pith. sign in

arxiv: 2604.17793 · v1 · submitted 2026-04-20 · ⚛️ physics.flu-dyn · cond-mat.soft

Tangential and normal partial slip at the liquid-fluid interfaces: application to a small liquid droplet, gas bubble, and aerosol

Peter Lebedev-Stepanov This is my paper

Pith reviewed 2026-05-10 04:28 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cond-mat.soft
keywords slip lengthHadamard-Rybczynski equationterminal velocitygas bubbleliquid dropletaerosolpartial slipStokes flow
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The pith

An analytical solution for the slow motion of small droplets and bubbles incorporates normal slip from density gradients and separate tangential slip lengths on each side of the interface.

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

The paper derives an exact low-Reynolds-number solution for a fluid sphere translating through an immiscible fluid when the spherical interface obeys both normal slip tied to a density gradient and tangential partial slip. Normal slip is allowed only when one phase is gaseous; each fluid is assigned its own independent slip length rather than a single friction coefficient. The resulting drag expression is matched to buoyancy to obtain revised terminal-velocity formulas for a rising gas bubble and a falling aerosol particle. These formulas reduce to the classic Hadamard-Rybczynski result only when both slip lengths vanish and they predict that gas density varies spatially inside the bubble or around the droplet, with the relative variation growing as size increases. The analysis indicates that the generalized model should apply most directly to water-hydrocarbon or water-alcohol interfaces.

Core claim

The central claim is that the Stokes equations for a small spherical drop or bubble moving through an unbounded immiscible fluid admit an exact solution once the interface conditions are written in terms of two independent slip lengths (one for each fluid) plus a normal-slip term proportional to the density gradient when one phase is a gas. Matching the resulting hydrodynamic force to the buoyancy force produces explicit expressions for terminal velocity that differ from the Hadamard-Rybczynski equation and that imply a non-uniform density distribution whose magnitude increases with bubble or droplet radius.

What carries the argument

The slip-length boundary condition applied independently on each side of the liquid-fluid interface, combining a tangential velocity discontinuity with a normal mass flux driven by the local density gradient.

If this is right

  • Terminal velocities of gas bubbles rising in liquids and aerosols falling in air are given by explicit formulas that include the two slip lengths and the density-gradient term.
  • Gas density is higher near the interface than at the center of a rising bubble, with the relative difference increasing for larger bubbles.
  • The same spatial density variation occurs in the gas surrounding a falling liquid droplet.
  • The generalized equations are expected to describe motion at hydrophobic-hydrophilic interfaces such as oil-water emulsions more accurately than no-slip treatments.

Where Pith is reading between the lines

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

  • If slip lengths can be extracted from terminal-velocity data for common liquid pairs, the same interface conditions could be inserted into calculations of droplet coalescence or breakup rates in emulsions.
  • The requirement that normal slip accompany a density gradient suggests the same mechanism may operate in other gas-liquid flows that possess temperature or concentration gradients.
  • Direct numerical simulation of the Stokes equations with the proposed interface conditions on a sphere would provide an independent check on the analytic drag formula.

Load-bearing premise

Normal slip at the interface requires a density gradient and is possible only when one of the two fluids is a gas.

What would settle it

An experiment that measures uniform gas density inside a rising bubble of any size while the measured terminal velocity still deviates from the classical Hadamard-Rybczynski prediction would falsify the density-gradient part of the model.

Figures

Figures reproduced from arXiv: 2604.17793 by Peter Lebedev-Stepanov.

Figure 1
Figure 1. Figure 1: FIG. 1. Liquid velocity profile [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Viscous flows of fluids 1 and 2 with partial slip condition at their interface, z [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Direction of velocities on the droplet surface in the presence of slip at the interface in the [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG 5. Polar coordinates that include the radial coordinate and polar angle, ( , ) r  [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG.6. Graphs of the dimensionless functions [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
read the original abstract

An analytical solution is obtained for the problem of the slow movement of a small drop of a fluid in another immiscible fluid in an infinitely large reservoir with the boundary condition of the normal slip and/or tangential partial slip at the interface. That generalizes the conventional Navier and Maxwellian boundary conditions of partial slip. Normal slip is accompanied by the density gradient in the fluid and is applicable only if one of the phases in contact at the interface is a gas. Although tangential partial slip and the associated generalization of the Hadamard-Rybczynski equation (HRE) have been considered previously, they were done using the friction coefficient formalism. Here, this issue is discussed within the more general formalism of slip lengths. It is proven that each of the two fluids separated by an interface has its own slip length. New equations describing the terminal velocity of gas bubble rise and aerosol falling have been obtained. The result is compared with experiment. It has been shown that the gas density within a rising bubble and around a falling droplet in the air is not uniform. The relative magnitude of the density increment increases with the size of the bubble or aerosol. Presumably, the best applicability of the generalized HRE should be expected for the interface of hydrophobic liquid and hydrophilic one (water and hydrocarbons, water and 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 considered.

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 manuscript derives an analytical solution for the slow (Stokes) motion of a small fluid droplet or bubble in an immiscible fluid under generalized interface boundary conditions that include both tangential partial slip and a novel normal slip (accompanied by a density gradient and asserted to be valid only when one phase is gaseous). This generalizes the Hadamard-Rybczynski equation; new closed-form terminal-velocity expressions for rising gas bubbles and falling aerosols are obtained, compared with experiment, and used to infer non-uniform gas density inside bubbles and around droplets. The work also discusses experimental determination of slip lengths and suggests optimal applicability for hydrophobic-hydrophilic liquid pairs.

Significance. If the normal-slip condition can be shown to be consistent with the continuity equation and if the terminal-velocity formulas are accompanied by quantitative error estimates and direct data comparisons, the results would supply a practical extension of classical two-phase Stokes solutions with potential utility for emulsion modeling in the oil and medical industries. The explicit use of slip lengths rather than friction coefficients and the claim that each fluid possesses its own slip length are useful clarifications.

major comments (2)
  1. [Boundary conditions and analytical solution] The central analytical solution and the new terminal-velocity formulas rest on the normal-slip boundary condition. The manuscript states that normal slip is accompanied by a density gradient and is applicable only when one phase is a gas, yet it does not demonstrate that the derived velocity field satisfies the full continuity equation once a normal-velocity discontinuity is introduced at the interface. Standard incompressible or weakly compressible two-phase derivations enforce continuity of normal velocity to satisfy mass conservation; without an explicit check (or variable-density terms retained throughout the domain), the solution risks violating mass balance. This issue is load-bearing for the claimed generalization of the Hadamard-Rybczynski equation.
  2. [Terminal velocity equations and experimental comparison] The terminal-velocity expressions are presented as new equations for bubble rise and aerosol fall, but they depend directly on three free parameters (tangential slip lengths for each fluid and the normal slip length). The manuscript asserts that each fluid has its own slip length and that density is non-uniform, yet no quantitative error estimates, sensitivity analysis, or tabulated comparison of predicted versus measured velocities (with residuals) are supplied in the abstract or the described results. This weakens the experimental validation claim.
minor comments (2)
  1. [Abstract] The abstract states that 'it is proven that each of the two fluids separated by an interface has its own slip length,' but the corresponding derivation or theorem is not cross-referenced to a specific section or equation in the main text.
  2. [Results and discussion] The claim that 'the relative magnitude of the density increment increases with the size of the bubble or aerosol' should be supported by an explicit plot or table of the density increment versus radius derived from the model.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our work. We address the major comments point by point below and have incorporated revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Boundary conditions and analytical solution] The central analytical solution and the new terminal-velocity formulas rest on the normal-slip boundary condition. The manuscript states that normal slip is accompanied by a density gradient and is applicable only when one phase is a gas, yet it does not demonstrate that the derived velocity field satisfies the full continuity equation once a normal-velocity discontinuity is introduced at the interface. Standard incompressible or weakly compressible two-phase derivations enforce continuity of normal velocity to satisfy mass conservation; without an explicit check (or variable-density terms retained throughout the domain), the solution risks violating mass balance. This issue is load-bearing for the claimed generalization of the Hadamard-Rybczynski equation.

    Authors: We acknowledge the referee's concern regarding the continuity equation. Our normal slip condition is formulated specifically for cases where one phase is gaseous, allowing for a density gradient that accommodates the normal velocity discontinuity while preserving mass conservation. To address this explicitly, we have added an appendix in the revised manuscript that derives and verifies the integrated form of the continuity equation across the interface, confirming consistency with the variable-density assumption in the gas phase. This supports the validity of our generalization of the Hadamard-Rybczynski equation. revision: yes

  2. Referee: [Terminal velocity equations and experimental comparison] The terminal-velocity expressions are presented as new equations for bubble rise and aerosol fall, but they depend directly on three free parameters (tangential slip lengths for each fluid and the normal slip length). The manuscript asserts that each fluid has its own slip length and that density is non-uniform, yet no quantitative error estimates, sensitivity analysis, or tabulated comparison of predicted versus measured velocities (with residuals) are supplied in the abstract or the described results. This weakens the experimental validation claim.

    Authors: We agree that providing quantitative error estimates and direct comparisons would enhance the validation. In the revised manuscript, we have added a dedicated section with tabulated comparisons of our predicted terminal velocities against experimental data for gas bubbles and aerosols. This includes calculated residuals, percentage errors, and a sensitivity analysis with respect to the slip length parameters. These additions demonstrate the model's predictive capability and support our conclusions regarding non-uniform density. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper presents an analytical solution to the Stokes flow problem for a droplet or bubble subject to generalized slip boundary conditions (normal slip accompanied by density gradient when one phase is gas, plus tangential partial slip). Terminal-velocity expressions are obtained directly from this solution and compared with experiment; slip lengths enter as model parameters analogous to viscosity ratios in the classical Hadamard-Rybczynski derivation. No step reduces a claimed prediction to a fitted input by construction, no uniqueness theorem is imported from self-citation, and the normal-slip condition is introduced as an explicit modeling choice rather than smuggled via prior ansatz. The derivation chain therefore does not collapse to its own inputs.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard low-Reynolds-number Stokes flow assumptions plus the introduction of two independent slip lengths as free parameters that must be supplied externally.

free parameters (3)
  • tangential slip length for fluid 1
    Introduced to generalize the tangential boundary condition; value determined by experiment or fitting.
  • tangential slip length for fluid 2
    Introduced to generalize the tangential boundary condition; value determined by experiment or fitting.
  • normal slip length
    Introduced to allow normal slip accompanied by density gradient; value determined by experiment or fitting.
axioms (2)
  • domain assumption Flow is steady, incompressible, and at very low Reynolds number (Stokes regime) in an infinite domain.
    Required for the analytical solution of the Navier-Stokes equations around a sphere.
  • domain assumption The two fluids are immiscible and the interface remains spherical.
    Stated in the problem setup for small droplets and bubbles.

pith-pipeline@v0.9.0 · 5587 in / 1549 out tokens · 39027 ms · 2026-05-10T04:28:00.178376+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

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    SPHERICAL DROPLET SLIDING Let us determine the sign distribution in the pair of slip lengths between a liquid droplet and an external fluid. The correct choice of signs should ensure positive energy dissipation due to friction at the interface between the two fluids. For the sake of clarity, let us assume that the droplet rises. Consider it in a coordinat...

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    The external liquid 1 32( ' ) 6 2 1 3 1 cos3 'r gR R RV r R r                                   , (S.70) 1 32( ' ) 6 2 2 3 2 sin6 ' gR R RV r R r                                     , (S.71) 3 2 ( ' ) cos3 gRp r      . (S.72) Stream function is (Table S1):  ...

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  43. [43]

    (S.76) Stream function is (Table S1): 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                            , (S.74) 1 22( ' ) 6 2' 3 2 1 sin3 ' ' gR rV R R                             , (S.75) 1 10( ' ) 6 2' 3 cos3 ' grp R               . (S.76) Stream function is (Table S1): 1 4 24 2( '...

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  44. [44]

    In this case, the normal slip condition (S.22) should be used instead of condition (S.13)

    Gas-liquid interface and small bubble rise Let us consider the motion of a gas bubble. In this case, the normal slip condition (S.22) should be used instead of condition (S.13). Taking into account Eq. (S.22), for the radial component of the velocity of the internal fluid (S.38) we have 1 0 ''( , ) 2 ' cos cos10r DfaV R c n R          . (S.95)...

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