Beris–Edwards Model: Nematic Hydrodynamics

• Beris–Edwards model is a hydrodynamic Q-tensor theory for nematic liquid crystals that couples Navier–Stokes dynamics to the evolution of the orientational order parameter.
• The model supports both co-rotational (ξ = 0) and full alignment (ξ ∈ ℝ) regimes, influencing eigenvalue preservation and the characteristics of flow-induced stress feedback.
• Advanced analytical and numerical studies reveal its versatility in applications ranging from free-energy formulations to asymptotic limits and surface-confined dynamic flows.

The Beris–Edwards model is a hydrodynamic $Q$-tensor theory for nematic liquid crystals in which incompressible Navier–Stokes flow is coupled to an evolution equation for a symmetric traceless order parameter $Q$. In the formulations treated across recent analytical, numerical, and asymptotic studies, $Q$ encodes orientational order, the flow both transports and rotates or aligns $Q$, and elastic and orientational stresses feed back into the momentum balance. The model is used on Euclidean domains, thin strips, half-spaces, and curved surfaces, and it supports both full alignment dynamics with parameter $\xi\in\mathbb R$ and the co-rotational specialization $\xi=0$ (Weber et al., 2023, Wu et al., 2017).

1. Continuum formulation and kinematics

In a standard incompressible $Q$-tensor formulation, the unknowns are the fluid velocity $\mathbf u$, the pressure $p$, and the tensor order parameter $\mathbf Q\in \mathcal S_0^d$, where

$Q$0

The model couples Navier–Stokes dynamics to a tensorial transport–reaction–diffusion equation,

$Q$1

with strain-rate and vorticity tensors

$Q$2

and alignment/corotation term

$Q$3

The elastic stress is commonly written in the equivalent form

$Q$4

while the molecular field is the variational derivative of the Landau–de Gennes free energy (Weber et al., 2023).

This formulation makes the coupling bidirectional. The flow transports $Q$5 through $Q$6, rotates and stretches it through $Q$7, and receives feedback through $Q$8 and $Q$9. In the co-rotational case $Q$0, the stretching term reduces to pure commutator form,

$Q$1

so the symmetric part $Q$2 no longer enters the $Q$3-kinematics. This restricted model is the setting for several strong-solution, invariant-region, and partial-regularity results (Abels et al., 2013, Du et al., 2019).

In uniaxial reductions one writes

$Q$4

with director $Q$5 and scalar order parameter $Q$6. That reduction is used both as a physical interpretation of $Q$7 and as a mathematical limiting or constrained regime in several later analyses (Abels et al., 2013, Feng et al., 2021).

2. Free energy, molecular field, and dissipation structure

A common free-energy choice is the Landau–de Gennes functional

$Q$8

with

$Q$9

where $Q$0. The molecular field then takes the form

$Q$1

Equivalently, defining

$Q$2

one has $Q$3 (Weber et al., 2023).

The model has a distinctive dissipative algebra. The fundamental identity

$Q$4

expresses the cancellation between the fluid stress and the tensorial stretching term. This is the mechanism behind the decrease of kinetic energy plus Landau–de Gennes energy in the continuous system and in structure-preserving discretizations (Weber et al., 2023). In the co-rotational setting, analogous commutator cancellations underlie global energy dissipation, local energy inequalities, and blow-up analysis (Du et al., 2019).

Beyond the one-constant elastic approximation, recent work treats general Landau–de Gennes elastic energies with four elastic constants,

$Q$5

or coercive modifications equivalent on the uniaxial manifold. These generalizations are mathematically significant because the $Q$6-term introduces noncoercive cubic elastic contributions for biaxial tensors, whereas in uniaxial settings the same energy can be reformulated in a coercive way (Feng et al., 2021, Liu et al., 2018).

The choice of bulk potential also changes the admissible-state mechanism. Polynomial Landau–de Gennes energies with $Q$7 provide quartic coercivity, while the Ball–Majumdar singular potential enforces physical eigenvalue constraints by assigning $Q$8 outside the admissible set. This distinction is analytically decisive in suitable weak-solution and physicality results (Du et al., 2019).

3. Co-rotational and non-co-rotational regimes

The parameter $Q$9 separates two qualitatively different kinematic regimes. When $\xi\in\mathbb R$0, the fluid acts on $\xi\in\mathbb R$1 only through rigid-body rotation, represented by $\xi\in\mathbb R$2 or $\xi\in\mathbb R$3. When $\xi\in\mathbb R$4, symmetric-gradient effects enter through

$\xi\in\mathbb R$5

so the flow can also align or stretch the orientational distribution (Wu et al., 2017, Murza et al., 2017).

This distinction has direct spectral consequences. For the co-rotational system, if the initial $\xi\in\mathbb R$6-tensor has eigenvalues in a prescribed interval, then those eigenvalues remain in the same interval during the evolution; a direct proof uses extremal eigenvalues, projection of the $\xi\in\mathbb R$7-equation onto eigenvectors, and the fact that the co-rotational commutator vanishes under that projection. The argument applies both in the whole space and in bounded domains with Dirichlet boundary data (Contreras et al., 2018). By contrast, in the full non-co-rotational model, eigenvalue-range preservation fails in general; in the two-dimensional periodic setting, the alignment/stretching terms can drive $\xi\in\mathbb R$8 outside the admissible interval even when the co-rotational system would preserve it (Wu et al., 2017).

Shear-flow reductions expose the same split dynamically. Under an imposed linear shear flow and spatial homogeneity, the Beris–Edwards PDE reduces to an ODE for $\xi\in\mathbb R$9. In the co-rotational case, after passing to a rotating frame $\xi=0$0 with $\xi=0$1, the reduced dynamics become the gradient flow

$\xi=0$2

Thus the nontrivial dissipation acts on the eigenvalues, while the eigenframe rotates periodically with the shear. In the non-co-rotational case, the paper identifies distinct short-time and long-time regimes through rescaling in $\xi=0$3: the large-$\xi=0$4 limit yields an integrable flow-dominated system, whereas the small-$\xi=0$5 regime approaches the stationary manifold of $\xi=0$6 (Murza et al., 2017).

The same separation governs rigorous model reduction. In dimension two, weak solutions of a Beris–Edwards system with stiff bulk penalization converge to weak solutions of the Ericksen–Leslie equations as the elastic coefficient tends to zero. The limit constrains $\xi=0$7 to the uniaxial manifold

$\xi=0$8

and yields explicit Leslie coefficients in terms of Beris–Edwards parameters. The limiting weak solutions may have singular points, so the result does not require smooth Ericksen–Leslie dynamics (Xin et al., 2021).

4. Existence, regularity, and solution frameworks

The analytical theory now spans weak, strong, suitable weak, and maximal-regularity settings. For the full three-dimensional incompressible Beris–Edwards system in $\xi=0$9, with the full alignment/tumbling parameter $Q$0 retained and the polynomial Landau–de Gennes bulk potential under the stable bulk assumption $Q$1, a global weak solution has been constructed in the natural energy class with

$Q$2

together with the distributional form of the equations and the expanded Leray–Hopf type energy inequality used in weak–strong uniqueness arguments. A key technical difficulty is the non-corotational term $Q$3, which prevents direct passage to the limit in the expanded inequality; the proof instead proceeds through a hyperviscous approximation, a localized tail estimate, and a low-order chain rule for the bulk energy (Zhang et al., 16 May 2026).

On bounded domains, strong solutions are known in several regimes. For a simplified co-rotational system with $Q$4-dependent viscosity $Q$5, homogeneous Dirichlet boundary conditions, and bounded $Q$6 domains in dimensions $Q$7 and $Q$8, local-in-time existence and uniqueness hold in the class

$Q$9

with the proof based on linearization, a variable-coefficient Stokes operator, a singular perturbation of the $\mathbf u$0-equation, and Banach’s fixed-point theorem (Abels et al., 2013). In a different direction, the Beris–Edwards system with general Landau–de Gennes energy depending on four non-zero elastic constants admits strong solutions for uniaxial $\mathbf u$1-tensors up to a maximal time, and strong biaxial solutions converge smoothly to the uniaxial solution up to that maximal existence time (Feng et al., 2021).

A Caffarelli–Kohn–Nirenberg type theory is available for the three-dimensional co-rotational model. Suitable weak solutions have been constructed globally for either the polynomial Landau–de Gennes bulk potential in $\mathbf u$2 or the Ball–Majumdar potential in $\mathbf u$3. These solutions satisfy a local energy inequality and are smooth away from a closed singular set $\mathbf u$4 with vanishing one-dimensional parabolic Hausdorff measure,

$\mathbf u$5

The proof uses retarded mollification, local energy estimates, pressure decomposition, blow-up analysis, and a linear coupled system arising in the blow-up limit (Du et al., 2019).

Several specialized settings are also well developed. In the half-space $\mathbf u$6, the linearized Beris–Edwards system has $\mathbf u$7-$\mathbf u$8 maximal regularity, derived from resolvent analysis, $\mathbf u$9-boundedness, and operator-valued Fourier multiplier theory; this yields local well-posedness for the nonlinear half-space model with small initial data (Barbera et al., 2024). In the two-dimensional periodic co-rotational case with general Landau–de Gennes elastic energy including the cubic $p$0-term, global existence and uniqueness of weak solutions hold provided the initial $p$1-norm of $p$2 is sufficiently small. The proof uses a specific approximating system preserving the $p$3-norm of $p$4, which keeps the noncoercive cubic elastic term perturbative (Liu et al., 2018).

5. Numerical schemes and asymptotic limits

Numerically, an important development is the convergence analysis of an IEQ-based first-order semi-discrete scheme. The Invariant Energy Quadratization method replaces the nonquadratic bulk potential by an auxiliary scalar variable

$p$5

leading to a linearly implicit scheme that is unconditionally energy-stable. The convergence proof shows not only stability of the semi-discrete approximation but also that the limit solves the original Beris–Edwards system, thereby establishing weak equivalence between the IEQ reformulation and the original model (Weber et al., 2023).

Thin-domain and hydrostatic limits reveal nontrivial anisotropic reorganizations. In dimension two, the scaled anisotropic co-rotational Beris–Edwards system in a thin strip admits global well-posedness for small analytic data, and as the thickness parameter tends to zero the solutions converge to a hydrostatic system. In that specific two-dimensional limit, the $p$6-tensor vanishes identically, so the effective dynamics reduce to a hydrostatic Navier–Stokes system around a shear profile (Li et al., 2022). In the three-dimensional thin-strip setting, the hydrostatic limit is more intricate: the fluid converges to a 3D Prandtl/hydrostatic system, while the $p$7-tensor does not satisfy a standard lower-dimensional Beris–Edwards equation. Instead, $p$8 satisfy transport–diffusion–reaction equations coupled to algebraic constraints, and $p$9 lose their own time dynamics and are reconstructed from explicit algebraic relations involving $\mathbf Q\in \mathcal S_0^d$0 and $\mathbf Q\in \mathcal S_0^d$1 (Anna et al., 2024).

Sharp-interface limits provide a complementary asymptotic regime. At critical temperature in a bounded three-dimensional domain, solutions of a diffuse-interface Beris–Edwards system converge, under well-prepared initial data and vanishing diffuse interfacial thickness, to a corresponding sharp interface model. The analysis uses the relative entropy method and elaborated energy estimates, and yields spatial decay estimates for the velocity field in the $\mathbf Q\in \mathcal S_0^d$2 sense together with error estimates for the phase field (Su, 2024).

These asymptotic results suggest that the Beris–Edwards model serves not only as a continuum description of nematohydrodynamics but also as a parent system for hydrostatic, director-based, and sharp-interface reductions. The precise limit depends sensitively on geometry, scaling, and whether the co-rotational or full alignment structure is retained.

6. Surface formulations, evolving geometry, and confined-flow dynamics

Surface versions of the Beris–Edwards model extend the theory from flat domains to curved manifolds. On a closed $\mathbf Q\in \mathcal S_0^d$3-dimensional hypersurface $\mathbf Q\in \mathcal S_0^d$4, $\mathbf Q\in \mathcal S_0^d$5, a thermodynamically consistent surface model couples tangent incompressible flow to a $\mathbf Q\in \mathcal S_0^d$6-tensor field through tangential differential operators and curvature-dependent corrections. The $\mathbf Q\in \mathcal S_0^d$7-kinematics includes the star-corotation term

$\mathbf Q\in \mathcal S_0^d$8

and the momentum equation contains the extra curvature force

$\mathbf Q\in \mathcal S_0^d$9

For such systems, global weak solutions exist on closed $Q$00 hypersurfaces, and the proof combines a Faedo–Galerkin scheme, spectral bases for tangent Stokes and tensor Laplace–Beltrami operators, and curvature-aware energy estimates (Benavides et al., 2 Jul 2026).

A broader geometric generalization concerns evolving surfaces. Using the Lagrange–D’Alembert principle, thermodynamically consistent surface Beris–Edwards models have been derived for viscous inextensible surface flow coupled to a Landau–de Gennes–Helfrich energy. These formulations distinguish a general model with three-dimensional surface $Q$01-tensor dynamics from a surface-conforming model with tangential anchoring, and they also distinguish material and Jaumann tensorial time derivatives. The total energy

$Q$02

satisfies the dissipation law

$Q$03

and optional constraints such as surface conformity, constant normal eigenvalue, uniaxiality, isotropy, no-normal-flow, and no-flow can be imposed by Lagrange multipliers (Nitschke et al., 2023).

Confined-channel applications show how scalar-order variability in the Beris–Edwards framework changes the dynamical landscape. In pressure-driven channel flows with strong homeotropic anchoring, the model supports multistability: a uniform nematic state coexists with states having spatially varying scalar order and director winding. Under forcing, these equilibria continue to Bowser and Dowser flow states; increasing the pressure gradient destabilizes them sequentially and produces periodic oscillations, defect-mediated transitions, and spatiotemporal chaos. The central distinction from Ericksen–Leslie theory is that the scalar order parameter $Q$04 is not fixed: modest spatial variation in $Q$05, including temporary collapse near topological transitions, organizes the observed nonequilibrium regimes (Valani et al., 18 Nov 2025).

Taken together, these developments show that the Beris–Edwards model is not a single PDE on a single domain class, but a family of closely related hydrodynamic $Q$06-tensor systems. Across whole-space analysis, bounded-domain strong theory, structure-preserving numerics, surface and thin-film reductions, and confined-flow applications, the defining features remain the Landau–de Gennes molecular field, flow-induced transport and corotation or alignment, and stress couplings compatible with a dissipative free-energy structure.