Landau–Levich problem

A thin-film fluid problem determining the coating deposited on a plate withdrawn slowly from a liquid bath through the balance of viscous, capillary and gravitational effects.

Version

v2 · 2026-09-19 · History

Domain-specific #

5256

Domain group

Natural Sciences

Origin domain

Physics

Subdomain

Fluid Dynamics → Physics

Core Idea

The Landau–Levich problem predicts the entrained film connecting a dynamic meniscus to a uniform coating. Lubrication flow and curvature pressure are matched between static and dynamic meniscus regions, yielding the characteristic capillary-number scaling of film thickness. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of fluid dynamics. It is A thin-film fluid problem determining the coating deposited on a plate withdrawn slowly from a liquid bath through the balance of viscous, capillary and gravitational effects.

How would you explain it like I'm…

The Honey Spoon Puzzle

When you pull a spoon slowly out of honey, a thin coat of honey stays stuck to it. The Landau–Levich problem figures out how thick that coat will be. It depends on how fast you pull, how sticky and gooey the liquid is, and how much the liquid's skin likes to pull itself flat.

How Thick Is the Coat?

Imagine dipping a flat plate into a bath of liquid and pulling it out slowly. A thin film of liquid comes up with the plate and coats it evenly. The Landau–Levich problem asks how thick that coating will be. The answer depends on a tug-of-war between the liquid's stickiness (viscosity), its surface tension, which tries to keep the surface smooth and curved, and gravity pulling it back down. It only works this way if you pull slowly and the liquid wets the plate.

Film Thickness from Slow Withdrawal

The Landau–Levich problem predicts the thickness of the liquid film left on a plate that is withdrawn slowly from a bath. Near the bath, the liquid surface forms a curved, nearly still meniscus; higher up, liquid is being dragged along in a moving region that connects to a flat, uniform film. The answer comes from balancing viscous drag, which carries liquid up, against surface tension and gravity, which push it back. Solving the flow in the thin moving region and matching its curvature to the static meniscus gives a scaling law: film thickness depends on the capillary number, a ratio of viscous to surface-tension effects. This is a specific coating calculation, not fluid dynamics in general, and its law only holds when the withdrawal is slow, the film is thin, and the liquid wets the plate.

 

The Landau–Levich problem is the classic thin-film problem of determining the coating deposited on a plate withdrawn slowly from a liquid bath. The domain splits into a static meniscus, governed by a hydrostatic balance of gravity and capillary pressure, and a dynamic meniscus where the plate drags fluid upward and lubrication theory applies. In the dynamic region, viscous stresses balance the curvature (Laplace) pressure gradient, and the profile must asymptote to a flat entrained film of uniform thickness. Matching the curvature of the dynamic region to that of the static meniscus fixes the film thickness, giving the characteristic power-law scaling with the capillary number (viscosity times speed over surface tension). The result is only as good as its assumptions: slow withdrawal (small capillary number), a thin film, complete wetting, and a valid asymptotic matching. A careful analysis states these assumptions and parameters explicitly, checks limiting cases, and avoids transferring the scaling to regimes, such as fast withdrawal or partial wetting, where it is not justified.

Scope of Application

Landau–Levich problem belongs to fluid dynamics and is useful where the analyst can specify a plate withdrawn at speed, Newtonian liquid bath, viscosity, density, surface tension, gravity, meniscus regions, capillary number and far-field film thickness, then evaluate the slow-withdrawal, thin-film, wetting and matching assumptions are explicit before using the classical scaling law. The scope is broad within that domain but bounded by the need for the slow-withdrawal, thin-film, wetting and matching assumptions are explicit before using the classical scaling law. Conceptual fluid-mechanics identity, not an industrial operating protocol.

Clarity

The abstraction clarifies a crowded vocabulary by making the slow-withdrawal, thin-film, wetting and matching assumptions are explicit before using the classical scaling law the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Landau–Levich problem can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Landau–Levich problem. Landau–Levich problem compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: a plate withdrawn at speed, Newtonian liquid bath, viscosity, density, surface tension, gravity, meniscus regions, capillary number and far-field film thickness. Reject examples whose alleged carrier belongs to a different problem.
2. Lock the constitutive rule. Express the slow-withdrawal, thin-film, wetting and matching assumptions are explicit before using the classical scaling law independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of fluid dynamics because they reuse a plate withdrawn at speed, Newtonian liquid bath, viscosity, density, surface tension, gravity, meniscus regions, capillary number and far-field film thickness, Lubrication flow and curvature pressure are matched between static and dynamic meniscus regions, yielding the characteristic capillary-number scaling of film thickness., and type the carrier, state every parameter and convention in the definition, test that the slow-withdrawal, thin-film, wetting and matching assumptions are explicit before using the classical scaling law, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Landau–Levich problem Domain-specific

Parents (1) — outgoing typed relations

• Landau–Levich problem is a kind of Equilibrium Prime

  The candidate literally instantiates prime:equilibrium.

Hierarchy path (1) — routes to 1 parentless root

• Landau–Levich problem → is a kind of Equilibrium → is a kind of, typical Fixed Point

Neighborhood in Abstraction Space

Landau–Levich problem sits in a moderately populated region (54^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Fluid Flow & Transport Phenomena (31 abstractions)

Nearest neighbors

• Marangoni number — 0.87
• Stefan adhesion — 0.86
• Chaotic mixing — 0.85
• Frenkel–Kontorova model — 0.85
• Immersed boundary method — 0.85

Computed from structural-signature embeddings · 2026-10-25