local errors near stokeslet singularities could accumulate.
minor comments (2)
  1. [Abstract] The abstract states that contributions 'depend only on the shape' but does not clarify whether this holds exactly for any stokeslet placement or only in the limit of dense coverage.
  2. [Introduction/Methods] Reference [1] (Teubner) is cited but the precise relation between the original continuous formulation and the discrete adaptation is not spelled out in a dedicated comparison section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed reading and helpful comments. We address each major point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Adaptation of the Lorentz Reciprocal Theorem (abstract and main text)] The central claim that net velocity/angular velocity is exactly a sum of per-point contributions determined by linear operations on the Oseen matrix rests on the adaptation of the Lorentz reciprocal theorem to discrete singular sources. No derivation is supplied showing how the theorem is modified for the discrete case, how the singularity on the diagonal is handled, or that the resulting mobility exactly enforces the prescribed velocity at each stokeslet (as opposed to a weak/integrated sense). This is load-bearing for the assertion that the discrete representation faithfully reproduces continuous surface profiles.

    Authors: We agree that the current manuscript provides only a brief statement of the adaptation without a self-contained derivation. In the revised version we will add an explicit section deriving the discrete form of the reciprocal theorem, including the regularization procedure used to remove the diagonal singularity and the demonstration that the resulting linear system enforces the prescribed velocity at each stokeslet location. revision: yes

  2. Referee: [Numerical method and Teubner implementation] No error analysis, continuum-limit proof, or numerical validation (e.g., recovery of known mobilities for a uniformly charged sphere or comparison against boundary-integral solutions for nonuniform profiles) is presented. Without such tests the mapping from continuous velocity profile to discrete sum remains unquantified, especially for nonuniform profiles where local errors near stokeslet singularities could accumulate.

    Authors: We accept that the manuscript lacks quantitative validation and error analysis. The revised manuscript will include (i) a brief continuum-limit argument relating the discrete sum to the continuous surface integral, (ii) numerical recovery of the known electrophoretic mobility for a uniformly charged sphere, and (iii) direct comparisons against boundary-integral solutions for selected nonuniform surface-velocity profiles. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation

full rationale

The paper's central derivation adapts the standard Lorentz Reciprocal Theorem to discrete stokeslet sources and then performs linear operations on the pre-existing Oseen interaction matrix between stokeslet pairs to obtain per-point contributions to net velocity and angular velocity. These steps are presented as direct extensions of known theorems and matrix algebra applied to a given shape representation; no parameters are fitted to target outputs, no predictions are defined in terms of themselves, and the cited Teubner method is invoked as an external reference rather than a self-referential load-bearing premise. The resulting mobility expressions therefore remain independent of the paper's own fitted values or redefinitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of adapting the Lorentz reciprocal theorem to discrete sources and on the linearity of operations on the Oseen matrix; no free parameters or new physical entities are introduced.

axioms (2)
  • domain assumption The Lorentz Reciprocal Theorem can be adapted to discrete stokeslet sources
    The paper states it first adapts the theorem to discrete sources as the foundation for the method.
  • domain assumption Contributions to net velocity from each surface point are obtained by linear operations on the Oseen interaction matrix
    The abstract presents this as the numerical implementation step.

pith-pipeline@v0.9.0 · 5672 in / 1386 out tokens · 24217 ms · 2026-05-24T20:12:43.971760+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Continuum-statistical dynamics of colloidal suspensions under kinematic reversibility

    cond-mat.soft 2026-02 unverdicted novelty 7.0

    A continuum framework shows Onsager reciprocity and colloidal diffusion coefficients emerge directly from the Lorentz reciprocal theorem applied to sedimentation flow problems.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · cited by 1 Pith paper

  1. [1]

    Teubner, The Journal of Chemical Physics 76, 5564 (1982)

    M. Teubner, The Journal of Chemical Physics 76, 5564 (1982)

    work page 1982

  2. [2]

    J. L. Moran and J. D. Posner, Annual Re- view of Fluid Mechanics 49, 511 (2017), https://doi.org/10.1146/annurev-fluid-122414-034456

    work page doi:10.1146/annurev- 2017

  3. [3]

    J. L. Anderson, Annual review of fluid mechanics 21, 61 (1989)

    work page 1989

  4. [4]

    H. A. Stone and A. D. Samuel, Physical review letters 77, 4102 (1996)

    work page 1996

  5. [5]

    K. A. Eslahian, A. Majee, M. Maskos, and A. W¨ urger, Soft Matter 10, 1931 (2014)

    work page 1931

  6. [6]

    Moyses, J

    H. Moyses, J. Palacci, S. Sacanna, and D. G. Grier, Soft Matter 12, 6357 (2016)

    work page 2016

  7. [7]

    Walther and A

    A. Walther and A. H. E. M¨ uller, Chemical Reviews , Chemical Reviews 113, 5194 (2013). 6

    work page 2013

  8. [8]

    C. C. Maass, C. Kr¨ uger, S. Herminghaus, and C. Bahr, Annual Review of Condensed Matter Physics 7, 171 (2016)

    work page 2016

  9. [9]

    M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Rev. Mod. Phys. 85, UNSP 1143 (2013)

    work page 2013

  10. [10]

    J. L. Anderson, Journal of Colloid and Interface Science 105, 45 (1985)

    work page 1985

  11. [11]

    Fair and J

    M. Fair and J. Anderson, Journal of Colloid and Interface Science 127, 388 (1989)

    work page 1989

  12. [12]

    Long and A

    D. Long and A. Ajdari, Phys. Rev. Lett. 81, 1529 (1998)

    work page 1998

  13. [13]

    K. S. Chae and A. M. Lenhoff, Biophysical Journal 68, 1120 (1995)

    work page 1995

  14. [14]

    Allison, BIOPHYSICAL CHEMISTRY 93, 197 (2001)

    S. Allison, BIOPHYSICAL CHEMISTRY 93, 197 (2001)

    work page 2001

  15. [15]

    ´A. V. Delgado, F. Gonz´ alez-Caballero, R. Hunter, L. Koopal, and J. Lyklema, Journal of colloid and inter- face science 309, 194 (2007)

    work page 2007

  16. [16]

    Moths and T

    B. Moths and T. A. Witten, Phys. Rev. E 88, 022307 (2013)

    work page 2013

  17. [17]

    M. Han, H. Wu, and E. Luijten, European Physical Journal-Special Topics 225, 685 (2016)

    work page 2016

  18. [18]

    Sacanna, W

    S. Sacanna, W. T. Irvine, L. Rossi, and D. J. Pine, Soft Matter 7, 1631 (2011)

    work page 2011

  19. [19]

    G. Meng, N. Arkus, M. P. Brenner, and V. N. Manoha- ran, Science 327, 560 (2010)

    work page 2010

  20. [20]

    Burelbach and H

    J. Burelbach and H. Stark, The European Physical Jour- nal E 42 (2019), 10.1140/epje/i2019-11769-y

    work page doi:10.1140/epje/i2019-11769-y 2019

  21. [21]

    Happel and H

    J. Happel and H. Brenner, Low Reynolds number hydro- dynamics: with special applications to particulate media , Mechanics of Fluids and Transport Processes (Springer Netherlands, 1983)

    work page 1983

  22. [22]

    A. J. Mowitz and T. A. Witten, Phys. Rev. E 96, 062613 (2017)

    work page 2017

  23. [23]

    To be published,

    L. Braverman, A. J. Mowitz, and T. A. Witten, “To be published,”

    work page

  24. [24]

    However the present derivation doesn’t depend on this explicit form

    The explicit form of L Lis well known and is called the Oseen tensor[21]. However the present derivation doesn’t depend on this explicit form. One may readily verify that the Oseen tensor satisfies the symmetry property shown for L Lij below

    work page

  25. [25]

    A better approximation for the velocity at ⃗ ri owing to the force at site i is required

    The diagonal elements L Lii are not defined for point stokeslets. A better approximation for the velocity at ⃗ ri owing to the force at site i is required. It is conve- nient to replace point the point force by distributing it uniformly over over some small region representing the continuum force on the fluid near site i. This procedure is not unique, but i...

    work page

  26. [26]

    A similar procedure treats motion caused by an external force. Here one requires that all the stokeslets move at a common velocity ⃗V and that the stokeslets provide suffi- cient screening that the interior of the object also moves at velocity ⃗V . Then the total of the stokeslet forces is the force required for this motion[22]

    work page

  27. [27]

    It depends on the local ionic environment of the surface in equilibrium without applied field ⃗E0

    The zeta potential that determines the slip velocity is the potential of the object surface relative to the bulk fluid. It depends on the local ionic environment of the surface in equilibrium without applied field ⃗E0. It also depends on the local charge density on the surface and is proportional to this density for weakly charged regions. Typical colloids ...

    work page

  28. [28]

    For general stokeslet objects, this locality may not be well defined, since an arbitrary set of stokeslets need not re- semble any smooth surface. However, if stokeslets are ar- ranged over a smooth surface with spacing much smaller than the local inverse curvature, the stokeslet object may approximate the corresponding smooth body, as noted above. Then th...

    work page

  29. [29]

    The functional form of this falloff is the well-known Oseen tensor mentioned above

    work page

  30. [30]

    Goldfriend, H

    T. Goldfriend, H. Diamant, and T. A. Witten, Phys. Rev. E 93, 042609 (2016)

    work page 2016

  31. [31]

    Goldfriend, H

    T. Goldfriend, H. Diamant, and T. A. Witten, Physics of Fluids 27, 123303 (2015)

    work page 2015

  32. [32]

    J. G. Kirkwood and J. Riseman, The Journal of Chemical Physics 16, 565 (1948)

    work page 1948

  33. [33]

    Zottl and H

    A. Zottl and H. Stark, Phys. Rev. Lett. 112, 118101 (2014)

    work page 2014