pith. sign in

arxiv: 2605.21567 · v1 · pith:2IEJL5DDnew · submitted 2026-05-20 · ⚛️ physics.flu-dyn · physics.bio-ph

Cilia-driven transport in confined ducts: an active porous media model

JP Raimondi , Feng Ling , Eva Kanso This is my paper

Pith reviewed 2026-05-22 08:40 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn physics.bio-ph
keywords active porous mediacilia transportconfined ductsmetachronal wavesNavier-Stokes-Brinkmanlow Reynolds numberfluid transportbio-inspired pumps
0
0 comments X

The pith

Cilia in confined ducts show a linear trade-off between flow rate and sustainable pressure.

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

The paper develops an active porous medium model for dense arrays of beating cilia lining duct walls, driven by prescribed metachronal waves. It identifies the ciliary confinement ratio and the mean ciliary fraction as the two morphological parameters that set transport limits. Solving the incompressible Navier-Stokes-Brinkman equations numerically in the low-Reynolds-number regime, along with a complementary mean-field analysis, yields flows that follow a decreasing linear relation between flow rate and pressure generation. This relation supplies a physical account of why ciliated ducts range from high-throughput carpets to pressure-generating flames. The framework sits between classical envelope theories and full filament-resolved simulations, supplying design rules for bio-inspired microfluidic pumps.

Core claim

We model dense arrays of beating cilia lining duct walls as an active porous medium driven by prescribed metachronal waves. The resulting flows are described by the incompressible Navier-Stokes-Brinkman equations. We find that transport is characterized by a decreasing linear relationship between flow rate and pressure generation, marking a fundamental trade-off between throughput and sustainable adverse pressure. These results provide a unified physical interpretation of the morphological diversity of ciliated ducts, from high-throughput ciliary carpets to pressure-generating ciliary flames, and offer guiding principles for the design of bio-inspired microfluidic pumps.

What carries the argument

Active porous medium model of ciliary arrays with prescribed metachronal waves, governed by the incompressible Navier-Stokes-Brinkman equations.

Load-bearing premise

Dense ciliary arrays can be represented accurately as a homogeneous active porous medium with prescribed metachronal waves, without resolving individual filament dynamics or boundary effects at the duct scale.

What would settle it

Measure the pressure-flow curve in a microfluidic channel lined with artificial cilia and test whether the relation is linear and matches the predicted slope set by the confinement ratio and ciliary fraction.

Figures

Figures reproduced from arXiv: 2605.21567 by Eva Kanso, Feng Ling, JP Raimondi.

Figure 1
Figure 1. Figure 1: Ciliated systems, corresponding geometric measures, and biological data. The ciliary confinement ratio, 𝑐, denotes the ratio of the ciliary layer thickness to the lumen diameter. The ciliary fraction, ⟨𝛷⟩, defines the corresponding fraction of cilia within the ciliated region. (a) Ciliary carpets tend to have large lumen (50–1000, 𝜇m) with short (5–10, 𝜇m), wall-normal cilia. Activity is distributed over t… view at source ↗
Figure 2
Figure 2. Figure 2: Reference configuration and ciliary kinematics. (a) Undeformed reference configuration of the active Brinkman channel, with channel length 𝐿, height 𝐻, and ciliary layer thickness ℓ. The ciliary confinement ratio is defined as 𝑐 = 2ℓ/𝐻. Inset: cilia of diameter 𝑑 are uniformly spaced with center-to-center distance 𝑏, giving an average ciliary fraction ⟨𝛷⟩ = 𝑑/𝑏. The Brinkman screening length ℓ𝐵 is proporti… view at source ↗
Figure 3
Figure 3. Figure 3: Discrete ciliary carpets and corresponding continuum fields. Left to right: time snapshots of the discrete rod representation (number of rods not to scale), Brinkman coefficient field, and ciliary velocity field (colormap showing streamwise component 𝑢𝑐 = 𝒗𝑐 ·e𝑥 ). (a) Translating rod model, with 𝑎 = 5 𝜇m. (b) Rotating rod model, with 𝛼0 = 𝜋/6. (c) Kinematics reconstructed from experimental measurements (S… view at source ↗
Figure 4
Figure 4. Figure 4: Flows driven by prescribed ciliary activity. Left: instantaneous active porous media force density, 𝐵𝒗𝑐, colormap indicates streamwise component 𝐵𝒗𝑐 · e𝑥 in units of pN/𝜇m3 . Middle: instantaneous velocity fields at zero applied pressure, scaled by the maximum speed (indicated). Grey rods denote the extent of the ciliary carpet (rod spacing not to scale). Right: streamwise velocity profiles averaged over o… view at source ↗
Figure 5
Figure 5. Figure 5: Quantifying inertial effects. Lag magnitude is denoted 𝛥 and lag penetration depth 𝛿. (a) Period￾averaged flow profile, normalized lag magnitude, and normalized penetration depth at fixed ⟨𝐵⟩ = 104 and varied Re. Note, 𝑢(𝑦) profiles for Re < 10−3 coincide with the profile for Re = 10−3 . (b) Period-averaged flow profile, normalized lag magnitude, and normalized penetration depth at fixed Re = 0.01 and vari… view at source ↗
Figure 6
Figure 6. Figure 6: Flow field and pumping performance for fixed channel geometry and ciliary fraction. (a) Flow field at zero applied pressure 𝛥𝑃 = 0, i.e. the no-load flow. (b) Flow field at stall, 𝛥𝑃 = 𝛥𝑃𝑠 with zero net flow 𝑄 = 0. (c) Period averaged flow profiles with corresponding mean-field analytical predictions (dashed). Blue curves indicate 𝛥𝑃 = 0, orange curves indicate 𝛥𝑃 = 𝛥𝑃𝑠 , and gray curves indicate intermedi… view at source ↗
Figure 7
Figure 7. Figure 7: Flow profiles and pumping performance for varied ciliary fraction and fixed confinement ratio. Analytical predictions are indicated with dashed lines, color indicates the ciliary fraction ⟨𝛷⟩, with warmer colors indicating more material and higher 𝐵¯. (a) Period averaged flow profiles at 𝛥𝑃 = 0. (b) Period averaged flow profiles at the stall pressure of the largest ⟨𝛷⟩ plotted. (c) Pump curves correspondin… view at source ↗
Figure 8
Figure 8. Figure 8: Flow profiles and pumping performance for fixed ciliary fraction and varied confinement ratio. (a) No load flow profiles plotted on normalized y axes. (b) Flow profiles at the stall pressure of the smallest 2ℓ/𝐻. Increasing the ciliated fraction allows the pumps to sustain larger adverse pressure before flow reversal. (c) Pump curves corresponding with the profiles in (a,b). Larger ciliated fraction, and t… view at source ↗
Figure 9
Figure 9. Figure 9: Numerical results for varied confinement ratio (a-c) and ciliary fraction (d-f). (a) No-load flow rate versus confinement ratio. Increased fraction increases the no load flow rate at every confinement ratio. (b) Stall pressure versus confinement ratio. Increased ciliary fraction increases the stall pressure at every confinement ratio. (c) Maximum pumping efficiency versus confinement ratio. The amount of c… view at source ↗
Figure 10
Figure 10. Figure 10: Model results overlaid with biological data. Numerical no-load flow rate (a), stall pressure (b), and maximum pumping efficiency (c). Analytical no-load flow rate (d), stall pressure (e), and maximum pumping efficiency (f). Contour lines in indicate equal elevation and are spaced evenly in percentiles of the data. Colored markers correspond with the biological data in figure 1 and table 4. Blue markers in… view at source ↗
Figure 11
Figure 11. Figure 11: Representative images of ciliary lumen from published literature. (a) Cross section of the larval zebrafish brain ventricle Olstad et al. (2019). Colored regions indicate regions of ciliary activity. We measured the confinement ratio as the average of ℓ and 𝐻 taken from multiple locations throughout this cross section. The ciliary fraction was estimated from the published cilia number density Olstad et al… view at source ↗
Figure 12
Figure 12. Figure 12: Convergence of the numerical solver. (a,b) Convergence to a periodic steady state. (c,d) Spatial resolution study. (a) Instantaneous flow rate obtained by spatial averaging over one metachronal wavelength (solid lines) is compared with the period-averaged flow rate computed from temporal averaging at a fixed location (dashed lines). Results are shown for 𝐿 = 100 𝜇m and 𝐻 = 50, 100, 200 𝜇m with 𝛷ref = 0.2,… view at source ↗
Figure 13
Figure 13. Figure 13: Validation tests. (a) Poiseuille–Brinkman flow driven by a constant body force 𝛥𝑃 𝒆𝑥 in a homogeneous porous medium with Brinkman coefficient 𝐵¯. (b) Womersley flow driven by an oscillatory body force 𝛥𝑃 cos(𝜔𝑡) 𝒆𝑥 in the absence of Brinkman drag (𝐵¯ = 0). (c) Kolmogorov–Brinkman flow driven by a spatial body force 𝛥𝑃 sin(𝑘𝑥) 𝒆𝑦 with homogeneous Brinkman coefficient 𝐵¯. 102 103 104 N_x = N_y (number of mo… view at source ↗
Figure 14
Figure 14. Figure 14: Validation using Poiseuille–Brinkman flow. (a) Normalized velocity profiles for pressure-driven flow with homogeneous Brinkman coefficient 𝐵 = 0–106 at the highest tested resolution 𝑀𝑥 = 𝑁𝑦 = 8000. Numerical solutions (solid lines) agree with analytical solutions (dashed lines). Each profile is normalized by its maximum; increasing 𝐵 sharpens the near-wall gradients (inset). (b) Relative 𝐿 2 error versus … view at source ↗
Figure 15
Figure 15. Figure 15: Validation using oscillatory Womersley flow. (a) Normalized maximum amplitude velocity profiles for Womersley numbers Wo = 0.4–40 at high temporal (1000 time steps per period) and spatial (𝑀𝑥 = 𝑁𝑦 = 800) resolution. Numerical (solid) and analytical (dashed) solutions are indistinguishable; the profiles for Wo = 0.4 and 1.3 overlap. (b) Relative 𝐿 2 error, averaged over one period at steady state, versus t… view at source ↗
Figure 16
Figure 16. Figure 16: Validation using wall-bounded Kolmogorov-Brinkman flow. (a) Analytical flow field (left) and numerical flow field (right). Color bar indicates the magnitude of the horizontal component of the flow 𝑢. (b) Percent error between the analytical and numerical flow fields, as measured by 100 × |𝒖𝑛𝑢𝑚 − 𝒖𝑎𝑛𝑎 |/(max(|𝒖𝑎𝑛𝑎 |)). (c) Vertical component of the flow 𝑣(𝑥) at 𝑦 = 𝐻/2. (d) Horizontal component of the flow… view at source ↗
read the original abstract

Ciliated organs transport viscous fluids through confined ducts, yet how duct morphology and ciliary activity jointly set the limits of flow rate and sustainable pressure remains unclear. Here, we model dense arrays of beating cilia lining duct walls as an active porous medium driven by prescribed metachronal waves, and identify two key morphological parameters that govern transport: the ciliary confinement ratio and the mean ciliary fraction. The resulting flows are described by the incompressible Navier-Stokes-Brinkman equations, which we solve numerically using a spectral method in the low-Reynolds-number regime. We also develop a complementary mean-field analytical model. The active porous medium framework provides an intermediate description between classical envelope theories and filament-resolved simulations and enables a systematic investigation of how fluid transport is shaped by confinement and packing of ciliary material. We find that transport is characterized by a decreasing linear relationship between flow rate and pressure generation, marking a fundamental trade-off between throughput and sustainable adverse pressure. These results provide a unified physical interpretation of the morphological diversity of ciliated ducts, from high-throughput ciliary carpets to pressure-generating ciliary flames, and offer guiding principles for the design of bio-inspired microfluidic pumps.

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 models dense ciliary arrays in confined ducts as a homogeneous active porous medium with prescribed metachronal waves. It solves the incompressible Navier-Stokes-Brinkman equations numerically via a spectral method at low Reynolds number and develops a complementary mean-field analytical model. Two morphological parameters—the ciliary confinement ratio and mean ciliary fraction—are identified as governing transport. The central result is a strictly decreasing linear relationship between flow rate Q and adverse pressure generation, interpreted as a fundamental trade-off between throughput and sustainable pressure head. The framework is positioned as an intermediate description between classical envelope models and filament-resolved simulations, offering a unified explanation for morphological diversity in ciliated ducts from high-throughput carpets to pressure-generating flames.

Significance. If the homogeneous active-porous-medium idealization remains quantitatively faithful when filament compliance, hydrodynamic interactions, and load-dependent metachronal adjustment are restored, the work supplies a useful reduced-order description that enables systematic exploration of confinement and packing effects. The linear Q–ΔP trade-off, if shown to be robust rather than an algebraic consequence of the chosen forcing, could rationalize observed duct morphologies and inform bio-inspired microfluidic pump design. The provision of both numerical spectral solutions and an analytical mean-field closure is a strength.

major comments (2)
  1. [Model formulation and governing equations] The linear relation Q(ΔP) = Q0 − α ΔP follows directly from the linearity of the Stokes-Brinkman equations once the active body-force (or velocity) term and permeability are fixed; any superposition of the prescribed metachronal drive with an imposed pressure gradient yields strict linearity at low Re. This algebraic property holds independently of the specific values of the ciliary confinement ratio and mean ciliary fraction. The manuscript should therefore qualify the claim that the relation marks a “fundamental trade-off” by demonstrating that the same linearity persists (or is modified) when the active term is allowed to depend on the local flow, e.g., through compliant filament dynamics or flow-induced changes in metachronal coordination.
  2. [Numerical methods and validation] The weakest assumption is the representation of dense ciliary arrays as a homogeneous active porous medium with waves prescribed independently of the flow. The paper should provide a quantitative test—e.g., comparison of the mean-field predictions against a limited set of filament-resolved simulations at the duct scale—for at least one value of the confinement ratio to establish the error incurred by the homogeneity approximation.
minor comments (2)
  1. [Model formulation] Clarify the precise definition of the active forcing term in the Brinkman equation (is it a body force or an effective velocity boundary condition?) and state whether the permeability is taken constant or spatially modulated by the local ciliary fraction.
  2. [Results] The abstract states that the model “enables a systematic investigation”; the results section should include a brief parameter sweep table or figure showing how the slope α and intercept Q0 vary with the two morphological parameters.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below, indicating planned revisions to the manuscript where appropriate.

read point-by-point responses

  1. Referee: [Model formulation and governing equations] The linear relation Q(ΔP) = Q0 − α ΔP follows directly from the linearity of the Stokes-Brinkman equations once the active body-force (or velocity) term and permeability are fixed; any superposition of the prescribed metachronal drive with an imposed pressure gradient yields strict linearity at low Re. This algebraic property holds independently of the specific values of the ciliary confinement ratio and mean ciliary fraction. The manuscript should therefore qualify the claim that the relation marks a “fundamental trade-off” by demonstrating that the same linearity persists (or is modified) when the active term is allowed to depend on the local flow, e.g., through compliant filament dynamics or flow-induced changes in metachronal coordination.

    Authors: We agree that the linear Q(ΔP) relation is a direct algebraic consequence of the linearity of the Stokes-Brinkman equations with fixed active forcing and permeability. Within our model, this linearity constitutes a trade-off between throughput and pressure generation under the assumption of prescribed, flow-independent metachronal waves. To address the referee's suggestion, we will revise the manuscript to qualify the claim explicitly, noting that the relation is specific to this modeling choice. We will also add a concise discussion of how the relation might be modified by compliant filament dynamics or flow-dependent metachronal adjustments, supported by references to existing literature on ciliary mechanics. These revisions will appear in the updated version. revision: yes

  2. Referee: [Numerical methods and validation] The weakest assumption is the representation of dense ciliary arrays as a homogeneous active porous medium with waves prescribed independently of the flow. The paper should provide a quantitative test—e.g., comparison of the mean-field predictions against a limited set of filament-resolved simulations at the duct scale—for at least one value of the confinement ratio to establish the error incurred by the homogeneity approximation.

    Authors: We acknowledge that the homogeneous active porous medium idealization is the primary modeling assumption whose accuracy merits scrutiny. Direct filament-resolved simulations at the duct scale remain computationally prohibitive and lie outside the scope of this work, which develops the reduced-order framework as an intermediate description. In the revised manuscript we will substantially expand the discussion of limitations to include order-of-magnitude estimates of the homogeneity error and comparisons with prior filament-based studies, thereby clarifying the expected range of validity. revision: partial

standing simulated objections not resolved
  • Quantitative validation of the homogeneous approximation via direct comparison against filament-resolved simulations at the duct scale for at least one confinement ratio.

Circularity Check

1 steps flagged

Linearity of flow rate vs. adverse pressure follows directly from linearity of Stokes-Brinkman equations with prescribed metachronal forcing

specific steps
  1. other [Abstract]
    "We find that transport is characterized by a decreasing linear relationship between flow rate and pressure generation, marking a fundamental trade-off between throughput and sustainable adverse pressure."

    The model is defined via the incompressible Navier-Stokes-Brinkman equations solved numerically in the low-Reynolds-number regime with prescribed metachronal waves as the active drive. At low Re these equations are linear, so the velocity (and thus flow rate Q) is a linear superposition of the active forcing term and any imposed pressure gradient. The reported linear Q(DeltaP) = Q0 - alpha DeltaP is therefore guaranteed by construction of the linear system and the independent prescription of wave kinematics; it does not depend on solving the equations or on the specific values of the ciliary confinement ratio and mean ciliary fraction.

full rationale

The paper solves the incompressible Navier-Stokes-Brinkman system at low Re with metachronal waves prescribed independently of the flow. The resulting linear Q vs. DeltaP relation is an algebraic consequence of the linearity of the governing equations once the active body force and permeability are fixed; it does not emerge from the morphological parameters or require numerical solution. While the active-porous-medium idealization may still be useful as an intermediate model, the central 'fundamental trade-off' claim reduces to a property of the chosen linear PDE system rather than a non-trivial prediction about ciliary physics.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on treating cilia as a continuum active porous medium whose effective forcing is set by two morphological parameters; these parameters function as the primary controls but are not derived from first principles within the paper.

free parameters (2)
  • ciliary confinement ratio
    Morphological parameter that governs transport limits; introduced as one of two key controls.
  • mean ciliary fraction
    Packing parameter for ciliary material; introduced as the second key morphological control.
axioms (2)
  • domain assumption Low-Reynolds-number regime governs the flows
    Explicitly stated for the numerical solutions of the incompressible Navier-Stokes-Brinkman equations.
  • domain assumption Dense ciliary arrays behave as a homogeneous active porous medium with prescribed metachronal waves
    Core modeling choice that enables the Brinkman description and mean-field reduction.

pith-pipeline@v0.9.0 · 5729 in / 1299 out tokens · 30587 ms · 2026-05-22T08:40:23.281774+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

188 extracted references · 188 canonical work pages · 2 internal anchors

  1. [1]

    Biophysical journal , volume=

    Force generation and dynamics of individual cilia under external loading , author=. Biophysical journal , volume=. 2010 , publisher=

    work page 2010

  2. [2]

    Nature , pages=

    3D-printed low-voltage-driven ciliary hydrogel microactuators , author=. Nature , pages=. 2026 , publisher=

    work page 2026

  3. [3]

    Annals of Anatomy-Anatomischer Anzeiger , volume=

    Lymphatic lacunae of the mucosal folds of human uterine tubes—A rediscovery of forgotten structures and their possible role in reproduction , author=. Annals of Anatomy-Anatomischer Anzeiger , volume=. 2018 , publisher=

    work page 2018

  4. [4]

    Journal of morphology , volume=

    Morphology of the kidney in the West African caecilian, Geotrypetes seraphini (Amphibia, Gymnophiona, Caeciliidae) , author=. Journal of morphology , volume=. 2004 , publisher=

    work page 2004

  5. [5]

    Zoomorphology , volume=

    Ultrastructure of the metanephridial system in Aeolosoma bengalense (Annelida) , author=. Zoomorphology , volume=. 1994 , publisher=

    work page 1994

  6. [6]

    Journal of Morphology , volume=

    Fine structure of dicyemid mesozoans, with special reference to cell junctions , author=. Journal of Morphology , volume=. 1997 , publisher=

    work page 1997

  7. [7]

    Acta Zoologica , volume=

    Dorsolateral ciliary folds in the polychaete foregut: structure, prevalence and phylogenetic significance , author=. Acta Zoologica , volume=. 1996 , publisher=

    work page 1996

  8. [8]

    Human Reproduction , volume=

    Ciliary function and motor protein composition of human fallopian tubes , author=. Human Reproduction , volume=. 2015 , publisher=

    work page 2015

  9. [9]

    Elife , volume=

    Regulation of cilia abundance in multiciliated cells , author=. Elife , volume=. 2019 , publisher=

    work page 2019

  10. [10]

    Current Biology , volume=

    Ciliary beating compartmentalizes cerebrospinal fluid flow in the brain and regulates ventricular development , author=. Current Biology , volume=. 2019 , publisher=

    work page 2019

  11. [11]

    Stem Cells and Development , volume=

    The number of stem cells in the subependymal zone of the adult rodent brain is correlated with the number of ependymal cells and not with the volume of the niche , author=. Stem Cells and Development , volume=. 2012 , publisher=

    work page 2012

  12. [12]

    Frontiers in neuroanatomy , volume=

    The neural elements in the lining of the ventricular-subventricular zone: making an old story new by high-resolution scanning electron microscopy , author=. Frontiers in neuroanatomy , volume=. 2015 , publisher=

    work page 2015

  13. [13]

    Scientific Reports , volume=

    High throughput screening of airway constriction in mouse lung slices , author=. Scientific Reports , volume=. 2024 , publisher=

    work page 2024

  14. [14]

    Cells , volume=

    Identification of cilia in different mouse tissues , author=. Cells , volume=. 2021 , publisher=

    work page 2021

  15. [15]

    Transactions of the American Microscopical Society , pages=

    Internal anatomy of Meiopriapulus fijiensis (Priapulida) , author=. Transactions of the American Microscopical Society , pages=. 1989 , publisher=

    work page 1989

  16. [16]

    Zoomorphology , volume=

    Ultrastructure and phylogenetic significance of the head kidneys in Thalassema thalassemum (Thalassematinae, Echiura) , author=. Zoomorphology , volume=. 2011 , publisher=

    work page 2011

  17. [17]

    Acta zoologica , volume=

    Ultrastructure of the flame bulbs and protonephridial capillaries of Artioposthia sp.(Platyhelminthes, Tricladida, Geoplanidae) , author=. Acta zoologica , volume=. 1992 , publisher=

    work page 1992

  18. [18]

    Cell and Tissue Research , volume=

    On the excretory system of the rotifer Habrotrocha rosa Donner , author=. Cell and Tissue Research , volume=. 1978 , publisher=

    work page 1978

  19. [19]

    Cell and Tissue Research , volume =

    Georg K. Cell and Tissue Research , volume =. 1962 , publisher =

    work page 1962

  20. [20]

    Ductules, collecting ducts, and osmoregulatory cells , author=

    Fine structure of the protonephridial system in planaria: II. Ductules, collecting ducts, and osmoregulatory cells , author=. Zeitschrift f. 1968 , publisher=

    work page 1968

  21. [21]

    Acta Zoologica , volume=

    The ciliated brain duct of Oikopleura dioica (Tunicata, Appendicularia) , author=. Acta Zoologica , volume=. 1982 , publisher=

    work page 1982

  22. [22]

    Applied scientific research , volume=

    Analytical investigation of the fluid flow in the interface region between a porous medium and a clear fluid in channels partially filled with a porous medium , author=. Applied scientific research , volume=. 1996 , publisher=

    work page 1996

  23. [23]

    International Journal of Heat and Fluid Flow , volume=

    Fluid mechanics of the interface region between a porous medium and a fluid layer—an exact solution , author=. International Journal of Heat and Fluid Flow , volume=. 1990 , publisher=

    work page 1990

  24. [24]

    Proceedings of the National Academy of Sciences , volume=

    Finding the ciliary beating pattern with optimal efficiency , author=. Proceedings of the National Academy of Sciences , volume=. 2011 , publisher=

    work page 2011

  25. [25]

    Physical Review Letters , volume=

    Multisynchrony in active microfilaments , author=. Physical Review Letters , volume=. 2020 , publisher=

    work page 2020

  26. [26]

    Nature Communications , volume=

    Flow physics of nutrient transport drives functional design of ciliates , author=. Nature Communications , volume=. 2025 , publisher=

    work page 2025

  27. [27]

    Journal of Fluid Mechanics , volume=

    Optimal feeding in swimming and attached ciliates , author=. Journal of Fluid Mechanics , volume=. 2025 , publisher=

    work page 2025

  28. [28]

    eLife , volume=

    Feeding rates in sessile versus motile ciliates are hydrodynamically equivalent , author=. eLife , volume=. 2026 , publisher=

    work page 2026

  29. [29]

    Physical Review Fluids , volume=

    Nutrient transport in concentration gradients , author=. Physical Review Fluids , volume=. 2025 , publisher=

    work page 2025

  30. [30]

    Nature communications , volume=

    Structure and function relationships of mucociliary clearance in human and rat airways , author=. Nature communications , volume=. 2025 , publisher=

    work page 2025

  31. [31]

    Transport in Porous Media , volume=

    Effects of cilia movement on fluid velocity: II numerical solutions over a fixed domain , author=. Transport in Porous Media , volume=. 2020 , publisher=

    work page 2020

  32. [32]

    Transport in Porous Media , volume=

    Effects of cilia movement on fluid velocity: I model of fluid flow due to a moving solid in a porous media framework , author=. Transport in Porous Media , volume=. 2021 , publisher=

    work page 2021

  33. [33]

    Journal of cell science , volume=

    Ciliary activity of cultured rabbit tracheal epithelium: beat pattern and metachrony , author=. Journal of cell science , volume=. 1981 , publisher=

    work page 1981

  34. [34]

    Nature Reviews Physics , volume=

    The multiscale physics of cilia and flagella , author=. Nature Reviews Physics , volume=. 2020 , publisher=

    work page 2020

  35. [35]

    Swirling Instability of the Microtubule Cytoskeleton , author =. Phys. Rev. Lett. , volume =. 2021 , month =. doi:10.1103/PhysRevLett.126.028103 , url =

    work page doi:10.1103/physrevlett.126.028103 2021

  36. [36]

    Nature Physics , volume=

    Self-organized intracellular twisters , author=. Nature Physics , volume=. 2024 , publisher=

    work page 2024

  37. [37]

    Journal of Fluid Mechanics , volume=

    Measurements of the unsteady flow field around beating cilia , author=. Journal of Fluid Mechanics , volume=. 2021 , publisher=

    work page 2021

  38. [38]

    Journal of theoretical Biology , volume=

    Muco-ciliary transport in the lung , author=. Journal of theoretical Biology , volume=. 1986 , publisher=

    work page 1986

  39. [39]

    Physical Review Research , volume=

    Lagrangian transport by ciliary arrays , author=. Physical Review Research , volume=. 2025 , publisher=

    work page 2025

  40. [40]

    2013 , publisher=

    Data-driven modeling & scientific computation: methods for complex systems & big data , author=. 2013 , publisher=

    work page 2013

  41. [41]

    Proceedings of the National Academy of Sciences , volume=

    Highly motile nanoscale magnetic artificial cilia , author=. Proceedings of the National Academy of Sciences , volume=. 2021 , publisher=

    work page 2021

  42. [42]

    Proceedings of the National Academy of Sciences , volume=

    Spontaneous phase coordination and fluid pumping in model ciliary carpets , author=. Proceedings of the National Academy of Sciences , volume=. 2022 , publisher=

    work page 2022

  43. [43]

    Nature physics , volume=

    Flow physics guides morphology of ciliated organs , author=. Nature physics , volume=. 2024 , publisher=

    work page 2024

  44. [44]

    arXiv preprint arXiv:2108.01693 , year=

    A multiscale biophysical model gives quantized metachronal waves in a lattice of cilia , author=. arXiv preprint arXiv:2108.01693 , year=

    work page arXiv

  45. [45]

    Lab on a Chip , volume=

    Magnetically-actuated artificial cilia for microfluidic propulsion , author=. Lab on a Chip , volume=. 2011 , publisher=

    work page 2011

  46. [46]

    Microsystems & nanoengineering , volume=

    Microfluidic pumping using artificial magnetic cilia , author=. Microsystems & nanoengineering , volume=. 2018 , publisher=

    work page 2018

  47. [47]

    Elife , volume=

    Stem cells and fluid flow drive cyst formation in an invertebrate excretory organ , author=. Elife , volume=. 2015 , publisher=

    work page 2015

  48. [48]

    Zeitschrift f

    Fine structure of the protonephridial system in Planaria , author=. Zeitschrift f. 1968 , publisher=

    work page 1968

  49. [49]

    Pumping fluids through conduits , author=

    Living in a physical world X. Pumping fluids through conduits , author=. Journal of biosciences , volume=. 2007 , publisher=

    work page 2007

  50. [50]

    Physical review letters , volume=

    Is the zero Reynolds number approximation valid for ciliary flows? , author=. Physical review letters , volume=. 2019 , publisher=

    work page 2019

  51. [51]

    1975 , publisher=

    On the formal structure of physical theories , author=. 1975 , publisher=

    work page 1975

  52. [52]

    Journal of Computational Physics , volume=

    An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology , author=. Journal of Computational Physics , volume=

    work page

  53. [53]

    Advances in imaging and electron physics , volume=

    A reference discretization strategy for the numerical solution of physical field problems , author=. Advances in imaging and electron physics , volume=. 2002 , publisher=

    work page 2002

  54. [54]

    2003 , school=

    Discrete exterior calculus , author=. 2003 , school=

    work page 2003

  55. [55]

    Discrete Exterior Calculus

    Discrete exterior calculus , author=. arXiv preprint math/0508341 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  56. [56]

    Bulletin of the American mathematical society , volume=

    Finite element exterior calculus: from Hodge theory to numerical stability , author=. Bulletin of the American mathematical society , volume=

    work page

  57. [57]

    International Conference on Large-Scale Scientific Computing , pages=

    A locally conservative mimetic least-squares finite element method for the Stokes equations , author=. International Conference on Large-Scale Scientific Computing , pages=. 2009 , organization=

    work page 2009

  58. [58]

    Compatible spatial discretizations , pages=

    Principles of mimetic discretizations of differential operators , author=. Compatible spatial discretizations , pages=. 2006 , publisher=

    work page 2006

  59. [59]

    Journal of Computational Physics , volume=

    Mixed mimetic spectral element method for Stokes flow: A pointwise divergence-free solution , author=. Journal of Computational Physics , volume=. 2013 , publisher=

    work page 2013

  60. [60]

    Measurement Science and Technology , volume=

    A bi-symmetric log transformation for wide-range data , author=. Measurement Science and Technology , volume=. 2012 , publisher=

    work page 2012

  61. [61]

    Journal of theoretical biology , volume=

    Flow patterns around ciliated microorganisms and in ciliated ducts , author=. Journal of theoretical biology , volume=. 1982 , publisher=

    work page 1982

  62. [62]

    Biophysical journal , volume=

    Ciliary motion modeling, and dynamic multicilia interactions , author=. Biophysical journal , volume=. 1992 , publisher=

    work page 1992

  63. [63]

    Journal of Fluid Mechanics , volume=

    The discrete-cilia approach to propulsion of ciliated micro-organisms , author=. Journal of Fluid Mechanics , volume=. 1976 , publisher=

    work page 1976

  64. [64]

    Journal of Fluid Mechanics , volume=

    Fluid transport by cilia between parallel plates , author=. Journal of Fluid Mechanics , volume=. 1978 , publisher=

    work page 1978

  65. [65]

    Physical Review Fluids , volume=

    Coarse graining the dynamics of immersed and driven fiber assemblies , author=. Physical Review Fluids , volume=. 2019 , publisher=

    work page 2019

  66. [66]

    Ciliary and flagellar membranes , pages=

    Introduction to cilia and flagella , author=. Ciliary and flagellar membranes , pages=. 1990 , publisher=

    work page 1990

  67. [67]

    Bioscience , volume=

    Cilia and diseases , author=. Bioscience , volume=. 2014 , publisher=

    work page 2014

  68. [68]

    Physics of Fluids , volume=

    Cilia beating patterns are not hydrodynamically optimal , author=. Physics of Fluids , volume=. 2014 , publisher=

    work page 2014

  69. [69]

    Science , volume=

    Cilia-based flow network in the brain ventricles , author=. Science , volume=. 2016 , publisher=

    work page 2016

  70. [70]

    British journal of diseases of the chest , volume=

    Ciliary function in health and disease , author=. British journal of diseases of the chest , volume=. 1985 , publisher=

    work page 1985

  71. [71]

    Science , volume=

    A human syndrome caused by immotile cilia , author=. Science , volume=. 1976 , publisher=

    work page 1976

  72. [72]

    Optimal feeding is optimal swimming for all P

    Michelin, S. Optimal feeding is optimal swimming for all P. Physics of Fluids , volume=. 2011 , publisher=

    work page 2011

  73. [73]

    Bulletin of mathematical biology , volume=

    Discrete cilia modelling with singularity distributions: application to the embryonic node and the airway surface liquid , author=. Bulletin of mathematical biology , volume=. 2007 , publisher=

    work page 2007

  74. [74]

    Respiratory physiology & neurobiology , volume=

    Modelling mucociliary clearance , author=. Respiratory physiology & neurobiology , volume=. 2008 , publisher=

    work page 2008

  75. [75]

    Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , pages=

    Mathematical modelling of cilia-driven transport of biological fluids , author=. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , pages=. 2009 , organization=

    work page 2009

  76. [76]

    Journal of Fluid Mechanics , volume=

    Mixing and transport by ciliary carpets: a numerical study , author=. Journal of Fluid Mechanics , volume=. 2014 , publisher=

    work page 2014

  77. [77]

    Physical Review Fluids , volume=

    Simulating cilia-driven mixing and transport in complex geometries , author=. Physical Review Fluids , volume=. 2020 , publisher=

    work page 2020

  78. [78]

    Philosophical Transactions of the Royal Society B , volume=

    Cilia oscillations , author=. Philosophical Transactions of the Royal Society B , volume=. 2020 , publisher=

    work page 2020

  79. [79]

    Biophysical journal , volume=

    Motor regulation results in distal forces that bend partially disintegrated Chlamydomonas axonemes into circular arcs , author=. Biophysical journal , volume=. 2014 , publisher=

    work page 2014

  80. [80]

    Physical review letters , volume=

    Cooperative molecular motors , author=. Physical review letters , volume=. 1995 , publisher=

    work page 1995

Showing first 80 references.