arxiv: 2605.01196 · v1 · submitted 2026-05-02 · ⚛️ physics.flu-dyn · cond-mat.soft

Recognition: unknown

Effects of surface viscosities on the motion of a droplet enclosing a translating particle

Ali G\"urb\"uz, Guangpu Zhu, Herv\'e Nganguia, Lailai Zhu, On Shun Pak, Y. N. Young

Authors on Pith no claims yet

Pith reviewed 2026-05-09 18:42 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cond-mat.soft

keywords surface viscositydroplet motioncompound particleinterfacial rheologyStokes flowBoussinesq-Scriven laweccentric configurationconfinement effects

0 comments

The pith

Surface shear viscosity leaves the velocity of a concentric droplet enclosing a particle unchanged, while dilatational viscosity and eccentricity both alter the motion.

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

The paper examines how surface viscosities at the interface affect the motion of a viscous droplet that encloses a rigid particle translating steadily through it under Stokes flow. It derives an exact analytical solution when the particle and droplet are concentric and extends the analysis to eccentric positions with a spectral boundary integral method. The central result is that surface shear viscosity exerts no influence on the induced droplet velocity in the concentric case, whereas surface dilatational viscosity can either increase or decrease that velocity depending on the degree of confinement and the viscosity ratio inside versus outside the droplet. Eccentric placement breaks the symmetry and introduces a dependence on shear viscosity that consistently speeds up the droplet, with the effect growing stronger as the offset increases. These findings supply a quantitative benchmark for understanding compound particles whose interfaces have rheological complexity.

Core claim

For concentric configurations, the induced droplet velocity is independent of surface shear viscosity, while surface dilatational viscosity can either enhance or suppress the droplet motion depending on the interplay between confinement and viscosity ratio. In contrast, when the particle is eccentrically positioned within the droplet, a dependence on surface shear viscosity emerges, leading to a consistent enhancement of droplet motion that becomes more pronounced with increasing eccentricity. The analytical and numerical results agree closely and reveal how interfacial rheology, confinement, and symmetry breaking jointly govern the dynamics.

What carries the argument

The Boussinesq-Scriven constitutive law applied to the droplet interface, which incorporates both surface shear and dilatational viscosities; solved exactly for concentric geometries and via spectral boundary integral method for eccentric cases.

Load-bearing premise

The droplet interface obeys the Boussinesq-Scriven constitutive law exactly, with no additional interfacial effects such as Marangoni stresses, bending rigidity, or non-Newtonian surface behavior.

What would settle it

Direct experimental measurement of droplet velocity for a centered particle at fixed particle speed but varying surface shear viscosity, which should show no change if the claim holds, or a measurable velocity shift if the independence assumption fails.

Figures

Figures reproduced from arXiv: 2605.01196 by Ali G\"urb\"uz, Guangpu Zhu, Herv\'e Nganguia, Lailai Zhu, On Shun Pak, Y. N. Young.

**Figure 1.** Figure 1: FIG. 1: Geometric setup and notation for the compound particle system. A spherical particle of radius view at source ↗

**Figure 2.** Figure 2: FIG. 2: Induced droplet speed view at source ↗

**Figure 3.** Figure 3: FIG. 3: The magnitude of force view at source ↗

**Figure 4.** Figure 4: FIG. 4: Droplet mobility view at source ↗

**Figure 5.** Figure 5: FIG. 5: Flow field inside and outside a viscous droplet enclosing a translating spherical particle in the concentric view at source ↗

**Figure 6.** Figure 6: FIG. 6: Induced droplet speed view at source ↗

**Figure 7.** Figure 7: FIG. 7: Flow field inside and outside a viscous droplet enclosing a translating spherical particle in an eccentric view at source ↗

**Figure 8.** Figure 8: FIG. 8: Scaled force, view at source ↗

read the original abstract

We investigate the influence of interfacial rheology on the motion of a compound particle consisting of a viscous droplet enclosing a translating rigid particle in the Stokes flow regime. The droplet interface is modeled using the Boussinesq-Scriven constitutive law, incorporating both surface shear and dilatational viscosities. An exact analytical solution is derived for the concentric configuration, and the analysis is extended to eccentric geometries using a spectral boundary integral method, enabling a systematic examination of confinement, viscosity contrast, and interfacial properties. For concentric configurations, we show that the induced droplet velocity is independent of surface shear viscosity, while surface dilatational viscosity can either enhance or suppress the droplet motion depending on the interplay between confinement and viscosity ratio. This behavior is rationalized in terms of competing effects between reduced interfacial mobility and increased driving force required to maintain the prescribed particle speed. In contrast, when the particle is eccentrically positioned within the droplet, a dependence on surface shear viscosity emerges, leading to a consistent enhancement of droplet motion that becomes more pronounced with increasing eccentricity. The analytical and numerical results are in excellent agreement and reveal how interfacial rheology, confinement, and symmetry breaking jointly govern the dynamics of compound particle systems. These findings provide mechanistic insight and establish a quantitative benchmark for future studies of active compound particles with complex interfaces.

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.

Desk Editor's Note private letter to a colleague

The paper cleanly separates how surface shear and dilatational viscosities affect droplet motion depending on whether the inner particle is centered or eccentric.

read the letter

The standout result is that surface shear viscosity has no effect on the droplet's translation speed in the concentric case, while dilatational viscosity can raise or lower it depending on confinement and the inner-to-outer viscosity ratio. Once the particle sits off-center, shear viscosity begins to increase the droplet speed, and that boost grows with eccentricity. They explain the shear independence as a symmetry consequence and the dilatational changes as a competition between lower interfacial mobility and the extra force needed to hold the particle speed fixed. Both effects follow directly from the Stokes equations plus the Boussinesq-Scriven law. They give an exact analytical solution for the concentric geometry and use a spectral boundary integral method for the eccentric cases, with the two agreeing closely where they can be compared. The study sweeps confinement ratio, viscosity contrast, and the two surface viscosity coefficients in a systematic way, which makes the trends easy to see. The symmetry-based separation of regimes and the reported agreement between analytics and numerics are the solid parts. The model assumes the interface obeys Boussinesq-Scriven exactly, with no Marangoni stresses, bending rigidity, or non-Newtonian surface behavior, and everything stays in the Stokes limit. Those are standard modeling choices but narrow the direct applicability to real interfaces. No experimental checks appear. This work is aimed at researchers who already work on low-Reynolds-number flows with complex interfaces or compound particles. Readers who need quantitative benchmarks or mechanistic explanations for how interfacial rheology and particle position interact will get value from it. The central claims rest on reproducible derivations and consistent numerics, so the paper deserves a serious referee.

Referee Report

0 major / 4 minor

Summary. The paper investigates the effects of surface shear and dilatational viscosities on the motion of a viscous droplet enclosing a translating rigid particle under Stokes flow, modeling the interface with the Boussinesq-Scriven law. It derives an exact analytical solution for concentric configurations showing droplet velocity independence from surface shear viscosity but dependence on dilatational viscosity (modulated by confinement and viscosity ratio, via competing mobility and driving-force effects), and employs a spectral boundary integral method for eccentric cases revealing shear-viscosity-induced enhancement that increases with eccentricity. Analytical and numerical results agree closely, highlighting the roles of interfacial rheology, confinement, and symmetry breaking.

Significance. If the results hold, the work delivers mechanistic insight into compound-particle dynamics with complex interfaces and establishes quantitative benchmarks via the exact concentric solution and validated spectral method. The symmetry-based separation of shear (null effect concentrically, enhancement eccentrically) and dilatational effects is a clear strength, with direct applicability to microfluidics and active soft-matter systems.

minor comments (4)

Abstract and §4 (numerical results): the claim of 'excellent agreement' between analytical and numerical solutions should be supported by a quantitative metric (e.g., relative error or L2 norm) rather than qualitative description alone.
§2 (model formulation): define the surface shear and dilatational viscosity coefficients (and their nondimensional groups) at first use and maintain consistent notation throughout the velocity expressions.
§3 (concentric analytical solution): a brief schematic or additional sentence clarifying the physical competition between reduced mobility and increased driving force would aid reader intuition without altering the derivation.
Figure captions (throughout): expand to list the specific parameter values (viscosity contrast, confinement ratio, eccentricity) used in each panel for reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, as well as for recognizing its significance and recommending minor revision. The referee's description correctly captures the key results on the independence from surface shear viscosity in concentric cases, the role of dilatational viscosity, and the enhancement due to eccentricity. Since the report lists no specific major comments, we have no individual points requiring rebuttal or clarification at this time. We will incorporate minor revisions to improve presentation, clarity, or any other aspects as appropriate.

Circularity Check

0 steps flagged

No significant circularity; derivation follows from governing equations

full rationale

The paper presents an exact analytical solution for the concentric configuration derived directly from the Stokes equations with the Boussinesq-Scriven constitutive law at the interface, demonstrating independence from surface shear viscosity and a dependence on dilatational viscosity via competing mobility and driving-force effects. The eccentric case is handled via a spectral boundary integral method. These steps are self-contained mathematical derivations from the stated model assumptions with no reduction to fitted parameters, self-referential definitions, or load-bearing self-citations. No enumerated circularity pattern is present; the central claims are independent of the inputs by construction.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard low-Reynolds-number hydrodynamics and the Boussinesq-Scriven interface model; no new entities are postulated and the varied quantities (viscosity ratio, confinement, surface viscosities) are treated as input parameters rather than fitted constants.

free parameters (3)

viscosity contrast
Ratio of internal droplet viscosity to external fluid viscosity, varied parametrically in both concentric and eccentric analyses.
confinement ratio
Ratio of rigid-particle radius to droplet radius, used to explore confinement effects.
surface shear and dilatational viscosity coefficients
Dimensionless surface viscosities appearing in the Boussinesq-Scriven law, treated as independent parameters.

axioms (2)

domain assumption Stokes flow (negligible inertia, Re = 0)
All analysis is performed under the Stokes-flow approximation stated in the abstract.
domain assumption Boussinesq-Scriven constitutive law for the interface
The droplet interface is modeled exclusively with this law incorporating both surface shear and dilatational viscosities.

pith-pipeline@v0.9.0 · 5554 in / 1662 out tokens · 77552 ms · 2026-05-09T18:42:08.992978+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

90 extracted references · 39 canonical work pages

[1]

xkVllwGD9iMTtKCz5I7A2z7IOXQ=

CONCLUSION In this work, we have examined the hydrodynamics of a compound particle system with interfacial rheology, focusing on how surface viscosities influence the droplet motion induced by a translating particle enclosed within the droplet. An analytical solution was derived for the concentric configuration, and the analysis was extended to eccentric ...
[2]

K. M. Wisdom, J. A. Watson, X. Qu, F. Liu, G. S. Watson, and C.-H. Chen. Self-cleaning of superhydrophobic surfaces by self-propelled jumping condensate.Proc. Natl. Acad. Sci. U.S.A., 110(20):7992–7997, 2013

2013
[3]

Boˇ ziˇ c and M

A. Boˇ ziˇ c and M. Kanduˇ c. Relative humidity in droplet and airborne transmission of disease.J. Biol. Phys., 47:1, 2021

2021
[4]

M. L. P¨ ohlker, C. P¨ ohlker, O. O. Kr¨ uger, J.-D. F¨ orster, T. Berkemeier, W. Elbert, J. Fr¨ ohlich-Nowoisky, U. P¨ oschl, G. Bagheri, E. Bodenschatz, J. A. Huffman, S. Scheithauer, and E. Mikhailov. Respiratory aerosols and droplets in the transmission of infectious diseases.Rev. Mod. Phys., 95:045001, 2023

2023
[5]

A. R. Abate, C.-H. Chen, J. J. Agresti, and D. A. Weitz. Beating poisson encapsulation statistics using close-packed ordering.Lab Chip, 9:2628–2631, 2009

2009
[6]

E. W. M. Kemna, R. M. Schoeman, F. Wolbers, I. Vermes, D. A. Weitz, and A. van den Berg. High-yield cell ordering and deterministic cell-in-droplet encapsulation using dean flow in a curved microchannel.Lab Chip, 12:2881–2887, 2012

2012
[7]

Brouzes, T

E. Brouzes, T. Kruse, R. Kimmerling, and H. H. Strey. Rapid and continuous magnetic separation in droplet microfluidic devices.Lab Chip, 15:908–919, 2015

2015
[8]

Y. Ding, F. Qiu, X. Casadevall i Solvas, F. W. Y. Chiu, B. J. Nelson, and A. DeMello. Microfluidic-based droplet and cell manipulations using artificial bacterial flagella.Micromachines, 7(2), 2016

2016
[9]

Ramos, M

G. Ramos, M. L. Cordero, and R. Soto. Bacteria driving droplets.Soft Matter, 16:1359–1365, 2020

2020
[10]

Rajabi, H

M. Rajabi, H. Baza, T. Turiv, and O. D. Lavrentovich. Directional self-locomotion of active droplets enabled by nematic environment.Nat. Phys., 17:260, 2021

2021
[11]

Kokot, H

G. Kokot, H. A. Faizi, G. E. Pradillo, A. Snezhko, and P. M. Vlahovska. Spontaneous self-propulsion and nonequilibrium shape fluctuations of a droplet enclosing active particles.Commun. Phys., 5:91, 2022

2022
[12]

M. He, J. S. Edgar, G. D. M. Jeffries, R. M. Lorenz, J. P. Shelby, and D. T. Chiu. Selective encapsulation of single cells and subcellular organelles into picoliter- and femtoliter-volume droplets.Anal. Chem., 77:1539, 2005

2005
[13]

A. D. Griffiths and D. S. Tawfik. Miniaturising the laboratory in emulsion droplets.Trends Biotechnol., 24:395, 2006

2006
[14]

B. T. Kelly, J.-C. Baret, V. Taly, and A. D. Griffiths. Miniaturizing chemistry and biology in microdroplets.Chem. Commun., pages 1773–1788, 2007

2007
[15]

Chabert and J.-L

M. Chabert and J.-L. Viovy. Microfluidic high-throughput encapsulation and hydrodynamic self-sorting of single cells. Proc. Natl. Acad. Sci. U.S.A., 105(9):3191–3196, 2008

2008
[16]

Clausell-Tormos, D

J. Clausell-Tormos, D. Lieber, J.-C. Baret, A. El-Harrak, O. J. Miller, L. Frenz, J. Blouwolff, K. J. Humphry, S. K¨ oster, H. Duan, C. Holtze, D. A. Weitz, A. D. Griffiths, and C. A. Merten. Droplet-based microfluidic platforms for the encapsulation and screening of mammalian cells and multicellular organisms.Chem. Biol., 15(5):427–437, 2008. 16

2008
[17]

Thiele and S

J. Thiele and S. Seiffert. Double emulsions with controlled morphology by microgel scaffolding.Lab Chip, 11:3188–3192, 2011

2011
[18]

H. Wen, Y. Yu, G. Zhu, L. Jiang, and J. Qin. A droplet microchip with substance exchange capability for the developmental study of c. elegans.Lab Chip, 15:1905–1911, 2015

1905
[19]

R. E. Johnson and S. S. Sadhal. Fluid mechanics of compound multiphase drops and bubbles.Annu. Rev. Fluid Mech., 17:289–320, 1985. doi:10.1146/annurev.fl.17.010185.001445

work page doi:10.1146/annurev.fl.17.010185.001445 1985
[20]

S. S. Sadhal and H. N. Oguz. Stokes flow past compound multiphase drops: the case of completely engulfed drops/bubbles. J. Fluid Mech., 160:511–529, 1985. doi:10.1017/S0022112085003585

work page doi:10.1017/s0022112085003585 1985
[21]

Daddi-Moussa-Ider, H

A. Daddi-Moussa-Ider, H. L¨ owen, and S. Gekle. Creeping motion of a solid particle inside a spherical elastic cavity.Eur. Phys. J. E, 41:104, 2018. doi:10.1140/epje/i2018-11715-7

work page doi:10.1140/epje/i2018-11715-7 2018
[22]

S. Y. Reigh, L. Zhu, F. Gallaire, and E. Lauga. Swimming with a cage: low-reynolds-number locomotion inside a droplet. Soft Matter, 13:3161–3173, 2017

2017
[23]

V. A. Shaik, V. Vasani, and A. M. Ardekani. Locomotion inside a surfactant-laden drop at low surface P´ eclet numbers.J. Fluid Mech., 851:187–230, 2018. doi:10.1017/jfm.2018.491

work page doi:10.1017/jfm.2018.491 2018
[24]

Kree and A

R. Kree and A. Zippelius. Controlled locomotion of a droplet propelled by an encapsulated squirmer.Eur. Phys. J. E, 44: 6, 2021

2021
[25]

Kree and A

R. Kree and A. Zippelius. Mobilities of a drop and an encapsulated squirmer.Eur. Phys. J. E, 45:15, 2022

2022
[26]

R. Kree, L. R¨ uckert, and A. Zippelius. Influence of heterogeneity or shape on the locomotion of a caged squirmer.J. Fluid Mech., 967:A7, 2023. doi:10.1017/jfm.2023.450

work page doi:10.1017/jfm.2023.450 2023
[27]

Nganguia, A

H. Nganguia, A. Adegbuyi, M. Uffenheimer, and O. S. Pak. Squirming inside a liquid droplet with surface viscosities. Phys. Rev. Fluids, 10:033104, Mar 2025. doi:10.1103/PhysRevFluids.10.033104

work page doi:10.1103/physrevfluids.10.033104 2025
[28]

A. R. Sprenger, V. A. Shaik, A. M. Ardekani, M. Lisicki, A. J. T. M. Mathijssen, F. Guzm´ an-Lastra, H. L¨ owen, A. M. Menzel, and A. Daddi-Moussa-Ider. Towards an analytical description of active microswimmers in clean and in surfactant- covered drops.Eur. Phys. J. E, 43:58, 2020

2020
[29]

R. Kree, L. R¨ uckert, and A. Zippelius. Dynamics of a droplet driven by an internal active device.Phys. Rev. Fluids, 6: 034201, Mar 2021. doi:10.1103/PhysRevFluids.6.034201

work page doi:10.1103/physrevfluids.6.034201 2021
[30]

Kawakami and P

S. Kawakami and P. M. Vlahovska. Microswimmer dynamics in a hele-shaw droplet.Phil. Trans. R. Soc. A, 383(2304): 20240254, 2025

2025
[31]

Kawakami and P

S. Kawakami and P. M. Vlahovska. Migration and deformation of a droplet enclosing an active particle.J. Fluid Mech., 1007:A41, 2025. doi:10.1017/jfm.2025.75

work page doi:10.1017/jfm.2025.75 2025
[32]

C. K. V. S. and S. P. Thampi. Dynamics and stability of a concentric compound particle – a theoretical study.Soft Matter, 15:7605, 2019

2019
[33]

P. K. Singeetham, K. V. S. Chaithanya, and S. P. Thampi. Dilute dispersion of compound particles: deformation dynamics and rheology.J. Fluid Mech., 917:A2, 2021. doi:10.1017/jfm.2021.233

work page doi:10.1017/jfm.2021.233 2021
[34]

G. G. Fuller and J. Vermant. Complex fluid-fluid interfaces: Rheology and structure.Annu. Rev. Chem. Biomol. Eng., 3 (1):519–543, 2012

2012
[35]

J. R. Samaniuk and J. Vermant. Micro and macrorheology at fluid–fluid interfaces.Soft Matter, 10:7023–7033, 2014

2014
[36]

Manikantan and T

H. Manikantan and T. M. Squires. Surfactant dynamics: hidden variables controlling fluid flows.J. Fluid Mech., 892:P1,
[37]

doi:10.1017/jfm.2020.170

work page doi:10.1017/jfm.2020.170 2020
[38]

N. O. Jaensson, P. D. Anderson, and J. Vermant. Computational interfacial rheology.J. Non-Newton. Fluid Mech., 290: 104507, 2021

2021
[39]

H. A. Stone and L. G. Leal. The effects of surfactants on drop deformation and breakup.J. Fluid Mech., 220:161–186,
[40]

doi:10.1017/S0022112090003226

work page doi:10.1017/s0022112090003226
[41]

P. M. Vlahovska, J. Blawzdziewicz, and M. Loewenberg. Small-deformation theory for a surfactant-covered drop in linear flows.J. Fluid Mech., 624:293–337, 2009. doi:10.1017/S0022112008005417

work page doi:10.1017/s0022112008005417 2009
[42]

Mandal, U

S. Mandal, U. Ghosh, and S. Chakraborty. Effect of surfactant on motion and deformation of compound droplets in arbitrary unbounded Stokes flows.J. Fluid Mech., 803:200–249, 2016. doi:10.1017/jfm.2016.497

work page doi:10.1017/jfm.2016.497 2016
[43]

Chembai Ganesh, J

S. Chembai Ganesh, J. Koplik, J. F. Morris, and C. Maldarelli. Dynamics of a surface tension driven colloidal motor based on an active janus particle encapsulated in a liquid drop.J. Fluid Mech., 958:A12, 2023. doi:10.1017/jfm.2023.5

work page doi:10.1017/jfm.2023.5 2023
[44]

J. G. Oldroyd. The effect of interfacial stabilizing films on the elastic and viscous properties of emulsions.Proc. R. Soc. Lond. A, 232:567–577, 1955. doi:10.1098/rspa.1955.0240

work page doi:10.1098/rspa.1955.0240 1955
[45]

Langevin

D. Langevin. Rheology of adsorbed surfactant monolayers at fluid surfaces.Annu. Rev. Fluid Mech., 46:47–65, 2014. doi:10.1146/annurev-fluid-010313-141403

work page doi:10.1146/annurev-fluid-010313-141403 2014
[46]

Manikantan and T

H. Manikantan and T. M. Squires. Surface viscosity and Marangoni stresses at surfactant laden interfaces.J. Fluid Mech., 792:712–739, 2016. doi:10.1017/jfm.2016.96

work page doi:10.1017/jfm.2016.96 2016
[47]

R. W. Flumerfelt. Effects of dynamic interfacial properties on drop deformation and orientation in shear and extensional flow fields.J. Colloid Interface Sci., 76:330–349, 1980. doi:10.1016/0021-9797(80)90377-X

work page doi:10.1016/0021-9797(80)90377-x 1980
[48]

M. D. Levan. Motion of a droplet with a newtonian interface.J. Colloid Interface Sci., 83(1):11–17, 1981

1981
[49]

Narsimhan

V. Narsimhan. Letter: The effect of surface viscosity on the translational speed of droplets.Phys. Fluids, 30(8):081703, 2018

2018
[50]

J. M. Rallison. The deformation of small viscous drops and bubbles in shear flows.Annu. Rev. Fluid Mech., 16:45–66,
[51]

doi:10.1146/annurev.fl.16.010184.000401

work page doi:10.1146/annurev.fl.16.010184.000401
[52]

Pozrikidis

C. Pozrikidis. Effects of surface viscosity on the finite deformation of a liquid drop and the rheology of dilute emulsions in simple shearing flow.J. Non-Newton. Fluid Mech., 51:161–178, 1994. doi:10.1016/0377-0257(94)85010-0. 17

work page doi:10.1016/0377-0257(94)85010-0 1994
[53]

Gounley, G

J. Gounley, G. Boedec, M. Jaeger, and M. Leonetti. Influence of surface viscosity on droplets in shear flow.J. Fluid Mech., 791:464–494, 2016. doi:10.1017/jfm.2016.39

work page doi:10.1017/jfm.2016.39 2016
[54]

Ponce-Torres, J

A. Ponce-Torres, J. M. Montanero, M. A. Herrada, E. J. Vega, and J. M. Vega. Influence of the surface viscosity on the breakup of a surfactant-laden drop.Phys. Rev. Lett., 118:024501, 2017. doi:10.1103/PhysRevLett.118.024501

work page doi:10.1103/physrevlett.118.024501 2017
[55]

Narsimhan

V. Narsimhan. Shape and rheology of droplets with viscous surface moduli.J. Fluid Mech., 862:385–420, 2019. doi: 10.1017/jfm.2018.960

work page doi:10.1017/jfm.2018.960 2019
[56]

Dandekar and A

R. Dandekar and A. M. Ardekani. Effect of interfacial viscosities on droplet migration at low surfactant concentrations. J. Fluid Mech., 902:A2, 2020. doi:10.1017/jfm.2020.551

work page doi:10.1017/jfm.2020.551 2020
[57]

Singh and V

N. Singh and V. Narsimhan. Deformation and burst of a liquid droplet with viscous surface moduli in a linear flow field. Phys. Rev. Fluids, 5:063601, Jun 2020. doi:10.1103/PhysRevFluids.5.063601

work page doi:10.1103/physrevfluids.5.063601 2020
[58]

D. P. Panigrahi, S. Santra, T. N. Banuprasad, S. Das, and S. Chakraborty. Interfacial viscosity-induced suppression of lateral migration of a surfactant laden droplet in a nonisothermal poiseuille flow.Phys. Rev. Fluids, 6:053603, May 2021

2021
[59]

M. A. Herrada, A. Ponce-Torres, M. Rubio, J. Eggers, and J. M. Montanero. Stability and tip streaming of a surfactant- loaded drop in an extensional flow. Influence of surface viscosity.J. Fluid Mech., 934:A26, 2022. doi:10.1017/jfm.2021.1118

work page doi:10.1017/jfm.2021.1118 2022
[60]

Singh and V

N. Singh and V. Narsimhan. Numerical investigation of the effect of surface viscosity on droplet breakup and relaxation under axisymmetric extensional flow.J. Fluid Mech., 946:A24, 2022. doi:10.1017/jfm.2022.601

work page doi:10.1017/jfm.2022.601 2022
[61]

L. E. Scriven. Dynamics of a fluid interface: Equation of motion for Newtonian surface fluids.Chem. Eng. Sci., 12:98–108,
[62]

doi:10.1016/0009-2509(60)87003-0

work page doi:10.1016/0009-2509(60)87003-0
[63]

D. A. Edwards, H. Brenner, and D. T. Wasan.Interfacial Transport Processes and Rheology. Butterworth-Heinemann, 1991

1991
[64]

Happel and H

J. Happel and H. Brenner.Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media. Noord- hoff International Publishing, 1973

1973
[65]

Masoud and H

H. Masoud and H. A. Stone. The reciprocal theorem in fluid dynamics and transport phenomena.J. Fluid Mech., 879:P1,
[66]

doi:10.1017/jfm.2019.553

work page doi:10.1017/jfm.2019.553 2019
[67]

H. Zhao, A. H. G. Isfahani, L. N. Olson, and J. B. Freund. A spectral boundary integral method for flowing blood cells. J. Comput. Phys., 229(10):3726–3744, 2010. ISSN 0021-9991. doi:10.1016/j.jcp.2010.01.024

work page doi:10.1016/j.jcp.2010.01.024 2010
[68]

I. G. Graham and I. H. Sloan. Fully discrete spectral boundary integral methods for Helmholtz problems on smooth closed surfaces inR 3.Numer. Math., 92(2):289–323, 2002. ISSN 0945-3245. doi:10.1007/s002110100343

work page doi:10.1007/s002110100343 2002
[69]

P.-C. Chao, A. G¨ urb¨ uz, F. Sachs, and M. V. Sivaselvan. Fully implicit spectral boundary integral computation of red blood cell flow.Phys. Fluids, 33(7):071909, 2021. doi:10.1063/5.0055036

work page doi:10.1063/5.0055036 2021
[70]

G¨ urb¨ uz.Simulations of Red Blood Cell Flow by Boundary Integral Methods

A. G¨ urb¨ uz.Simulations of Red Blood Cell Flow by Boundary Integral Methods. PhD thesis, State University of New York at Buffalo, 2021

2021
[71]

Rahimian, S

A. Rahimian, S. K. Veerapaneni, D. Zorin, and G. Biros. Boundary integral method for the flow of vesicles with viscosity contrast in three dimensions.J. Comput. Phys., 298:766–786, 2015. ISSN 0021-9991. doi:10.1016/j.jcp.2015.06.017

work page doi:10.1016/j.jcp.2015.06.017 2015
[72]

G. K. Youngren and A. Acrivos. Stokes flow past a particle of arbitrary shape: a numerical method of solution.J. Fluid Mech., 69:377–403, 1975. doi:10.1017/S0022112075001486

work page doi:10.1017/s0022112075001486 1975
[73]

Pozrikidis.Boundary integral and singularity methods for linearized viscous flow, volume 8

C. Pozrikidis.Boundary integral and singularity methods for linearized viscous flow, volume 8. Cambridge University Press, New York, 1992. ISBN 0521406935

1992
[74]

Pozrikidis

C. Pozrikidis. Interfacial dynamics for Stokes flow.J. Comput. Phys., 169:250–301, 2001. doi:10.1006/jcph.2000.6582

work page doi:10.1006/jcph.2000.6582 2001
[75]

Barthes-Biesel

D. Barthes-Biesel. Motion and deformation of elastic capsules and vesicles in flow.Annu. Rev. Fluid Mech., 48:25–52,
[76]

doi:10.1146/annurev-fluid-122414-034345

work page doi:10.1146/annurev-fluid-122414-034345
[77]

K. E. Atkinson.The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, 1997. ISBN 9780521583916

1997
[78]

O. A. Ladyzhenskaya.The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York, 2nd edition, 1969

1969
[79]

Kim and S

S. Kim and S. J. Karrila.Microhydrodynamics: Principles and Selected Applications. Dover, 1991

1991
[80]

J. C. Adams and P. N. Swarztrauber. SPHEREPACK 3.0: A model development facility.Mon. Weather Rev., 127(8): 1872–1878, 1999. ISSN 1520-0493

1999

Showing first 80 references.