Ericksen Number¶
A dimensionless liquid-crystal flow parameter comparing a declared viscous scale with orientational elasticity.
Core Idea¶
The Ericksen number compares a flow-driven viscous scale with the orientational elasticity of a liquid crystal.[1][2][3] One common form is \(\mathrm{Er}=\eta U L/K\): viscosity coefficient \(\eta\), speed \(U\), characteristic length \(L\) and elastic constant \(K\) must all be defined for the same model.[1][2][3] The units cancel, giving a dimensionless quantity.[1][2][3]
The symbols are not universal. Two experimental studies use rotational viscosity, while a separate confined-flow study uses a declared apparent dynamic viscosity.[1][2][3] A higher Er says the chosen viscous scale is larger relative to the chosen elastic scale; it does not by itself identify a director texture or a universal transition threshold.
Scope of Application¶
The ratio is used in liquid-crystal flow studies in which an orientational director field interacts with motion.[1][2] Larson and Mead apply it to a polymeric nematic solution sheared between glass plates.[1] Vásquez-Montoya and colleagues apply it to the small-molecule nematic 5CB in a pressure-driven microchannel.[2] Both compare viscous flow and orientational elasticity, but their coefficients, lengths, geometry and observed states differ.[1][2]
The number is one control among several. The polymeric experiment also invokes Deborah number at high velocities; the 5CB study describes acoustic forcing with a separate streaming Reynolds number.[1][2]
Clarity¶
Before comparing two reported values, ask which viscosity, speed, length and elastic constant each author chose. In the polymeric plate-shear case the speed is a shear velocity and the length is the plate gap.[1] In the 5CB channel case the speed is measured mean flow velocity and the length is hydraulic diameter.[2] A numerical match using different conventions is not automatically a match in physical forcing.
“Small Er” and “large Er” describe the stated ratio. A low-Er particle-drag approximation can treat the director as little changed by the induced flow, but that conclusion has its own conditions.[4] The 5CB study's bowser, chiral and dowser ranges belong to its channel and should not be copied to every nematic flow.[2]
Manages Complexity¶
The number combines several dimensional inputs into a single comparison, making an experimental sweep easier to organize.[1][2] In the plate-shear study, changing speed and gap changes the reported Ericksen number; in the microchannel study, measured flow supplies an axis for comparing observed director configurations.[1][2]
Compression does not remove the remaining physics. Material coefficients, surface anchoring, geometry and independent forcing may still determine which state occurs. A single Er value is a useful coordinate, not a complete model of the liquid crystal.
Abstract Reasoning¶
First identify the ordered liquid crystal and the flow question. Choose the velocity and length that belong to that setup, and use a viscosity coefficient and orientational elastic constant from the same declared model.[1][2][3] Calculate the dimensionless quotient, then ask which response the experiment or theory actually connects to that value.[1][2][3]
At fixed other inputs, increasing the selected speed raises Er. Changing the characteristic length or the coefficient convention also changes Er, but may change what the number means. To infer a director state, check the other controls instead of treating the quotient as a universal rule.
Knowledge Transfer¶
The comparison transfers between polymeric plate shear and 5CB channel flow because each supplies a director-bearing medium, a justified viscous coefficient, velocity, length and orientational elastic reference.[1][2] Their numerical thresholds and particular textures do not transfer automatically. Outside orientational liquid-crystal flow, the general idea of dividing competing scales may be useful, but that does not make a different quotient the Ericksen number.
Examples¶
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Polymeric plate shear. Larson and Mead shear a poly(gamma-benzyl-glutamate) nematic solution between glass surfaces.[1] They use a plate velocity, plate gap, rotational viscosity and Frank splay elastic constant in their Ericksen-number formula.[1] Mapped back: the polymeric nematic supplies orientation; plate motion and gap supply the flow scales; rotational viscosity and splay elasticity supply the compared coefficients.[1] Their publisher abstract, rather than the full paper, supplies this bounded example.[1]
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5CB channel flow. Vásquez-Montoya and colleagues use mean velocity, hydraulic diameter, rotational viscosity and a single elastic constant for 5CB under pressure-driven microchannel flow.[2] Mapped back: 5CB supplies orientation; channel flow and hydraulic diameter supply the motion scales; the rotational and elastic coefficients supply the comparison.[2] Their observed flow states belong to that apparatus, with acoustic forcing tracked separately.[2]
Relationships to Other Abstractions¶
Current abstraction Ericksen Number Domain-specific
Parents (2) — outgoing typed relations
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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.
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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
- Ericksen Number → is a kind of Dimensionless Quantity → is a kind of Physical quantity → is a kind of Measurement
- Ericksen Number → is a kind of Ratio → is a kind of Comparison
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
- Korteweg Stress — 0.81
- Ellipsoid packing — 0.81
- Laminar flow — 0.80
- Hartmann Number — 0.80
- Free Surface — 0.80
Computed from structural-signature embeddings · 2026-10-25
Not to Be Confused With¶
- Reynolds number: a different nondimensional control considered separately in liquid-crystal-flow analysis.[5]
- One fixed viscosity: the two examples use rotational coefficients, while another published Ericksen-number calculation uses apparent dynamic viscosity.[1][2][3]
- A universal texture threshold: a number from one material and geometry does not establish a transition in every liquid crystal.[1][2]
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
[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
[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
[4] 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 ↩
[5] 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 ↩