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Sliding of a liquid spherical droplet in an external insoluble liquid at low Reynolds numbers

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper derives a generalized Hadamard-Rybczynski drag law in which a single interfacial slip length interpolates between Stokes drag for a rigid sphere and the classic circulating-droplet result.

desk verdict A clean derivation of a slip-modified Hadamard-Rybczynski formula that is mathematically fine but physically unconvincing as an explanation of experimental droplet settling. read the letter →

arxiv 2502.02814 v2 pith:AUVRTZH5 submitted 2025-02-05 physics.flu-dyn cond-mat.soft

Peter Lebedev-Stepanov This is my paper · ORCID
classification physics.flu-dyncond-mat.soft PACS 47.15.G47.55.D
keywords Hadamard-RybczynskiequationNavierslipboundaryconditionliquid-liquidinterfaceStokesdraglengthlowReynoldsnumberflowimmiscibleliquids
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Small droplets falling or rising through another liquid are observed to follow Stokes drag for a rigid sphere, not the Hadamard-Rybczynski (HR) formula for a circulating drop. The paper argues that a fundamental ingredient is missing from the HR derivation: the two immiscible liquids need not have equal tangential velocities at their interface. It replaces the no-slip condition with the Navier partial-slip condition, characterized by a slip length $\lambda$, and derives a generalized HR equation for the terminal velocity. At $\lambda=0$ the classic HR result is recovered, and for an infinitely viscous drop the formula becomes the known partial-slip Stokes drag law for a solid sphere. If the derivation is right, the model offers a one-parameter physical explanation of why small droplets behave like rigid spheres, with surfactants entering as modifiers of the slip length rather than as the only possible cause.

What carries the argument

The carrier of the argument is the Navier partial-slip boundary condition at the liquid-liquid interface, Eq. (118): $\lambda(\partial V_\theta/\partial r - V_\theta/r) = V_\theta - V'_\theta$, evaluated at the drop surface. This condition postulates that the relative tangential velocity of the two liquids is proportional to the tangential shear rate through a constant slip length $\lambda$. Inserting it into the $l=1$ mode of the general axisymmetric Stokes solution in spherical coordinates yields an algebraic equation for the flow amplitude, from which the terminal velocity follows. The same machinery produces the continuous-stress model as a special case and makes contact with the surface-viscosity formulation of interfacial rheology.

What would settle it

Measure terminal velocities of carefully cleaned, surfactant-free droplets of one liquid pair over at least a factor of ten in radius and several viscosity ratios, and fit Eq. (124) with constant $\lambda$; if the inferred $\lambda$ drifts systematically with radius or shear rate, or if the normalized velocity $V_0/R^2$ does not follow the predicted functional form, the constant-slip-length model is falsified.

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Extended reading notes

Core claim

The central result is the generalized Hadamard-Rybczynski equation, Eq. (124)/(137), for the terminal velocity of a spherical drop: $$V_0=\frac{2}{3}\,\frac{(\rho-\rho')$gR^{2}$}{\eta}\, \frac{\eta+\eta'+\frac{3\eta'\$\lambda$}{R}}{2\eta+3\eta'+\frac{6\eta'\$\lambda$}{R}}.$$ It is obtained by solving the axisymmetric Stokes equations with the usual conditions of vanishing radial velocity and continuous tangential viscous stress, replacing the no-slip equality of tangential velocities by the Navier condition that the velocity jump is proportional to the local shear rate through the slip length $\lambda$. The equation reduces to the classic HR formula at $\lambda=0$, to the continuous-stress model at a particular slip length, and to the published partial-slip Stokes drag formula when the drop viscosity tends to infinity. The paper presents this as the first application of the Navier condition to a liquid-liquid boundary and as a rationale for the observed Stokes-like motion of small droplets.

Load-bearing premise

The result stands on the assumption that a clean, immiscible liquid-liquid interface obeys a constant scalar Navier slip relation, so that the tangential velocity jump is proportional to the local shear rate with one slip length $\lambda$ that does not depend on drop size, shear rate, or contamination; if the interface is actually no-slip or the relation is nonlinear, the generalized equation becomes a curve-fitting expression rather than a physical drag law.

Editorial extensions

If this is right

  • At slip length $\lambda=0$ the generalized equation reduces exactly to the classical Hadamard-Rybczynski formula, so the standard theory is contained as a special case.
  • For an infinitely viscous drop the same equation reduces to the known partial-slip Stokes drag formula for a solid sphere, which has appeared in earlier work on microfluidic slip.
  • In the continuous-stress special case the terminal velocity is $6/5$ times the Stokes velocity and is independent of the internal viscosity, a qualitative signature that differs from both HR and Stokes behavior.
  • Because the equation is structurally close to the surface-viscosity formulation, it offers a hydrodynamic interpretation of interfacial effects in aqueous emulsions and water-hydrocarbon systems.
  • Surfactant effects, which are known to shift droplet settling from HR toward Stokes behavior, can be understood as changes in the effective slip length.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If $\lambda$ is a genuine material constant, the formula predicts a definite size and viscosity-ratio dependence of the terminal velocity; fitting it to droplets of different radii would show whether $\lambda$ stays fixed or varies, distinguishing a molecular slip length from an effective fitting parameter.
  • For the common case in which the internal liquid is less viscous than the external one, the continuous-stress special case corresponds to a negative slip length, which would mean the interface resists rather than aids sliding; measuring the sign of the inferred $\lambda$ could discriminate between true sliding and a proxy for surfactant-induced rigidity.
  • The model could be extended by letting $\lambda$ depend on surfactant coverage or on position over the drop surface, which would connect the slip-length picture directly to the well-documented Stokes-to-HR transition in contaminated systems.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper derives a generalized Hadamard-Rybczynski (HR) formula for a spherical liquid droplet rising or sinking in another immiscible liquid, replacing the usual no-slip tangential velocity condition at the liquid-liquid interface with a Navier partial-slip condition characterized by a slip length λ. The author first re-derives the standard HR equation, then analyzes a continuous-viscous-stress model that yields a specific slip length, and finally obtains a one-parameter generalized formula, Eq. (124)/(137). The formula reduces to the classical HR result at λ=0 and to the known partial-slip Stokes drag formula for an infinitely viscous drop. The paper argues that this generalization can reconcile experimental observations of droplet velocities that lie between the Stokes and HR predictions.

Significance. If the physical premise were established, the paper would offer a simple one-parameter generalization of the HR equation that could be useful for interpreting droplet sedimentation and creaming experiments. The algebraic derivation is transparent and the limiting checks are correct: Eq. (137) indeed reduces to the HR formula at λ=0 and to the published slip-corrected Stokes drag in the infinite-internal-viscosity limit. The manuscript also correctly notes a formal similarity to the Boussinesq surface-viscosity model. However, the physical interpretation of λ as a liquid-liquid material property is not supported by independent evidence, and, more importantly, the model predicts a droplet-size dependence that is opposite to the experimental trend cited in the paper. These issues prevent acceptance in the current form.

major comments (3)
  1. [Sec. 4, Eq. (137)] The opening of Sec. 4 states that experiments show small droplets obey Stokes drag and larger droplets approach the Hadamard-Rybczynski description. However, Eq. (137), written as V0/V_Stokes = 3(η+η'+3η'Λ)/(2η+3η'+6η'Λ) with Λ=λ/R, gives the opposite dependence: for fixed λ>0, smaller R (larger Λ) monotonically increases V0/V_Stokes from the HR value toward the full-slip value 3/2, never toward the Stokes value 1; larger R approaches the HR value. The closing claim of Sec. 5 that "the slip length allows the experiment and theory to be reconciled" is therefore unsupported by the formula as written unless λ is allowed to depend on droplet radius, which would make it a fitting parameter rather than a physical slip length.
  2. [Eq. (118)] The Navier partial-slip condition at the liquid-liquid interface is introduced by analogy with liquid flow over hydrophobic solid surfaces, without independent justification. This is the constitutive assumption on which the generalized formula rests. If the clean interface is actually no-slip, or if the observed Stokes-like behavior of small droplets is instead due to surfactants, then the λ appearing in Eq. (137) is a curve-fitting parameter. The manuscript should either provide direct supporting evidence for slip at clean immiscible-liquid interfaces (e.g., molecular simulations or measurements of the velocity jump) or explicitly frame λ as an empirical interfacial drag coefficient.
  3. [Sec. 3, Eqs. (113)-(115)] The continuous-viscous-stress model yields λ = (R/3)(η/η' − 1), which is negative whenever η' > η. Since the Navier condition in Eq. (118) is naturally interpreted with a positive slip length, the sign issue is left unexplained. The range of validity of the continuous-stress model and the physical meaning of a negative slip length need to be addressed before Eq. (115) can support the claim that the continuous-stress model corresponds to a particular admissible value of λ.
minor comments (5)
  1. [Abstract and Sec. 1] The sentence "No slip condition may be may be unnatural at the droplet interface" is garbled and duplicated; the abstract also repeats the introduction almost verbatim.
  2. [Sec. 3, after Eq. (96)] "exceeds the velocity of the solid sphere ... by 65times" should read "by a factor of 6/5"; the numerical ratio from Eqs. (49) and (96) is 6/5, not 65.
  3. [Sec. 4, after Eq. (128)] "substituting Eq. (95)" appears to be a typo; the arbitrary-slip velocity field follows from substituting the d obtained in Eq. (121), not the d of the continuous-stress model in Eq. (95).
  4. [General notation] The viscosities η and η' are not explicitly defined at first use; a brief definition at the start of Sec. 2 would improve readability.
  5. [Sec. 5, Boussinesq comparison] The comparison with Eq. (138) is qualitative; a sentence showing the formal mapping between λ and e (e.g., e proportional to λη') would make the equivalence precise.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (124)/(137) is derived from Stokes equations plus an explicitly stated Navier slip condition; the slip length is a free constitutive parameter, and the known limits are recovered as algebraic limits, not imported from the assumptions.

full rationale

The derivation chain is self-contained: the general axisymmetric Stokes solution (Table 1) is a standard mathematical result; the boundary conditions (5)-(8), (10), (12) and the Navier condition (118) are stated ab initio; solving for the unknown coefficients gives Eq. (124). The λ=0 limit reproduces Hadamard-Rybczynski because (118) reduces exactly to the no-slip condition (9); the infinite-internal-viscosity limit reproduces the published partial-slip solid-sphere drag formula [8,9]. The continuous-stress model of Sec. 3 is an independent derivation whose slip length (115) is then offered as a special case, not used to force Eq. (124). The structural similarity to Boussinesq's formula is acknowledged in the text rather than hidden. The only self-citation (Ref. [6], an appendix by the same author) supplies the standard solution of the Stokes equations; it is not the origin of the new boundary-condition physics and is not load-bearing in a circular sense. The empirical concern that fixed λ predicts the wrong droplet-size trend (small droplets should slip more, not less) is a physical correctness objection, not a logical circularity, because λ is not fitted to those data and then re-predicted. Therefore no circular step can be exhibited with a specific reduction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central model introduces no new physical objects, only a free parameter, the slip length. The load-bearing unmeasured ingredients are the Navier-slip constitutive law and the use of the author's own general solution for the Stokes eigenfunctions.

free parameters (1)
  • lambda (slip length at liquid-liquid interface) = none given
    Introduced in Section 4, Eq. (118), as an arbitrary constant. It is not measured or derived from first principles. It is the knob that interpolates between Stokes and HR behavior and can absorb the effects of surfactants.
assumptions (5)
  • standard math Stokes equations and the axisymmetric l=1 truncation describe the flow.
    The derivation relies on the general solution of axisymmetric Stokes equations in spherical coordinates (Table 1) and discards l>1 modes by orthogonality. This is standard for low-Reynolds-number droplets.
  • domain assumption The droplet remains spherical and does not deform.
    Used through boundary conditions (7)-(8), where the radial velocity vanishes at the drop surface, and throughout the surface stress balance. This is valid for small capillary numbers but is not stated quantitatively.
  • ad hoc to paper Immiscible, low-affinity liquids permit tangential velocity slip at their interface.
    Section 1 and Eq. (118) posit this from the intuition that immiscible liquids repel each other. No molecular, experimental, or direct numerical evidence is offered for a constant scalar slip length.
  • ad hoc to paper The boundary conditions can include both continuity of tangential viscous stress and Navier slip simultaneously.
    In Section 4, the internal solution from Section 3 uses stress continuity (Eq. 78) and the Navier condition is used in Eq. (119). Whether both can hold at a sharp interface depends on an interfacial friction model that is not supplied.
  • domain assumption The cited general solution in the author's own Appendix [6] is correct.
    The paper does not re-derive Table 1; it imports the solution from a self-cited preprint. If that solution contains errors, the present results would change.

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Pith. "Pith review of Sliding of a liquid spherical droplet in an external insoluble liquid at low Reynolds numbers." pith.science (2026). https://pith.science/paper/AUVRTZH5

@misc{pith2026250202814,
  author       = {Pith},
  title        = {Pith review of: Sliding of a liquid spherical droplet in an external insoluble liquid at low Reynolds numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AUVRTZH5}},
  note         = {Machine review of arXiv:2502.02814}
}
read the original abstract

The experiment shows that small liquid droplets under the action of gravity and the Archimedes force move in the external viscous liquid practically according to the Stokes drag force equation, and not in accordance with the Hadamard-Rybczynski (HR) formula, which was specially developed to describe the motion of a liquid droplet in an external viscous liquid. Various mechanisms are proposed to explain this: increased viscosity at the interface between two liquids and the presence of unaccounted surfactants. However, there is another fundamental mechanism that has not been taken into account. It can be expected that the velocities of such liquids, insoluble in each other, may not equalize at the boundary of the droplet. No slip condition may be may be unnatural at the droplet interface. In this paper, the Navier condition is applied to the liquid-liquid boundary for the first time. A generalized HR equation is obtained. If slip length {\lambda}=0 that equation transforms into the usual HR equation. At certain {\lambda}, we arrive at a model with continuity of the components of the viscous stress tensor at the interface of two fluids. For infinite viscosity of the drop, it becomes a well-known relation generalizing the Stokes drag force for a solid sphere, taking into account the boundary condition of partial slip.

Figures

Figures reproduced from arXiv: 2502.02814 by the authors.

Figure 1
Figure 1. Spherical coordinates ( , , ) r   , which include the radial coordinate, polar and azimuthal angles, respectively. The hydrostatic pressure in the external liquid on the surface of the drop is determined by the formula 0 p g h R     ( cos ), (1) where h is the depth of immersion of the center of the drop. The minus sign before the expression on the right takes into account that the hydrostatic pressure is com… view at source ↗

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Reference graph

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