Sliding of a liquid spherical drop in an external fluid: a generalization of the Hadamard-Rybczynski equation

Peter Lebedev-Stepanov

REVIEW 4 major objections 5 minor 35 references

Sliding of a liquid spherical drop in an external fluid: a generalization of the Hadamard-Rybczynski equation

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper derives a generalized Hadamard–Rybczynski equation in which a single slip length at the liquid–liquid interface controls the terminal velocity of a spherical drop, recovering the classical no-slip result at zero slip and the…

desk verdict The generalized Hadamard-Rybczynski formula (Eq. 74) is a legitimate and clean mathematical extension, but the paper's own experimental illustration uses slip lengths that violate its consistency condition, and the special stress-continuity model lacks a Stokes limit. read the letter →

arxiv 2507.05947 v1 pith:J45XIWZZ submitted 2025-07-08 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.Dr

keywords Hadamard–RybczynskiequationpartialslipNavierboundaryconditionliquid–liquidinterfaceStokesflowlengthdropsedimentationemulsions

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

The paper tries to establish that the slow steady motion of a spherical liquid drop in another immiscible liquid is governed by a one-parameter generalization of the Hadamard–Rybczynski equation, in which the no-slip condition is replaced by a linear partial-slip law at the liquid–liquid interface. The extra parameter is a slip length $\lambda$, and the terminal velocity takes the closed form of Eq. (74). This matters because the standard HRE often deviates from emulsion experiments, and the usual explanations invoke surfactants or interfacial viscosity; the paper offers a purely hydrodynamic alternative that can be tested by measuring slip lengths directly. If the equation is right, the terminal velocity depends on $\lambda$ in addition to the two viscosities, densities, radius, and gravity, with the classical HRE and Stokes law recovered as limiting cases.

What carries the argument

The load-bearing object is the generalized Navier partial-slip boundary condition at the liquid–liquid interface, Eqs. (14)–(15): the tangential viscous stress of each liquid is proportional to the relative tangential velocity between the two fluids across the interface, with slip lengths $\lambda$ (external) and $\lambda'$ (internal) linked by $\eta/\lambda = -\eta'/\lambda'$ through stress continuity. This replaces no-slip with a linear friction law. The derivation then uses the standard $l=1$ axisymmetric Stokes solution in spherical coordinates, enforces vanishing radial velocity at the drop surface, shear-stress continuity, and the force balance with the Archimedes force, which fixes all coefficients and yields Eq. (74). In the appendix, requiring continuity of all components of the viscous stress tensor fixes the slip length in terms of the viscosities and radius, producing the special parameter-free formula.

What would settle it

Measure terminal velocities for drops of liquid A in liquid B and liquid B in liquid A at the same radius and extract the two slip lengths from Eqs. (109)–(110); if $\eta_A/\lambda_A + \eta_B/\lambda_B$ is not zero within experimental error, the generalized Navier condition with linked slip lengths is falsified for that pair.

Watch

Extended reading notes

Core claim

The central claim is that partial slip at a liquid–liquid interface, written as a generalized Navier condition, produces a closed-form terminal velocity $$V_0 = \frac{2(\rho-\rho')$gR^{2}$}{3\eta}\,\frac{\$\alpha$ + 1 + 3\$\lambda$/R}{2\$\alpha$ + 3 + 6\$\lambda$/R},\tag{74}$$ with $\alpha=\eta'/\eta$, which reduces to the Hadamard–Rybczynski result when $\lambda=0$ and to the slip-modified Stokes drag law for a solid sphere when the droplet viscosity is infinite. Choosing the slip length $\lambda = \frac{R}{3}(\eta/\eta'-1)$ enforces continuity of all components of the viscous stress tensor at the interface and gives the parameter-free speed $V_0 = 4(\rho-\rho')gR^2/(15\eta)$, independent of the internal viscosity. The paper applies the model to silicate-oil drops falling in castor oil: the experimental points sit between the HRE curve and the stress-continuous partial-slip curve, and the two models differ by an order of magnitude in the predicted speed of the internal toroidal flow.

Load-bearing premise

The load-bearing premise is that the liquid–liquid interface obeys a linear partial-slip law with constant slip lengths, linked by $\eta/\lambda = -\eta'/\lambda'$, and with no surfactant effects, surface viscosity, or dependence of slip on the local flow.

Editorial extensions

If this is right

At $\lambda=0$ the generalized equation reduces exactly to the Hadamard–Rybczynski equation, so the classical result is contained as a special case.
For a droplet of infinite viscosity the generalized equation becomes the known partial-slip correction to Stokes drag for a solid sphere; at $\lambda=0$ this is the ordinary Stokes law.
If the slip mechanism is real, swapping the roles of the two liquids gives two measured velocities from which slip lengths can be extracted, and the model requires $\eta_A/\lambda_A = -\eta_B/\lambda_B$ for the same pair.
The stress-continuous version predicts a drop terminal velocity $4(\rho-\rho')gR^2/(15\eta)$ that is independent of the droplet viscosity and exceeds the solid-sphere Stokes value by a factor of $6/5$, providing a direct quantitative target for experiment.
In the silicate-oil/castor-oil comparison, the partial-slip model and HRE bracket the measured fall speeds; the decisive difference is that the partial-slip model predicts internal droplet circulation roughly an order of magnitude faster than HRE.

Reading between the lines

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

Beyond the paper: the cleanest test of the generalized Navier law is a micro-particle-image-velocimetry measurement of the tangential velocity jump at a settling drop's interface; if the jump is absent while the terminal velocity still deviates from HRE, the missing physics is not linear partial slip.
Beyond the paper: the structural similarity between the generalized HRE and the Boussinesq surface-viscosity formula suggests that a measured slip length may absorb surfactant and Marangoni effects; comparing $\lambda$ extracted from sedimentation with independent surface-viscosity measurements would probe whether slip is a genuine material property.
Beyond the paper: because $\lambda$ enters through $\lambda/R$, measuring the terminal velocity of the same liquid pair over a range of drop radii would give a direct radius-dependence signature; a constant $\lambda$ would support the model, while $\lambda$ growing with $R$ would point to a surface-viscosity interpretation instead.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The formula in Eq. (74) is worth knowing. The algebra is straightforward but the reduction to the classical HRE at λ=0 and to the slip-modified Stokes drag for a solid sphere in the infinite-viscosity limit is done carefully, and the result is a compact one-parameter extension that could be useful for emulsion sedimentation problems. Compared to the Boussinesq surface-viscosity model, the slip-length picture is at least as natural, and the paper makes that point without overselling it. The derivation itself is standard l=1 Stokes flow, and the checks against known limits are the right checks. That part is solid.

The soft spots are real and load-bearing for the paper's own claims. First, the numerical values in Fig. 3 do not satisfy the paper's own relation Eq. (16). For castor oil and silicate oil, Eq. (16) demands λ' = -0.0036 mm if λ = 0.108 mm, or λ = 4.75 mm if λ' = -0.159 mm. The values quoted in the text, λ = 0.108 mm and λ' = -0.159 mm, violate that condition by a factor of roughly 40. So the velocity profiles and the claim of an order-of-magnitude internal flow are not solutions of the model they are supposed to illustrate. This is an arithmetic defect, but it sits in the central example of the paper.

Second, the special stress-continuity model in the Appendix is questionable. Imposing continuity of all viscous stress components leads to a terminal velocity (Eq. A42) that is independent of the internal viscosity and does not reduce to Stokes drag for an infinitely viscous droplet. The paper never flags this missing limit. Since that model is the one used for the experimental comparison in Fig. 2, the comparison is not on firm ground.

Third, the physical justification for a constant Navier slip length at a liquid-liquid interface is thin. The paper is honest that this is a model, but it also implies that emulsions should exhibit slip without offering any direct molecular or experimental evidence. That is a softer concern than the arithmetic one, but it limits the paper's reach. The experimental data are adapted without error bars, so the agreement in Fig. 2 is suggestive at best.

Who gets value from this? A researcher working on low-Reynolds-number multiphase flow who wants a compact slip-modified drop velocity formula. The formula can be cited as a mathematical result, though the experimental and physical conclusions need correction and external validation.

Recommendation: deserve a serious referee, not a desk reject, but it needs major revision before publication. The arithmetic inconsistency and the missing Stokes limit in the Appendix model should be fixed or explicitly discussed, and the experimental claims toned down.

Referee Report

4 major / 5 minor

Summary. The paper derives an analytical solution for the slow motion of a spherical liquid drop in another immiscible liquid under a generalized Navier partial-slip boundary condition at the liquid-liquid interface. The main result, Eq. (74), is a generalized Hadamard-Rybczynski equation that reduces to the classical HRE for zero slip length and to the slip-modified Stokes law for a solid sphere in the infinite-drop-viscosity limit. The paper also proposes a specific 'continuous viscous stress tensor' model in Appendix A, which yields closed-form slip lengths (93)-(94) and a terminal velocity formula (95), and applies this model to published experimental data for silicate-oil drops falling in castor oil, presenting velocity profiles and streamlines. The central formula (74) appears mathematically consistent and correctly reproduces the stated limits; however, the special model and the experimental application contain serious internal inconsistencies that undermine the paper's secondary claims.

Significance. If Eq. (74) were the only contribution, the paper would constitute a useful and conceptually interesting extension of the Hadamard-Rybczynski theory, with the strength of an explicit solution and clear asymptotic reductions. The idea of introducing a liquid-liquid slip length and linking it to the Boussinesq surface-viscosity model is appealing. However, the paper's additional claims—the continuous-stress-tensor closure, the specific slip-length formulas, and the experimental interpretation—are currently not sound because the derived slip lengths violate the consistency relation (16) that the same model establishes. These errors are load-bearing for the experimental sections and for the claim that a particular partial-slip model explains deviations from HRE. The central generalization itself is not invalidated, but the manuscript as a whole requires major revision before the secondary claims can be accepted.

major comments (4)

[Appendix A, Eqs. (93)-(94) and Eq. (16)] The slip lengths derived in the Appendix do not satisfy the consistency condition (16) that the main model imposes. From λ = R(1-η'/η)/3 and λ' = R(1-η/η')/3, one obtains η/λ = 3η²/[R(η-η')] and -η'/λ' = 3η'²/[R(η-η')], which are equal only when η=η'. The statement in Section 6 that 'obviously, Eqs. (93) and (94) satisfy the condition given by Eq. (16)' is therefore false. More seriously, substituting λ from Eq. (93) into the main formula (74) gives V₀ = (8/15)(ρ-ρ')gR²/η, whereas the Appendix's Eq. (95) gives V₀ = (4/15)(ρ-ρ')gR²/η; these results differ by a factor of two. The Appendix model is thus not a special case of the generalized HRE, contradicting the paper's central narrative.
[Fig. 3 and Eq. (16)] The numerical values used in Fig. 3, λ = 0.108 mm and λ' = -0.159 mm for R = 1 mm, η = 0.693 Pa·s, η' = 0.0232 Pa·s, violate the model's own consistency relation. Using Eq. (16), η/λ ≈ 6417 Pa·s/m while -η'/λ' ≈ 146 Pa·s/m; starting from λ = 0.108 mm requires λ' = -0.0036 mm, and starting from λ' = -0.159 mm requires λ = 4.75 mm. Moreover, these values do not even follow from Eqs. (93)-(94), which for the stated parameters give λ ≈ 0.322 mm and λ' ≈ -9.62 mm. Consequently, the velocity profiles in Fig. 3(b) and the streamlines in Fig. 4(b) are not solutions of the generalized Navier model, and the reported order-of-magnitude enhancement of the internal droplet velocity is not a prediction of the model.
[Appendix A, boundary conditions (91)-(92)] The additional boundary conditions (91)-(92), which require continuity of the diagonal stress components σ_θθ and σ_φφ, are not physically appropriate for a fluid-fluid interface. For two immiscible fluids, the correct interfacial conditions are continuity of the traction vector (with the normal-stress jump balanced by surface tension) and, in the absence of surfactant gradients, continuity of tangential stress. There is no physical mechanism requiring σ_θθ and σ_φφ to be continuous across the interface. This over-constraint is the source of the unphysical behavior seen in the Appendix, including a terminal velocity independent of the internal viscosity η' (Eq. (95)) and slip lengths that diverge as η'/η→0 or η/η'→0. This is a modeling error rather than a mere presentation issue.
[Fig. 2 and Discussion] The experimental comparison in Fig. 2 uses the Appendix's Eq. (95) as the 'partial slip model' curve. Since Eq. (95) is not a consequence of the generalized HRE (74) with any slip length that satisfies Eq. (16), the orange curve in Fig. 2 does not represent the partial-slip model that the paper proposes. The Discussion's claim that the partial-slip model gives a deviation from the experiment comparable to HRE is therefore unsupported. The experimental evidence presented does not validate the generalized HRE or the continuous-stress closure; at most, it illustrates a family of curves that could be obtained if λ were fit freely, but no such fit is performed.

minor comments (5)

[Eqs. (72)-(75) and text around them] The algebraic forms of Eqs. (72)-(75) are hard to read because of spacing and missing parentheses; please re-typeset these formulas with clear denominators and brackets, and verify that Eq. (74) is exactly equivalent to Eq. (73) after substituting α = η'/η.
[Sec. 5, after Eq. (81)] There is a typo: 'on can obtain' should be 'one can obtain'. Also, in Eq. (86), the drag coefficient expression should be checked for dimensional consistency, as C_D = β/(π ρ R² V₀) appears to have units of Pa·s/(kg·m⁻³·m·m/s) = 1, but the stated formula may have an error in the prefactor.
[References] The reference to the Navier boundary condition is numbered inconsistently: [5-6] in the introduction but [14-15] in the Introduction and Section 6. Please standardize the citation numbering throughout.
[Fig. 2 caption] The caption contains a typo: 'liquids-liquid interface' should be 'liquid-liquid interface'.
[Appendix A, Eq. (A5)] The rearrangement from Eq. (A3) to Eq. (A4) is not immediately transparent; the derivation of Eq. (A5) appears to contain typographical errors in the Legendre-function identities and should be carefully checked.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the generalized HRE is derived from stated slip boundary conditions without fitting, and the special stress-continuity model fixes slip lengths from viscosity ratios rather than from the compared terminal-velocity data.

full rationale

The paper derives Eq. (74) by solving the Stokes equations with the generalized Navier slip conditions (14)-(15) and the consistency relation (16); the slip lengths λ and λ' are free parameters of the model, not fitted to the experimental data. The no-slip limit λ=0 gives the classical HRE (76), and the α→0 limit with bounded λ gives the slip-modified Stokes result (78), both by substitution into the same formula, not by construction. The special case in the Appendix imposes continuity of the diagonal stress components, which determines λ and λ' via Eqs. (93)-(94) from the viscosity ratio and radius; the terminal velocity in Eq. (95) follows from the same Stokes solution, and the comparison with the silicate-oil/castor-oil experiment uses that independently derived value rather than a best-fit slip length. The only self-citations (Refs. [15]-[16]) supply the standard separable solution of the Stokes equations in spherical coordinates, which is textbook material and not an unverified premise unique to this paper; the central claim therefore does not reduce to a fit, a renamed known result, or a self-citation chain. The apparent inconsistency of the Fig. 3 numerical slip lengths with Eq. (16) would be an arithmetic or correctness issue, not circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The main result rests on a new interfacial boundary condition (free λ) plus standard Stokes-flow and sphericity assumptions. The Appendix adds a stronger, ad hoc stress-continuity axiom that fixes λ but creates an unphysical solid-sphere limit. No numerical constants are fitted to the experiment; λ is either free or determined by the added axiom.

free parameters (2)

slip length λ for the external liquid
Introduced in Eqs. (14)-(15) as the liquid-liquid slip length. The main generalized HRE, Eq. (74), keeps λ free; it is fixed only in the Appendix special model via Eqs. (93)-(94).
slip length λ' for the internal liquid
Related to λ through Eq. (16), η/λ = -η'/λ', so it is not independent once λ is chosen.

assumptions (6)

domain assumption The flow is steady, axisymmetric, inertialess Stokes flow with Reynolds number much less than one.
Used throughout the derivation; Section 7 states Re << 1 and gives the droplet-size restriction in Eqs. (98)-(100).
domain assumption The drop remains spherical, with zero radial velocity at the interface, and surface tension only stabilizes the shape and is otherwise ignored.
Stated in Section 2 and encoded in Eqs. (7)-(8); no deformation dynamics are considered.
ad hoc to paper The generalized Navier partial-slip conditions in Eqs. (14)-(15) hold at the liquid-liquid interface, with σ_rθ proportional to the relative tangential velocity and η/λ = -η'/λ'.
This is the central new boundary condition of the paper; no molecular or experimental justification is provided for its validity at liquid-liquid interfaces.
ad hoc to paper The normal viscous stress σ_rr is continuous across the interface, Eq. (9), without a surface-tension jump.
Used in Eq. (13) to determine the coefficient d and hence the terminal velocity. This is not the standard HRE boundary condition and is physically questionable for an interface with surface tension.
ad hoc to paper In the Appendix, the diagonal stress components σ_θθ and σ_φφ are also required to be continuous, Eqs. (91)-(92).
This closure fixes λ through Eqs. (93)-(94) and produces Eq. (95), which is independent of the internal viscosity and leads to a zero-velocity solid limit when combined with Eq. (78), so the axiom is not physically safe.
standard math The general solution of the Stokes equations in spherical coordinates listed in Table 1 is complete, and the l=1 truncation suffices for these boundary conditions.
Taken from Refs. [15-16]; the orthogonality argument is used to discard higher Legendre modes.

invented entities (1)

liquid-liquid slip lengths λ and λ'
purpose: Parameterize partial slip at the interface between two immiscible liquids, generalizing Navier's condition from solid-liquid to liquid-liquid interfaces.
No direct measurement is presented. The paper proposes future experiments in Section 7 to infer λ from terminal-velocity measurements, and the special model derives λ from an extra stress-continuity axiom rather than from independent data.

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Pith. "Pith review of Sliding of a liquid spherical drop in an external fluid: a generalization of the Hadamard-Rybczynski equation." pith.science (2026). https://pith.science/paper/J45XIWZZ

@misc{pith2026250705947,
  author       = {Pith},
  title        = {Pith review of: Sliding of a liquid spherical drop in an external fluid: a generalization of the Hadamard-Rybczynski equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J45XIWZZ}},
  note         = {Machine review of arXiv:2507.05947}
}

read the original abstract

An analytical solution is obtained for the problem of the slow movement of a small drop of liquid in another immiscible liquid in an infinitely large reservoir with the boundary condition of partial slip at the liquid-liquid interface. That generalizes the conventional Navier condition of partial slip that given at the liquid-solid interface, since the solid (rigid) state can be considered as a liquid with infinite viscosity. A generalized Hadamard-Rybczynski equation (HRE) is obtained. If slip length {\lambda}=0 that equation transforms into the conventional HRE. For infinite viscosity of the droplet, generalized HRE becomes a well-known relation generalizing the Stokes drag force for a solid sphere, taking into account the boundary condition of partial slip. At certain {\lambda}, we arrive at a model with continuity of the all components of the viscous stress tensor at the interface of two fluids, including diagonal tensor components. Generalized HRE is applied to the interpretation of the experiment in which the velocity of falling of a spherical droplet of silicate oil in the castor oil is investigated. Streamlines have been built corresponding to both the Hadamard-Rybczynski model and the partial slip approach. Presumably, the best applicability of the generalized HRE should be expected for the interface of hydrophobic liquid and hydrophilic one (water - hydrocarbons, water - higher alcohols, in general: aqueous emulsions, water - lipophilic organic liquids and oils, etc.). These are quite important emulsions in practical terms, for example, for the oil industry and medicine. Experimental methods for determining the slip length are discussed.

Figures

Figures reproduced from arXiv: 2507.05947 by the authors.

**Figure 2.** FIG. 2. The velocity of the fall of a spherical drop of silicate oil in castor oil. The blue [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** FIG.3. Polar component of the velocity of the liquid inside the drop with a radius of 1 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG.4. The falling of a spherical drop of silicate oil with a radius of 1 mm in the castor oil: [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

35 extracted references · 24 canonical work pages

[1]

HRE is a generalization of the Stokes equation that corresponds to the motion of a solid sphere

INTRODUCTION The Hadamard–Rybczynski equation (HRE) describes the slow motion of a small spherical liquid drop in an external liquid [1-2]. HRE is a generalization of the Stokes equation that corresponds to the motion of a solid sphere. HRE turns into the Stokes equation in the limit of infinite viscosity of the liquid drop [3]. Thus, a solid body is trea...

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[2]

Both liquids are insoluble in each other, do not mix with each other, and have a clear interface

MODEL WITH ARBITRARY SLIP LENGTH AT LIQUID-LIQUID INTERFACE Let us consider a liquid droplet placed inside another liquid. Both liquids are insoluble in each other, do not mix with each other, and have a clear interface. The drop has a spherical shape stabilized by interfacial surface tension. The z axis is oriented vertically upwards (Fig. 1). If the dro...

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[3]

(66) Stream function is (Table 1):   12 24 2( ') 6 2, sin 36 ' gR r r R rr R R R r R                                         

The external liquid 1 32( ') 6 2 1 3 1 cos3 'r gR R RV r R r                                 , (64) 1 32( ') 6 2 2 3 2 sin6 ' gR R RV r R r                                   , (65) 3 2 ( ') cos3 gRp r   . (66) Stream function is (Table 1):   12 24 2( ') 6 2,...

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[4]

(70) Stream function is (Table 1): 1 4 24 2( ') 6 2' sin 36 ' ' gR r r R R R                               

The internal liquid 1 22( ') 6 2' 3 1 cos3 ' 'r gR rV R R                         , (68) 1 22( ') 6 2' 3 2 1 sin3 ' ' gR rV R R                          , (69) 1 10( ') 6 2' 3 cos3 ' grp R             . (70) Stream function is (Table 1): 1 4 24 2( ') 6 2' sin 36 ' ' ...

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[5]

(60) describes the steady-state velocity of a spherical liquid drop in an external fluid

DRAG COEFFICIENT DC Eq. (60) describes the steady-state velocity of a spherical liquid drop in an external fluid. It can also be represented in the equivalent form 11 2 1 1 0 1 1 2( ') 1 ' 3 3 3 2 ' 6 gR RV R              (72) or 2 1 0 1 2( ') '(1 3 ) 3 2 3 '(1 2 ) gR RV R            , (73) or 2 1 0 1 2( ') 1 3 3 2 3 6 ...

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[6]

(9) and (11)

MODEL WITH CONTINUOUS VISCOUS STRESS TENSOR In the framework of the axisymmetric problem considered above, the continuities of only two non-zero components of the viscous stress tensor were ensured, rr and r , according to conditions that are given by Eqs. (9) and (11). But there are two more non-zero stresses that do not vanish in the axisymmetric pro...

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[7]

coefficient of surface viscosity

DISCUSSION The HRE is usually applied to describe the motion of a small droplet of liquid in another liquid. In its derivation, the no-slip boundary condition at the liquid-liquid interface, Eq.(77), was used. This somewhat contradicts the initial assumption that both liquids are immiscible (poorly soluble in each other). Therefore, a more natural and gen...

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[8]

CONCLUSIONS It is currently accepted that the HRE should describe the system under consideration well. The existing deviations, according to Levich [3] and many other researchers [23-24], should be associated with the difficulty of ensuring sufficient purity of liquids, and unaccounted surfactants distort the interpretation of experimental result. This pa...

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Show all 35 references

[9]

The external liquid 3 25 ( ')4 cos 15r R R gRV r r                 , (A44) 3 2( ')5 8 sin 30 R R gRV r r                 , (A45) 2 ( ') cos3 gR Rp r        . (A46) 26 Stream function is (Table 1): 1 2 2 1 1sin 2 b R ...
[10]

(A50) 2( ')' cos 3 grp  

The internal liquid 22( ')' 1 cos15 'r gR rV R               , (A49) 22( ')' 1 2 sin15 ' gR rV R               . (A50) 2( ')' cos 3 grp   . (A51) Stream function is (Table 1): 4 2 2 1 1 '' sin ' 10 a r rR c P R R      ...
[11]

Hadamard, Compt

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

HRE is a generalization of the Stokes equation that corresponds to the motion of a solid sphere

INTRODUCTION The Hadamard–Rybczynski equation (HRE) describes the slow motion of a small spherical liquid drop in an external liquid [1-2]. HRE is a generalization of the Stokes equation that corresponds to the motion of a solid sphere. HRE turns into the Stokes equation in the limit of infinite viscosity of the liquid drop [3]. Thus, a solid body is trea...

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[2] [2]

Both liquids are insoluble in each other, do not mix with each other, and have a clear interface

MODEL WITH ARBITRARY SLIP LENGTH AT LIQUID-LIQUID INTERFACE Let us consider a liquid droplet placed inside another liquid. Both liquids are insoluble in each other, do not mix with each other, and have a clear interface. The drop has a spherical shape stabilized by interfacial surface tension. The z axis is oriented vertically upwards (Fig. 1). If the dro...

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[3] [3]

(66) Stream function is (Table 1):   12 24 2( ') 6 2, sin 36 ' gR r r R rr R R R r R                                         

The external liquid 1 32( ') 6 2 1 3 1 cos3 'r gR R RV r R r                                 , (64) 1 32( ') 6 2 2 3 2 sin6 ' gR R RV r R r                                   , (65) 3 2 ( ') cos3 gRp r   . (66) Stream function is (Table 1):   12 24 2( ') 6 2,...

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[4] [4]

(70) Stream function is (Table 1): 1 4 24 2( ') 6 2' sin 36 ' ' gR r r R R R                               

The internal liquid 1 22( ') 6 2' 3 1 cos3 ' 'r gR rV R R                         , (68) 1 22( ') 6 2' 3 2 1 sin3 ' ' gR rV R R                          , (69) 1 10( ') 6 2' 3 cos3 ' grp R             . (70) Stream function is (Table 1): 1 4 24 2( ') 6 2' sin 36 ' ' ...

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[5] [5]

(60) describes the steady-state velocity of a spherical liquid drop in an external fluid

DRAG COEFFICIENT DC Eq. (60) describes the steady-state velocity of a spherical liquid drop in an external fluid. It can also be represented in the equivalent form 11 2 1 1 0 1 1 2( ') 1 ' 3 3 3 2 ' 6 gR RV R              (72) or 2 1 0 1 2( ') '(1 3 ) 3 2 3 '(1 2 ) gR RV R            , (73) or 2 1 0 1 2( ') 1 3 3 2 3 6 ...

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[6] [6]

(9) and (11)

MODEL WITH CONTINUOUS VISCOUS STRESS TENSOR In the framework of the axisymmetric problem considered above, the continuities of only two non-zero components of the viscous stress tensor were ensured, rr and r , according to conditions that are given by Eqs. (9) and (11). But there are two more non-zero stresses that do not vanish in the axisymmetric pro...

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[7] [7]

coefficient of surface viscosity

DISCUSSION The HRE is usually applied to describe the motion of a small droplet of liquid in another liquid. In its derivation, the no-slip boundary condition at the liquid-liquid interface, Eq.(77), was used. This somewhat contradicts the initial assumption that both liquids are immiscible (poorly soluble in each other). Therefore, a more natural and gen...

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[8] [8]

CONCLUSIONS It is currently accepted that the HRE should describe the system under consideration well. The existing deviations, according to Levich [3] and many other researchers [23-24], should be associated with the difficulty of ensuring sufficient purity of liquids, and unaccounted surfactants distort the interpretation of experimental result. This pa...

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