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

This paper presents an analytical solution for the slow movement of small liquid droplets, gas bubbles, and aerosols by generalizing the Hadamard-Rybczynski equation to include both tangential and normal partial slip conditions at fluid-fluid interfaces, demonstrating that each fluid possesses its own slip length and deriving new equations for terminal velocity that account for non-uniform gas density.

The Gist

The Big Picture: Why Droplets Don't Always Behave Like Solid Balls

Imagine you are watching a tiny bubble rise in a glass of water, or a speck of dust fall through the air. Physics has a famous rulebook (the Hadamard-Rybczynski equation) that predicts exactly how fast these things should move.

For a long time, scientists assumed that when a liquid droplet moves through another liquid, the two fluids stick together perfectly at the boundary, like two pieces of tape pressed together. This is called the "no-slip" condition.

However, this paper argues that reality is messier. Sometimes, the fluids don't stick perfectly; they slide past each other a little bit. The author, Peter Lebedev-Stepanov, has updated the rulebook to account for this "slip," and he discovered two distinct ways this sliding happens: Sideways Sliding and Head-on Sliding.

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1. The Two Types of "Slip"

To understand this, imagine two people walking side-by-side on a moving walkway at an airport.

A. Tangential Slip (The "Sideways Slide")

This is the most common type of slip. Imagine the two people are walking in the same direction, but one is wearing slippery socks and the other has rubber soles. They are moving together, but one is sliding slightly faster than the other along the surface where they touch.

• The Physics: When a liquid droplet moves, the liquid inside and the liquid outside don't always move at the exact same speed right at the surface. They "slip" past each other.
• The Discovery: The author proves that both liquids have their own "slip length." Think of this as a measure of how "greasy" the surface feels to that specific liquid.
  • If the outside liquid is the "driver" (pushing the droplet), it has a positive slip length.
  • The inside liquid (being dragged along) has a negative slip length.
  • Analogy: It's like a tug-of-war where the rope is slippery. The person pulling (outside fluid) gets a better grip, while the person being pulled (inside fluid) slides a bit more. The math shows these two "grips" are perfectly linked.

B. Normal Slip (The "Head-on Slide")

This is the paper's big new insight. It only happens when a gas (like a bubble or air around a dust speck) meets a liquid.

Imagine a crowd of people (gas molecules) running toward a wall (the liquid surface).

• The Old View: Scientists thought the crowd just stopped dead at the wall.
• The New View: Because gas is squishy (compressible), the people in the front of the crowd get slightly squished together, and the people in the back spread out. This creates a tiny "density wave." The gas doesn't just stop; it "slips" through the wall slightly by changing its density.
• Analogy: Think of a spring. If you push a spring against a wall, it compresses. The gas molecules near the bubble compress slightly at the top and stretch slightly at the bottom. This compression creates a tiny "normal slip" (sliding perpendicular to the surface).

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2. Why Does This Matter? (The Real-World Impact)

The author applies these new rules to three specific scenarios:

1. Gas Bubbles Rising:

   • Old Theory: A bubble rises because it's light.
   • New Theory: A bubble rises faster than we thought because the air inside isn't uniform. The air gets slightly denser at the bottom of the bubble and lighter at the top. This tiny density change helps the bubble move.
   • Result: The new math predicts the speed of tiny bubbles more accurately than the old rules.

2. Aerosols Falling (Dust/Mist in Air):

   • Old Theory: A tiny water droplet falls through air at a specific speed.
   • New Theory: The air around the droplet isn't just flowing; it's getting slightly squished in front of the droplet and stretched behind it.
   • Result: When the author compared his new formula to real-world experiments, it matched much better, especially for very tiny droplets (micro-droplets). The "normal slip" (the squishing of air) explained the small errors in previous theories.

3. Oil and Water Emulsions:

   • This is crucial for the oil industry and medicine. Think of salad dressing (oil and water) or drug delivery systems (liposomes).
   • The author suggests that when you mix a hydrophobic liquid (oil) with a hydrophilic one (water), they slide past each other easily.
   • Analogy: Imagine trying to mix oil and water. Usually, they separate. But if you look closely at the boundary, the oil molecules are "sliding" over the water molecules. The new equations help engineers predict how fast these tiny droplets will separate or mix, which is vital for making better medicines or refining oil.

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3. The "Slip Length" Concept

The paper introduces a concept called Slip Length.

• Imagine: You are sliding on a floor. If the floor is icy, you slide a long way before stopping. If the floor is carpet, you stop immediately.
• In Physics: The "Slip Length" is a theoretical distance. It asks: "If we extended the speed of the fluid inside the droplet straight out, how far would it have to go before it matched the speed of the fluid outside?"
• The Twist: The author shows that for two liquids, there are two slip lengths, and they are mathematically tied together like a seesaw. If one goes up, the other goes down.

Summary: The Takeaway

This paper is like updating the instruction manual for how tiny things move in fluids.

• Before: We thought liquids stuck together perfectly or slid perfectly like ice.
• Now: We know they slide in a complex, linked way.
  • Sideways: They slide past each other (Tangential Slip).
  • Head-on: Gases compress and stretch against liquids (Normal Slip).

By accounting for these tiny "slips," scientists can now predict the speed of bubbles, raindrops, and medical droplets with much higher accuracy. It's a small adjustment to the math, but it makes a big difference in understanding the invisible world of fluids.

Technical Summary

1. Problem Statement

The paper addresses the hydrodynamics of small spherical particles (liquid droplets, gas bubbles, and aerosols) moving slowly (low Reynolds number) through an immiscible fluid. While the classical Hadamard-Rybczynski equation (HRE) describes the terminal velocity of such particles under the assumption of a no-slip boundary condition, experimental deviations often occur.
The author identifies two specific gaps in current theory:

1. Tangential Slip: Previous models often used a "friction coefficient" formalism which lacked symmetry between the two fluids. The paper argues for a more general slip length formalism where both fluids at the interface possess distinct slip lengths.
2. Normal Slip: Standard hydrodynamics assumes impermeability (no normal flow) at the interface. However, for gas-liquid interfaces (bubbles and aerosols), the compressibility of the gas allows for normal partial slip, accompanied by density gradients, which has been largely ignored in low-Reynolds-number analyses.

2. Methodology

The author employs analytical solutions to the Stokes equations (linearized Navier-Stokes equations) in spherical coordinates. The methodology involves:

• Generalized Boundary Conditions:
  • Tangential Slip: The author introduces a pair of slip lengths ($\lambda$ for the external fluid and $\lambda'$ for the internal fluid) rather than a single slip coefficient. By analyzing energy dissipation and momentum transfer, it is proven that these lengths are related by the viscosity ratio: $\lambda' = -\kappa \lambda$ (where $\kappa = \eta'/\eta$). This ensures that the leading fluid (driven by external forces) has a positive slip length relative to the trailing fluid.
  • Normal Slip (Gas-Liquid): Using Molecular Kinetic Theory, the author derives a boundary condition for the normal component of the velocity at a gas-liquid interface. This accounts for the mean free path of gas molecules ($L$) and results in a non-zero normal velocity compensated by diffusion, leading to a density gradient near the interface. The derived slip length for the gas is $\lambda \approx \frac{4}{3}L$ (or $L$ depending on accommodation), differing slightly from classical Maxwellian conditions.
• Analytical Solution: The Stokes equations are solved for the velocity and pressure fields inside and outside the sphere, applying the generalized boundary conditions (tangential and normal slip) and continuity of viscous stresses.
• Comparison: The derived terminal velocity equations are compared against experimental data for falling aerosols and existing theoretical limits (Stokes, HRE).

3. Key Contributions

A. Duality of Slip Lengths at Liquid-Liquid Interfaces

The paper establishes that at a liquid-liquid interface, two distinct slip lengths exist, one for each fluid.

• They are not independent; they are coupled by the viscosity ratio and must have opposite signs to satisfy thermodynamic requirements (positive energy dissipation).
• The fluid directly driven by the external force (the "leading" fluid) has a positive slip length, while the "trailing" fluid has a negative slip length.
• This generalizes the Navier boundary condition from solid-liquid to liquid-liquid interfaces.

B. Derivation of Normal Slip for Gases

A novel contribution is the derivation of the normal slip condition for gas bubbles and aerosols.

• The author proves that normal slip is inevitable for gases due to compressibility and molecular diffusion.
• This leads to a non-uniform gas density distribution around the particle (higher density at the "front" of a rising bubble, lower at the "back," and vice versa for falling droplets).
• The slip length for the gas is shown to be of the order of the molecular mean free path ($L$).

C. Generalized Hadamard-Rybczynski Equation (GHRE)

The author derives a new expression for the terminal velocity ($V_0$) that incorporates:

1. Tangential partial slip (linear in slip length $\lambda$).
2. Normal partial slip (quadratic in slip length $\lambda^2$ for bubbles/aerosols).
3. The viscosity ratio between the two phases.

The resulting equation (Eq. 148 in the text) modifies the classical HRE and Stokes formula, providing a correction factor that depends on the ratio of the particle radius to the slip length.

4. Results

• Liquid Droplets in Liquid: The generalized equation reduces to the classical HRE when slip lengths are zero. For hydrophobic-hydrophilic interfaces (e.g., oil-water emulsions), the model predicts deviations from HRE that can be used to experimentally determine slip lengths.
• Gas Bubbles: The contribution of normal slip to the rise velocity of small bubbles is found to be very small (second-order effect) compared to tangential slip, but the model confirms that gas density is non-uniform within the bubble.
• Aerosols (Liquid Droplets in Air):
  • The derived equation for falling aerosols (Eq. 148/151) shows excellent agreement with experimental data (specifically the semi-empirical relation by Otung) for droplet radii between 0.25 $\mu$m and 9.5 $\mu$m.
  • The model using the derived slip length ($\lambda = 4L/3$) fits the data better than models using the classical Maxwellian slip length ($\lambda = L$).
  • The inclusion of normal slip significantly improves the theoretical convergence with experimental data at the smallest droplet radii, where the Knudsen number is highest.
• Density Gradients: The paper quantifies the relative density increment of the gas around the particle. For a 10 $\mu$m bubble, this is $\sim 10^{-6}$, but it increases with radius, potentially causing shape deformation for larger bubbles (where the low-Re assumption breaks down).

5. Significance and Applications

• Theoretical Advancement: The paper resolves the ambiguity in defining slip at liquid-liquid interfaces by introducing the concept of dual, sign-coupled slip lengths. It bridges the gap between macroscopic hydrodynamics and microscopic molecular kinetic theory.
• Practical Applications:
  • Emulsions: The generalized HRE is particularly relevant for oil-water and water-alcohol emulsions (hydrophobic-hydrophilic interfaces), which are critical in the oil industry (enhanced oil recovery) and medicine (drug delivery systems).
  • Aerosol Physics: The improved accuracy in predicting terminal velocities of micro-droplets aids in understanding atmospheric processes, cloud formation, and pollution dispersion.
• Experimental Validation: The paper proposes a specific experimental protocol (swapping the internal and external fluids) to verify the relationship between the two slip lengths and the viscosity ratio, offering a method to distinguish partial slip effects from surfactant contamination (a common source of error in such experiments).

In conclusion, Lebedev-Stepanov provides a rigorous analytical framework that unifies tangential and normal slip phenomena, offering a more accurate description of micro-scale fluid dynamics in multiphase systems, particularly where gas-liquid interactions and hydrophobic interfaces are involved.