Skip to content

Ericksen Number

A dimensionless liquid-crystal flow parameter comparing a declared viscous scale with orientational elasticity.

Version
v1 · 2026-10-25 · History
Domain-specific #
14387
Origin domain
Liquid Crystal Physics
Subdomains
Nematic Hydrodynamics, Dimensional Analysis → Liquid Crystal Physics

Core Idea

The Ericksen number compares a viscous scale imposed by flow with an orientational elastic scale in a liquid crystal.[1][2][3] In a common convention, \(\mathrm{Er}=\eta U L/K\): \(U\) is a declared flow speed, \(L\) a declared length, \(\eta\) a model-appropriate viscosity coefficient, and \(K\) an orientational elastic constant.[1][2][3] The dimensions of these four quantities cancel, so \(\mathrm{Er}\) is dimensionless.[1][2][3] The symbol \(\eta\) does not identify a universal material coefficient.[1][2][3] Larson and Mead use a rotational viscosity and a Frank splay constant for a sheared polymeric nematic; Vásquez-Montoya and colleagues use a rotational viscosity and a single elastic constant for 5CB in a microchannel.[1][2] A separate confined-flow study defines its coefficient as an apparent dynamic viscosity and uses a bulk elastic constant.[3]

The number is a balance coordinate, not a complete prediction of director texture. A smaller value means the chosen viscous scale is smaller relative to the chosen elastic one; a larger value reverses that comparison. To infer a flow state, one must also specify the liquid crystal, geometry, boundary anchoring, the coefficient convention and any other active controls.[1][2][3] The two experiments cited here exhibit different responses, and one explicitly tracks Deborah number at high shear while the other tracks acoustic streaming separately.[1][2]

Structural Signature

Sig role-phrases: orientationally ordered medium — declared speed and length — model-appropriate viscous coefficient — orientational elastic constant — bounded state interpretation.

  • Orientationally ordered medium. The ratio is posed for a liquid crystal whose director or related order responds to flow. Without orientational order, the same arithmetic could produce a different dimensionless flow parameter, but not this director-flow comparison.[1][2]

  • Declared speed and length. The speed and geometric scale make the modeled deformation concrete. Larson and Mead choose plate-shear velocity \(V\) and gap \(d\); the 5CB microchannel study chooses measured mean velocity \(u\) and hydraulic diameter \(l\).[1][2] Replacing one length by another without changing the physical model alters the reported value.[1][2]

  • Model-appropriate viscous coefficient. Both worked experiments use rotational viscosity, written γ₁ or γ.[1][2] Lesniewska and colleagues instead use an apparent dynamic viscosity \(\eta\) in their confined-flow model.[3] Reporting a value without the coefficient's definition can conceal a change of convention.[1][2][3]

  • Orientational elastic constant. Larson and Mead use Frank splay K₁; the 5CB study uses a single elastic constant K; the confined particle-flow calculation uses a bulk elastic constant κ.[1][2][3] An unspecified elastic modulus multiplied by area is not a faithful replacement for these stated coefficients.[1][2][3]

  • Bounded state interpretation. The ratio organizes a comparison of flow forcing and elastic response. Anchoring, material response, geometry and additional forcing remain free to change the observed director field. The number's arithmetic can be calculated without a state model; a state claim cannot be justified from the number alone.[1][2][3]

What It Is Not

Ericksen number is not Reynolds number: they are distinct nondimensional controls in a liquid-crystal-flow model, and the Ericksen parameter here concerns viscous flow relative to orientational elasticity.[4]

It is not a universal order-one transition rule. Even when a small value supports an approximation that flow weakly disturbs a director field, the conclusion belongs to a particular model. Stark and Ventzki use such a low-Er approximation for particle drag; the 5CB channel study observes bowser, chiral and dowser states over its own parameter ranges.[5][2] Neither establishes a threshold for every liquid crystal or channel.[5][2]

It is not necessarily a rotational-viscosity formula. That coefficient occurs in the two primary examples below, but a published confined-flow model uses a declared apparent dynamic viscosity.[1][2][3] Conversely, substituting ordinary shear viscosity into either rotational-viscosity example without its constitutive justification does not reproduce the source's Ericksen number.[1][2][3]

Scope of Application

In sheared polymeric nematics, an Ericksen number organizes how imposed plate motion competes with the orientational elasticity of a liquid-crystalline solution.[1] Larson and Mead vary their plate gap and shear speed and report a sequence of textures across their explored range.[1] Their high-velocity interpretation also invokes Deborah number, which concerns a different time-scale comparison.[1]

In pressure-driven microfluidic nematics, the selected channel length and mean speed make the ratio a coordinate for flow-induced director states.[2] Vásquez-Montoya and colleagues examine 5CB first without acoustic forcing, then combine pressure-driven flow with surface acoustic waves.[2] They assign the latter forcing its own streaming Reynolds number, so a changed acoustic response cannot be credited to Ericksen number alone.[2]

In theoretical and numerical flow studies, the chosen viscosity and elastic coefficients can differ. Calderer and Liu analyze a model with distinct Ericksen, Reynolds and interface numbers; Lesniewska and colleagues define an apparent dynamic viscosity and a bulk elastic constant for a particle in a confined liquid-crystal flow.[4][3] The shared identity is the declared viscous-to-orientational-elastic scaling, not one universal list of numerical parameter values.[4][3]

Clarity

Writing \(\mathrm{Er}=\eta U L/K\) is compact, but the letters should be expanded before values are compared. Ask whether \(\eta\) is rotational viscosity, an apparent dynamic viscosity or another justified effective coefficient; whether \(U\) is plate, mean or centerline speed; which physical dimension supplies \(L\); and which elastic mode or model supplies \(K\).[1][2][3] The two primary experiments use the same four functional slots but different concrete assignments.[1][2]

The parameter also separates a force-scale comparison from a state label. In the 5CB study, “bowser,” “chiral” and “dowser” denote observed director configurations in that channel.[2] Ericksen number helps organize their occurrence, but their numeric boundaries belong to that apparatus and its anchoring and forcing conditions.[2]

Manages Complexity

The ratio compresses several dimensional inputs into one coordinate, allowing a controlled sweep to be ordered by relative flow and orientational-elastic scales. Larson and Mead can compare plate-shear conditions with different gaps and speeds; the 5CB study can place microchannel observations along a measured-flow axis.[1][2] This reduces a multivariable problem to a first comparison without claiming that all its variables have disappeared.[1][2]

Compression hides the choice of coefficients and other controls. In the polymeric experiment, Deborah number matters at its fastest shear; in the acoustically driven channel, streaming Reynolds number separately parameterizes wave forcing.[1][2] A one-number plot is therefore useful for organizing observations, while a prediction of a particular texture still requires the relevant material, geometry and boundary conditions.[1][2]

Abstract Reasoning

To use the abstraction, first identify the director-bearing medium and the flow being compared. Choose \(U\), \(L\), a viscosity coefficient and an orientational elastic constant from the same model, then verify that the resulting quotient is dimensionless.[1][2][3] A larger \(U\) at fixed other declared inputs raises Er and strengthens the chosen viscous scale relative to the elastic reference; changing \(L\) or a material coefficient may do the same without producing an equivalent experiment.[1][2][3]

Then state the intended inference separately: “flow is expected to disturb this director configuration more strongly under these conditions” is a bounded hypothesis; “Er alone guarantees this texture” is not. Compare data at like conventions, check what else varied, and use an independent control when a second physical process is present. The 5CB experiment's acoustic streaming measure and the polymeric study's Deborah-number qualification illustrate that extra step.[1][2]

Knowledge Transfer

The ratio transfers literally between unlike liquid-crystal flow settings when each setting supplies the director-bearing medium, a justified viscous coefficient, velocity, length and orientational elasticity.[1][2] A polymeric solution between moving plates and small-molecule 5CB in a pressure-driven channel fill those roles differently.[1][2] The formula's structure travels, while fitted coefficients and state boundaries do not automatically travel with it.[1][2]

Outside liquid-crystal or related orientational-flow models, saying that “viscous influence competes with elasticity” may be an analogy or may motivate another nondimensional group. It is not enough to import the Ericksen number name. The generic operation of forming a dimensionless ratio is broader than this domain-bound parameter.

Examples

Polymeric plate shear. Larson and Mead shear a poly(gamma-benzyl-glutamate) nematic solution between glass surfaces.[1] They use \(E\approx Vd\gamma_1/K_1\), with plate-shear velocity \(V\), gap \(d\), rotational viscosity \(\gamma_1\) and Frank splay constant \(K_1\).[1] Their publisher abstract reports director rotation and developing textures across a broad experimental sweep, and warns that Deborah number also matters at the highest velocities.[1] The available support here is the publisher abstract, not a claim to have checked every result in the full paper.[1] Mapped back: medium = polymeric nematic solution; speed and length = \(V,d\); viscous coefficient = \(\gamma_1\); elastic constant = \(K_1\); interpretation = that experiment's texture response, with an additional time-scale control.[1]

5CB pressure-driven microchannel. Vásquez-Montoya and colleagues flow the small-molecule nematic 5CB through a microchannel.[2] With acoustic forcing switched off, they define \(\mathrm{Er}=\gamma u l/K\), where \(u\) is measured mean velocity, \(l\) the hydraulic diameter, \(\gamma\) rotational viscosity and \(K\) a single elastic constant.[2] They observe bowser, chiral and dowser configurations over ranges of Er specified for their device; when acoustic forcing is introduced, they use a separate streaming Reynolds number.[2] Mapped back: medium = 5CB; speed and length = \(u,l\); viscous coefficient = \(\gamma\); elastic constant = \(K\); interpretation = channel-specific director states with an independently measured acoustic axis.[2]

The examples differ in material and flow geometry while instantiating the same declared viscous-to-orientational-elastic comparison. Their observed textures and thresholds are not interchangeable.[1][2]

Structural Tensions

T1: Scalar economy versus a complete director-flow account. One Er value compresses several inputs and makes a controlled sweep legible. The cost is that anchoring, geometry, relaxation and other forcing may change the state at the same Er. Adding every relevant parameter improves predictive specificity but loses the simplicity of a one-coordinate screen. Diagnostic: Which untracked control could reverse the state inference at this Er in this apparatus?[1][2]

T2: Shared formula versus coefficient fidelity. A symbolic \(\eta U L/K\) invites comparison across studies. But treating \(\eta\) or \(K\) as interchangeable erases distinctions between rotational and apparent dynamic viscosities and among elastic conventions.[1][2][3] Insisting on all coefficient definitions makes comparisons slower, but prevents false numerical equivalence. Diagnostic: Do the two reports normalize the same physical mechanism with compatible coefficients and lengths?[1][2][3]

Structural–Framed Character

The parameter has a structural physical core: a dimensionless quotient of a flow-viscous scale and an orientational-elastic scale. Its evaluative weight is neutral; high Er is not intrinsically good or bad. Its human-practice dependence lies in selecting a model, coefficients and geometry before interpreting a reported number. It has no institutional rule that creates the ratio, although reporting conventions and experimental protocols determine how values can be compared. Its vocabulary travels across liquid-crystal flow experiments, but the same name does not license importing one experiment's state labels or cutoffs into another. The ratio is recognized by its source-defined physical roles, not by a metaphorical resemblance to competing influences.[1][2][3]

Its character: a reusable orientational-flow comparison whose formula is compact and whose empirical meaning is model-bound.

Structural Core vs. Domain Accent

The core is the declared ratio of viscous flow scale to orientational elasticity in a director-bearing medium. The accents are constitutive choices and experimental conditions: polymeric versus small-molecule nematic, rotational versus apparent dynamic viscosity, splay versus single or bulk elasticity, gap versus hydraulic diameter, and plate shear versus pressure-driven confinement.[1][2][3] Those choices determine the measured value and the response that can be inferred from it.[1][2][3]

Removing the director and elastic referent leaves the broader idea of a dimensionless physical ratio. That broader form may be useful elsewhere, but it is no longer the Ericksen number's domain-specific identity. Two independently reviewed strict genera are recorded: Dimensionless Quantity for dimension-one physical quantity and Ratio for the ordered nonzero-denominator quotient. Other dimensionless groups are siblings with other physical differentiae, not alternate names for this one.

This entry is a kind of Dimensionless Quantity and is a kind of Ratio.

Ericksen number is a kind of Dimensionless Quantity: a physical quantity of dimension one, distinguished by its liquid-crystal flow and orientational-elastic roles. It is also a strict instance of Ratio: a declared viscous-flow numerator is divided by a nonzero orientational-elastic denominator. These parents are independent because dimensionless quantities can be formed without division and ratios can retain units. Neither parent alone predicts a director texture. Reynolds Number is a neighboring flow control and not an Ericksen synonym or parent.

Relationships to Other Abstractions

Local relationship map for Ericksen NumberParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Ericksen NumberDOMAINDomain-specific abstraction: Dimensionless Quantity — is a kind ofDimensionlessQuantityDOMAINPrime abstraction: Ratio — is a kind ofRatioPRIME

Current abstraction Ericksen Number Domain-specific

Parents (2) — outgoing typed relations

  • Ericksen Number is a kind of Dimensionless Quantity Domain-specific

    Ericksen number is a dimension-one physical quantity distinguished by a declared director-flow viscous-to-orientational-elastic comparison.

  • Ericksen Number is a kind of Ratio Prime

    Ericksen number is an ordered quotient of a declared viscous flow scale by a nonzero orientational-elastic reference scale.

Hierarchy paths (2) — routes to 2 parentless roots

Neighborhood in Abstraction Space

Ericksen Number sits in a sparse region of the domain-specific corpus (87th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

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

Not to Be Confused With

  • Reynolds number: a different nondimensional flow comparison, separately tracked in liquid-crystal-flow analysis.[4]
  • Deborah number: a time-scale comparison additionally relevant at high velocities in Larson and Mead's polymeric shear study.[1]
  • A universal \(\mathrm{Er}=1\) texture transition: no such all-material conclusion follows from the cited experiments.[1][2]
  • A fixed viscosity convention: rotational coefficients are used in the two worked examples; an apparent dynamic coefficient occurs in another published model.[1][2][3]
  • The 2002 Stark–Ventzki EPL paper as a citation for the 2001 PRE low-Er approximation: these are distinct works. The 2002 frozen-source reference is routed in the research packet and is not used as support for a 2001 claim.[5]

References

[1] Ronald G. Larson and Donald W. Mead, “The Ericksen number and Deborah number cascades in sheared polymeric nematics,” Liquid Crystals 15, no. 2 (1993): 151–169, https://doi.org/10.1080/02678299308031947. Original publisher abstract and metadata consulted for the formula, polymeric plate-shear case and reported cascade; full article was not available for this audit. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w ↩x ↩y ↩z ↩27 ↩28 ↩29 ↩30 ↩31 ↩32 ↩33 ↩34 ↩35 ↩36 ↩37 ↩38 ↩39 ↩40 ↩41 ↩42 ↩43 ↩44 ↩45 ↩46 ↩47

[2] Vásquez-Montoya et al., “Control of liquid crystals combining surface acoustic waves, nematic flows, and microfluidic confinement,” Soft Matter 20 (2024): 397–406, https://doi.org/10.1039/D3SM01443F. Original publisher article, §3.2 and Figure 3, accessed through indexed section text; the direct publisher HTML request was restricted. Supports the 5CB formula and bounded channel-state/acoustic-control claims. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w ↩x ↩y ↩z ↩27 ↩28 ↩29 ↩30 ↩31 ↩32 ↩33 ↩34 ↩35 ↩36 ↩37 ↩38 ↩39 ↩40 ↩41 ↩42 ↩43 ↩44 ↩45 ↩46 ↩47 ↩48 ↩49

[3] Magdalena Lesniewska, Nigel Mottram and Oliver Henrich, “Defect-influenced particle advection in highly confined liquid crystal flows,” Soft Matter 20 (2024): 2218–2231, https://doi.org/10.1039/D3SM01297B. Original author PDF, §3.2, printed p. 2221 (PDF fifth page, zero-index page 4), formula and parameter definitions for apparent dynamic viscosity, characteristic centerline speed and narrowest gap, and bulk elastic constant, https://eprints.gla.ac.uk/334853/1/334853.pdf. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w ↩x

[4] M. Carme Calderer and Chun Liu, “Liquid Crystal Flow: Dynamic and Static Configurations,” SIAM Journal on Applied Mathematics 60, no. 6 (2000): 1925–1949, https://doi.org/10.1137/S0036139998336249. Original publisher abstract consulted for its three distinct nondimensional controls and model-specific large-Er configurations; full article was not consulted for scientific claims here. Author order follows the original article's printed first page and the Penn State institutional record, https://pure.psu.edu/en/publications/liquid-crystal-flow-dynamic-and-static-configurations; SIAM's web listing reverses the displayed order. registry ↩a ↩b ↩c ↩d

[5] H. Stark and D. Ventzki, “Stokes drag of spherical particles in a nematic environment at low Ericksen numbers,” Physical Review E 64 (2001): 031711, https://doi.org/10.1103/PhysRevE.64.031711. Original paper abstract indexed at https://pubmed.ncbi.nlm.nih.gov/11580359/; supports only the stated low-Er approximation in that particle-drag setting. Distinct from the frozen 2002 Europhysics Letters paper, DOI 10.1209/epl/i2002-00541-0. registry ↩a ↩b ↩c