Generalized Hadamard–Rybczynski Equation

• Generalized Hadamard–Rybczynski Equation is a family of models extending the classical creeping-flow law by including effects such as partial slip, gas–liquid interfaces, and surfactant-induced Marangoni stresses.
• It employs both analytical and numerical methodologies, deriving terminal velocity laws and nonlocal PDEs that govern axisymmetric droplet surface evolution.
• The framework serves as both a predictive tool for hydrodynamic behavior and an inverse framework for characterizing interfacial transport properties in emulsions and aerosols.

Searching arXiv for the specified papers to ground the article in current records. arXiv search query: (Mecherbet, 2020) The generalized Hadamard–Rybczynski equation (HRE) denotes a family of extensions of the classical Hadamard–Rybczynski law for creeping-flow motion of a small spherical droplet. In the classical setting, the problem is the steady Stokes-flow motion of a small spherical liquid droplet in an infinitely large immiscible fluid reservoir, with no slip at the interface. In recent arXiv literature, the same label is used for several non-equivalent generalizations: a terminal-velocity law modified by tangential partial slip at a liquid–liquid interface; a further extension for gas–liquid systems that includes normal slip and gas-density gradients; and a nonlocal one-dimensional surface evolution equation for axisymmetric droplets in a mesoscopic sedimentation model. A related line of work treats surfactant-induced Marangoni stresses as a modified HR relation that interpolates between the Stokes and clean-drop limits (Lebedev-Stepanov, 8 Jul 2025, Lebedev-Stepanov, 20 Apr 2026, Mecherbet, 2020, Ervik et al., 2017).

1. Classical baseline and the scope of generalization

The classical HRE is the exact creeping-flow terminal-velocity result for a clean spherical drop with internal circulation. In the notation of the partial-slip formulation, it is

$$V_{HR}= \frac{2(p'-p)gR^2}{3\eta}\,\frac{1+K}{2+3K},$$

with $K=\eta'/\eta$, where $\eta$ is the viscosity of the external liquid and $\eta'$ is the droplet viscosity. In the notation used for surfactant-covered drops, the same clean-drop velocity is written as

$$U_{\text{HR}}=\frac{3(\mu_1+\mu_2)}{3\mu_1+2\mu_2}U_{\text{HS}} =\frac{2\Delta\rho |g| R^2}{3\mu_2}\,\frac{\mu_1+\mu_2}{3\mu_1+2\mu_2},$$

while the hard-sphere Stokes velocity is

$U_{\text{HS}}=\frac{2\Delta\rho |g| R^2}{9\mu_2}.$

The rigid-sphere limit is therefore obtained when internal circulation is suppressed, whereas the opposite $K\to 0$ limit gives the familiar clean-bubble result (Lebedev-Stepanov, 20 Apr 2026, Ervik et al., 2017).

Taken together, the recent literature indicates that “generalized HRE” is not a single equation but a problem class. Two recurring usages can be distinguished: the “terminal-velocity GHRE” (Editor’s term), in which the unknown is a scalar settling or rising speed, and the “surface-evolution GHRE” (Editor’s term), in which the unknown is an evolving axisymmetric interface profile.

Usage	Dependent variable	Representative relation
Classical clean-drop HRE	$V_0$ or $U$	terminal velocity of a spherical drop
Partial-slip GHRE	$V_0$	terminal velocity with interfacial slip
Surface-evolution GHRE	$K=\eta'/\eta$0	$K=\eta'/\eta$1

This multiplicity is substantive rather than terminological. In one branch, generalization means modifying the interfacial boundary condition while retaining a spherical steady drop. In another, it means reducing a moving-boundary Stokes problem to a nonlocal hyperbolic PDE for an axisymmetric shape function.

2. Mesoscopic sedimentation and the nonlocal surface-evolution formulation

In the mesoscopic sedimentation model, the starting point is an inertialess suspension in a viscous fluid under gravity, in a mean-field or mesoscopic scaling where a very large number of small particles generates a continuous density $K=\eta'/\eta$2. After a Galilean change of variables, the normalized transport–Stokes system is

$K=\eta'/\eta$3

with $K=\eta'/\eta$4. The velocity is incompressible, decays at infinity, and is represented through the Oseen tensor

$K=\eta'/\eta$5

A global well-posedness result holds for $K=\eta'/\eta$6 with finite first moment, yielding a unique solution

$K=\eta'/\eta$7

for all $K=\eta'/\eta$8, with a measure-preserving flow map $K=\eta'/\eta$9 satisfying

$\eta$0

For characteristic-function initial data, this propagates domain regularity; if $\eta$1 is homeomorphic to a sphere, the model rules out finite-time torus formation at this PDE level (Mecherbet, 2020).

The classical Hadamard–Rybczynski result is recovered exactly in the spherical case. If $\eta$2, then the solution is a rigid translation,

$\eta$3

with

$\eta$4

The key boundary compatibility condition is

$\eta$5

which implies that the normal velocity of the boundary is exactly that of a translating sphere.

The more novel reduction is the axisymmetric surface law. Writing the droplet as

$\eta$6

with

$\eta$7

the interface dynamics reduce to

$\eta$8

Here $\eta$9 and $\eta'$0 are nonlocal operators obtained by evaluating the Stokes velocity on the boundary and projecting it onto tangential and radial directions,

$\eta'$1

$\eta'$2

The boundary velocity is written through a singular boundary integral,

$\eta'$3

and in spherical coordinates depends on the kernel

$\eta'$4

This generalized HRE is a one-dimensional nonlocal hyperbolic PDE. For $\eta'$5 with $\eta'$6, there is local existence and uniqueness of

$\eta'$7

together with a unique center law

$\eta'$8

When $\eta'$9 and $$U_{\text{HR}}=\frac{3(\mu_1+\mu_2)}{3\mu_1+2\mu_2}U_{\text{HS}} =\frac{2\Delta\rho |g| R^2}{3\mu_2}\,\frac{\mu_1+\mu_2}{3\mu_1+2\mu_2},$$0, one has $$U_{\text{HR}}=\frac{3(\mu_1+\mu_2)}{3\mu_1+2\mu_2}U_{\text{HS}} =\frac{2\Delta\rho |g| R^2}{3\mu_2}\,\frac{\mu_1+\mu_2}{3\mu_1+2\mu_2},$$1, so $$U_{\text{HR}}=\frac{3(\mu_1+\mu_2)}{3\mu_1+2\mu_2}U_{\text{HS}} =\frac{2\Delta\rho |g| R^2}{3\mu_2}\,\frac{\mu_1+\mu_2}{3\mu_1+2\mu_2},$$2 is an exact solution. If a different center is used, the radius function is no longer constant, but the geometric surface remains the same sphere under reparametrization. Numerically, an upwind finite-difference scheme confirms that the spherical state is stable when $$U_{\text{HR}}=\frac{3(\mu_1+\mu_2)}{3\mu_1+2\mu_2}U_{\text{HS}} =\frac{2\Delta\rho |g| R^2}{3\mu_2}\,\frac{\mu_1+\mu_2}{3\mu_1+2\mu_2},$$3, and also shows breakdown of the radius parametrization when the imposed center motion violates $$U_{\text{HR}}=\frac{3(\mu_1+\mu_2)}{3\mu_1+2\mu_2}U_{\text{HS}} =\frac{2\Delta\rho |g| R^2}{3\mu_2}\,\frac{\mu_1+\mu_2}{3\mu_1+2\mu_2},$$4.

3. Tangential partial slip and generalized terminal-velocity laws

A distinct generalization retains a spherical droplet and modifies the interfacial boundary condition. In this approach, the interface is allowed to support tangential partial slip, expressed through slip lengths rather than strict no-slip continuity. For a small spherical drop moving slowly in another immiscible liquid, the analytical solution in the drop frame leads to

$$U_{\text{HR}}=\frac{3(\mu_1+\mu_2)}{3\mu_1+2\mu_2}U_{\text{HS}} =\frac{2\Delta\rho |g| R^2}{3\mu_2}\,\frac{\mu_1+\mu_2}{3\mu_1+2\mu_2},$$5

In the notation used in the two-slip-length formulation, the corresponding result is

$$U_{\text{HR}}=\frac{3(\mu_1+\mu_2)}{3\mu_1+2\mu_2}U_{\text{HS}} =\frac{2\Delta\rho |g| R^2}{3\mu_2}\,\frac{\mu_1+\mu_2}{3\mu_1+2\mu_2},$$6

Both formulas reduce to the classical HRE when $$U_{\text{HR}}=\frac{3(\mu_1+\mu_2)}{3\mu_1+2\mu_2}U_{\text{HS}} =\frac{2\Delta\rho |g| R^2}{3\mu_2}\,\frac{\mu_1+\mu_2}{3\mu_1+2\mu_2},$$7, and both recover a partial-slip version of the Stokes law in the rigid-sphere limit. In particular, for $$U_{\text{HR}}=\frac{3(\mu_1+\mu_2)}{3\mu_1+2\mu_2}U_{\text{HS}} =\frac{2\Delta\rho |g| R^2}{3\mu_2}\,\frac{\mu_1+\mu_2}{3\mu_1+2\mu_2},$$8 or $$U_{\text{HR}}=\frac{3(\mu_1+\mu_2)}{3\mu_1+2\mu_2}U_{\text{HS}} =\frac{2\Delta\rho |g| R^2}{3\mu_2}\,\frac{\mu_1+\mu_2}{3\mu_1+2\mu_2},$$9,

$U_{\text{HS}}=\frac{2\Delta\rho |g| R^2}{9\mu_2}.$0

which further reduces to the classical rigid-sphere Stokes velocity when $U_{\text{HS}}=\frac{2\Delta\rho |g| R^2}{9\mu_2}.$1 (Lebedev-Stepanov, 8 Jul 2025, Lebedev-Stepanov, 20 Apr 2026).

The boundary conditions are formulated symmetrically. For the liquid–liquid interface, the tangential slip condition is written as

$U_{\text{HS}}=\frac{2\Delta\rho |g| R^2}{9\mu_2}.$2

with continuity of normal velocity. A central conceptual point is that each of the two fluids has its own slip length. In one formulation the relation is

$U_{\text{HS}}=\frac{2\Delta\rho |g| R^2}{9\mu_2}.$3

and in another it appears as

$U_{\text{HS}}=\frac{2\Delta\rho |g| R^2}{9\mu_2}.$4

These expressions encode the same viscosity-ratio coupling but with different sign conventions. One paper makes explicit that the sign assignment is coordinate-dependent and that the physically relevant criterion is positive interfacial dissipation, whereas another remarks that negative $U_{\text{HS}}=\frac{2\Delta\rho |g| R^2}{9\mu_2}.$5 can occur in the formalism.

The partial-slip generalization also yields secondary transport quantities. With buoyancy force

$U_{\text{HS}}=\frac{2\Delta\rho |g| R^2}{9\mu_2}.$6

the friction coefficient is

$U_{\text{HS}}=\frac{2\Delta\rho |g| R^2}{9\mu_2}.$7

and the drag coefficient satisfies

$U_{\text{HS}}=\frac{2\Delta\rho |g| R^2}{9\mu_2}.$8

One special choice of slip length,

$U_{\text{HS}}=\frac{2\Delta\rho |g| R^2}{9\mu_2}.$9

enforces continuity of all viscous stress tensor components, including $K\to 0$0 and $K\to 0$1, and simplifies the speed to

$K\to 0$2

That result is independent of the inner viscosity $K\to 0$3, which distinguishes it sharply from the classical no-slip HRE.

4. Gas–liquid interfaces, normal slip, bubbles, and aerosols

The 2026 extension adds a mechanism absent from the liquid–liquid problem: normal partial slip at gas–liquid interfaces. The motivation is that gas density is compressible and nonuniform near the interface, so a purely tangential-slip description is incomplete. The gas flux is written as

$K\to 0$4

and in the diffusion-dominated regime

$K\to 0$5

the density perturbation satisfies

$K\to 0$6

A molecular-kinetic analysis then yields a normal stress relation and a normal slip length of order the mean free path,

$K\to 0$7

with a specific derivation for air giving

$K\to 0$8

The corresponding liquid-side slip length has magnitude

$K\to 0$9

The impermeability condition is also expressed as

$V_0$0

at the dense wall (Lebedev-Stepanov, 20 Apr 2026).

For a gas bubble, the terminal velocity acquires a correction factor involving both $V_0$1 and $V_0$2: $V_0$3 The linear term is attributed to tangential slip and the quadratic term to normal slip. In the small-$V_0$4 regime relevant for air bubbles in water, the normal-slip correction is very small, and the rise speed is very well approximated by the classical clean-bubble HRE limit,

$V_0$5

The associated gas-density profile is

$V_0$6

so the bubble is slightly denser at the bottom and less dense at the top.

For a falling liquid droplet in air, the gas-side slip length is taken as

$V_0$7

and the aerosol terminal velocity is expressed as a correction to the Stokes prefactor, with linear terms from tangential slip and quadratic terms from normal slip. The theory is compared with the empirical fit

$V_0$8

and the reported discrepancy with experiment is about $V_0$9 at the smallest radii considered, comparable to the experimental error. The paper states that inclusion of normal slip improves agreement, whereas the classical Maxwell condition $U$0 gives worse agreement. The aerosol gas-density perturbation is again

$U$1

implying higher air density ahead of the falling droplet and lower density behind it.

5. Surfactant-covered drops and Marangoni-retarded HR motion

A separate generalization attributes deviations from the clean-drop HRE to surfactant-induced Marangoni stresses. The flow is modeled as incompressible Newtonian Stokes flow in both phases, with interfacial conditions

$U$2

For a spherical interface, the exact axisymmetric solution implies that the interfacial tension must take the form

$U$3

with

$U$4

The resulting terminal velocity is

$U$5

so the clean-drop value is reduced by the first harmonic of the interfacial-tension field (Ervik et al., 2017).

The paper emphasizes that not every mathematical solution of the Stokes plus Marangoni problem is physically admissible. An interfacial energy balance shows that passive interfaces constrain the terminal speed to the interval

$U$6

Within this framework, a hovering drop is a formal solution of the stress equations but violates conservation of energy unless energy is supplied directly to the interface. This energy-based admissibility criterion is one of the paper’s main departures from the stagnant-cap picture.

The proposed “continuous-interface model” does not solve an interfacial advection–diffusion equation for surfactant. Instead, it imposes a mechanical balance on a 2D surfactant layer with interfacial concentration $U$7. Neglecting surface viscosities, the surface stress is taken to be purely isotropic,

$U$8

In the low-concentration regime, the relation between concentration and tension is linearized to

$U$9

A surface-force bound then implies a critical radius

$V_0$0

below which drops behave like hard spheres. Writing $V_0$1 and $V_0$2, the model gives $V_0$3 for $V_0$4 and $V_0$5 as $V_0$6. The critical radius is further predicted to be proportional to the interfacial surfactant concentration, and the average interfacial concentration is related to the bulk concentration by the Langmuir form

$V_0$7

The paper positions this continuous-interface model against the stagnant-cap model, arguing that the former is more appropriate for amphiphilic surfactant molecules, whereas the latter is natural when the interfacially active species behave more like adsorbed particles.

6. Applicability, interpretation, and recurrent points of confusion

The generalized HRE formulations share a creeping-flow core but differ sharply in what is being generalized. The partial-slip and gas–liquid theories assume a small spherical droplet or bubble, an unbounded reservoir, immiscible phases, and steady terminal motion under gravity and buoyancy. The surfactant model also retains sphericity and Stokes flow, but the constitutive closure is interfacial rather than kinematic. The mesoscopic surface-evolution theory abandons the steady spherical ansatz and instead derives a nonlocal PDE for an axisymmetric interface profile. These are therefore complementary, not interchangeable, generalizations (Lebedev-Stepanov, 8 Jul 2025, Lebedev-Stepanov, 20 Apr 2026, Ervik et al., 2017, Mecherbet, 2020).

A common misconception is to treat “generalized HRE” as a universal correction factor multiplying the classical terminal velocity. That interpretation fits the partial-slip and normal-slip branches, but not the surface-evolution branch, where the governing object is the radius function $V_0$8. Another frequent source of confusion is the sign of the slip length. The recent slip-length literature explicitly states that each phase has its own slip length and that one may appear positive while the other appears negative; the sign convention depends on coordinates, whereas the physical constraints are continuity relations tied to viscosity ratio and nonnegative interfacial dissipation.

The literature also makes the limits of validity explicit. For larger bubbles or droplets, $V_0$9 or $K=\eta'/\eta$00 can exceed unity, and then higher spherical modes, shape deformation, or both must be included. In the surface-evolution formulation, local existence requires that the spherical-graph parametrization remain nondegenerate, enforced by $K=\eta'/\eta$01. Numerical experiments further indicate that maximal existence can depend on the chosen moving center, because the parametrization may fail even when the underlying sphere remains geometrically intact.

The applied relevance repeatedly emphasized in the recent work concerns interfaces where partial slip is physically plausible: water–oil and oil–water emulsions, water–hydrocarbon systems, water–higher alcohol systems, aqueous emulsions, and lipophilic organic liquids and oils. These are cited as important in the oil industry and medicine. The same literature proposes using terminal-velocity measurements to infer slip lengths by measuring the motion of liquid $K=\eta'/\eta$02 in liquid $K=\eta'/\eta$03, then swapping the phases and checking the predicted reciprocity condition. In this sense, the generalized HRE serves both as a hydrodynamic theory and as an inverse characterization framework for interfacial transport properties.