Third-Grade Fluids: Constitutive Modeling

• Third-grade fluids are advanced non-Newtonian fluids characterized by constitutive laws incorporating terms up to third-order in kinematic or thermodynamic gradients.
• They are modeled via multiple frameworks, including Rivlin–Ericksen polynomial expansions, weakly nonlocal Korteweg-type formulations, and relativistic third-order gradient expansions.
• These models yield improved predictions for shear flow behavior, stability analyses, and optimal control, offering practical insights for computational fluid dynamics.

Third-grade fluids are non-Newtonian fluids whose constitutive description goes beyond the Newtonian relation between stress and the instantaneous rate of strain, but the exact meaning of “third-grade” depends on the theoretical setting. In the incompressible Rivlin–Ericksen framework, third-grade fluids are characterized by constitutive laws containing terms up to third order in the Rivlin–Ericksen kinematic tensors; in weakly nonlocal Korteweg-type theories, grade $3$ refers to dependence on second spatial gradients of fields such as density; and in relativistic hydrodynamics, the analogous notion is a constitutive expansion carried out to third order in spacetime gradients (Nouira et al., 2023, Gorgone et al., 2021, Diles et al., 2023).

1. Terminology and constitutive hierarchy

The literature represented here uses “third-grade” in several distinct but related senses. In the sense of Rivlin–Ericksen fluids of differential type, the constitutive law is written as a polynomial in the kinematic tensors $A_1,A_2,A_3,\dots$, with third-grade models retaining terms up to cubic order in these tensors. In weakly nonlocal thermodynamics, especially for Korteweg-type fluids, a material is of grade $N$ if its constitutive quantities depend on spatial gradients up to order $N$, and the state space may therefore include $\nabla\rho$, $\nabla^2\rho$, $\nabla\varepsilon$, and related variables. In relativistic hydrodynamics, the corresponding usage is “third-order hydrodynamics,” meaning a constitutive gradient expansion through $\mathcal{O}(\partial^3)$ (Gouin, 2018, Gorgone et al., 2021, Diles et al., 2023).

A pedagogical bridge between Newtonian constitutive laws and higher-grade models is provided by the analysis of simple shear and Stokes drag in non-Newtonian fluids. In that setting, Newton’s law of viscosity is written as $F=\eta A\,dv/dy$, or $\tau=\eta\dot\gamma$, and departures from linear drag at low Reynolds number motivate constitutive refinements in which stress depends on more than a constant viscosity. The supplied account explicitly connects this progression to higher-grade descriptions involving the first and second Rivlin–Ericksen tensors, such as

$A_1,A_2,A_3,\dots$0

and to constitutive expansions containing higher powers of $A_1,A_2,A_3,\dots$1 (Dounas-Frazer et al., 2012).

Context	Meaning of “third-grade”	Representative dependence
Rivlin–Ericksen incompressible fluids	Terms up to third order in kinematic tensors	$A_1,A_2,A_3,\dots$2, and cubic terms in $A_1,A_2,A_3,\dots$3
Weakly nonlocal Korteweg fluids	Constitutive dependence on second spatial gradients	$A_1,A_2,A_3,\dots$4
Relativistic hydrodynamics	Third-order gradient expansion	$A_1,A_2,A_3,\dots$5

A further terminological point is explicit in multi-gradient fluid theory: “perfect fluids of grade $A_1,A_2,A_3,\dots$6” are defined by an internal energy depending on $A_1,A_2,A_3,\dots$7, $A_1,A_2,A_3,\dots$8, and their gradients up to order $A_1,A_2,A_3,\dots$9,

$N$0

Accordingly, a model with gradients up to third order corresponds to $N$1 in that framework (Gouin, 2018).

2. Rivlin–Ericksen third-grade fluids of differential type

For incompressible fluids of differential type, the first Rivlin–Ericksen tensor is

$N$2

and higher tensors are defined recursively by

$N$3

A Newtonian fluid has stress

$N$4

whereas a third-grade fluid is modeled by

$N$5

with $N$6 the pressure, $N$7 the viscosity, $N$8, $N$9, and $N$0 (Coulaud, 2014).

In bounded-domain incompressible formulations, the same structure appears after enforcing incompressibility and eliminating pressure gradients from the constitutive representation. A representative deterministic third-grade equation is

$N$1

with $N$2, supplemented by $N$3 and homogeneous Dirichlet data (Nouira et al., 2023). A characteristic feature is the modified inertia operator $N$4, which makes the $N$5-metric intrinsic to the dynamics.

Thermodynamic and physical consistency impose algebraic restrictions on the material parameters. One set of conditions used in the stochastic Dirichlet theory is

$N$6

together with the sharper monotonicity condition

$N$7

In the Navier-boundary stochastic theory, the corresponding admissibility conditions are

$N$8

(Nouira et al., 2023, Tahraoui et al., 2023).

The two-dimensional whole-space vorticity reduction is especially important. Writing $N$9, one obtains a closed vorticity equation in which the third-grade term generates nonlinear diffusive contributions such as

$\nabla\rho$0

while certain $\nabla\rho$1-dependent terms vanish from the vorticity dynamics because they are gradients in two dimensions (Coulaud, 2014).

3. Weakly nonlocal and Korteweg-type third-grade fluids

In weakly nonlocal thermodynamics, third-grade Korteweg fluids are compressible fluids with internal capillarity, modeled by constitutive dependence on density gradients up to second order. A representative state space is

$\nabla\rho$2

and the constitutive mappings $\nabla\rho$3 depend on $\nabla\rho$4 (Gorgone et al., 2021).

For third-grade viscous Korteweg fluids, the stress ansatz takes the form

$\nabla\rho$5

while the heat flux is assumed linear in $\nabla\rho$6 and $\nabla\rho$7,

$\nabla\rho$8

The extended Liu procedure then reduces the entropy density to

$\nabla\rho$9

with $\nabla^2\rho$0, and determines the capillarity coefficients by

$\nabla^2\rho$1

$\nabla^2\rho$2

while $\nabla^2\rho$3 enforce nonnegative viscous dissipation (Gorgone et al., 2021).

A distinctive outcome is the nonclassical entropy flux

$\nabla^2\rho$4

where $\nabla^2\rho$5. The extra term is required for thermodynamic admissibility when $\nabla^2\rho$6 depends on $\nabla^2\rho$7 and $\nabla^2\rho$8. Under the same entropy constraints, the heat flux reduces to a Fourier law in $\nabla^2\rho$9,

$\nabla\varepsilon$0

(Gorgone et al., 2021).

The binary-mixture extension enlarges the state space to

$\nabla\varepsilon$1

with concentration $\nabla\varepsilon$2, diffusional mass flux

$\nabla\varepsilon$3

and entropy expansion

$\nabla\varepsilon$4

The corresponding entropy flux contains classical and nonlocal pieces,

$\nabla\varepsilon$5

showing that higher-grade mixture theories require explicit entropy transport beyond $\nabla\varepsilon$6 (Gorgone et al., 2021).

The equilibrium theory of two-dimensional third-grade Korteweg fluids makes this constitutive structure concrete. Under constant temperature and zero velocity, the mechanical equilibrium equations reduce to a single nonlinear elliptic equation,

$\nabla\varepsilon$7

provided Serrin’s compatibility condition is satisfied. For power-law choices $\nabla\varepsilon$8, $\nabla\varepsilon$9, this becomes the dimensionless nonlinear elliptic PDE

$\mathcal{O}(\partial^3)$0

on a rectangle with Dirichlet data, and the paper reports preliminary numerical solutions obtained by finite differences and MATLAB fsolve (Gorgone et al., 28 May 2025).

4. Mathematical well-posedness, asymptotics, and stability

The analytical theory of third-grade fluids is strongly conditioned by dimension, boundary conditions, and the notion of solution. In the deterministic two-dimensional whole-space problem, introducing self-similar variables converts the vorticity equation to a perturbation of the Ornstein–Uhlenbeck operator

$\mathcal{O}(\partial^3)$1

with the third-grade corrections multiplied by $\mathcal{O}(\partial^3)$2 and $\mathcal{O}(\partial^3)$3. For small data in $\mathcal{O}(\partial^3)$4, the solution admits the first-order asymptotic expansion

$\mathcal{O}(\partial^3)$5

and

$\mathcal{O}(\partial^3)$6

In physical variables, the vorticity converges to the self-similar heat kernel $\mathcal{O}(\partial^3)$7, and the velocity converges to the corresponding Oseen profile (Coulaud, 2014).

For stochastic equations with Navier slip boundary conditions in a two-dimensional non-axisymmetric bounded domain, strong well-posedness is available at $\mathcal{O}(\partial^3)$8-level. With initial data $\mathcal{O}(\partial^3)$9, $F=\eta A\,dv/dy$0, multiplicative white noise satisfying

$F=\eta A\,dv/dy$1

there exists a unique strong solution

$F=\eta A\,dv/dy$2

to the stochastic third-grade equation (Cipriano et al., 2021).

Higher regularity and pathwise local well-posedness have also been established on bounded domains with Navier boundary conditions. For $F=\eta A\,dv/dy$3, nonlinear multiplicative Wiener noise, and initial data in $F=\eta A\,dv/dy$4, there exists a unique maximal local strong pathwise solution

$F=\eta A\,dv/dy$5

obtained by a cut-off approximation, stochastic compactness, and a Yamada–Watanabe–Engelbert theorem (Tahraoui et al., 2023).

At lower regularity, the Dirichlet problem admits global martingale solutions in both two and three dimensions. On a bounded, simply connected domain $F=\eta A\,dv/dy$6, $F=\eta A\,dv/dy$7, with homogeneous Dirichlet boundary conditions, multiplicative Wiener noise, and initial data in

$F=\eta A\,dv/dy$8

the stochastic third-grade system possesses a martingale solution

$F=\eta A\,dv/dy$9

constructed by Galerkin approximation, stochastic compactness, and a Minty–Browder identification of the monotone operator

$\tau=\eta\dot\gamma$0

Under exponentially decaying forcing and diffusion and a large-viscosity condition,

$\tau=\eta\dot\gamma$1

the paper proves mean-square exponential stability,

$\tau=\eta\dot\gamma$2

and almost-sure exponential stability,

$\tau=\eta\dot\gamma$3

(Nouira et al., 2023).

5. Optimal control and stochastic forcing

Optimal control theory for third-grade fluids has developed in parallel with the well-posedness theory. In the deterministic two-dimensional Navier-slip setting, the control acts as a distributed body force $\tau=\eta\dot\gamma$4, the admissible set is

$\tau=\eta\dot\gamma$5

and the tracking functional is

$\tau=\eta\dot\gamma$6

The control-to-state map is Gâteaux differentiable, its derivative is the solution of the linearized third-grade equation, and the adjoint equation yields the variational inequality

$\tau=\eta\dot\gamma$7

The coupled optimality system is unique under the condition

$\tau=\eta\dot\gamma$8

(Tahraoui et al., 2022).

The multiplicative-noise control problem retains the same tracking structure but is necessarily local in time because strong solutions are available only up to a stopping time. For a predictable distributed random force $\tau=\eta\dot\gamma$9, the cost is

$A_1,A_2,A_3,\dots$00

where

$A_1,A_2,A_3,\dots$01

For each fixed $A_1,A_2,A_3,\dots$02, there exists a unique minimizer $A_1,A_2,A_3,\dots$03; the linearized forward equation coincides with the Gâteaux derivative of the control-to-state mapping; and the backward adjoint equation yields, in two dimensions, the first-order condition

$A_1,A_2,A_3,\dots$04

In three dimensions, the adjoint equation is proved only in a weaker sense, and uniqueness is not established (Tahraoui et al., 2023).

A global-in-time stochastic control theory is available on the two-dimensional torus for additive infinite-dimensional white noise. The controlled third-grade equation is posed on $A_1,A_2,A_3,\dots$05, the control belongs to a compact convex set

$A_1,A_2,A_3,\dots$06

and the cost functional is

$A_1,A_2,A_3,\dots$07

An infinite-dimensional Ornstein–Uhlenbeck process is used to rewrite the SPDE as a pathwise deterministic random PDE, leading to global well-posedness in

$A_1,A_2,A_3,\dots$08

The linearized state and adjoint equations are uniquely solvable, the control-to-state map is Gâteaux differentiable, and the optimal control satisfies

$A_1,A_2,A_3,\dots$09

(Kinra et al., 17 Jun 2025).

6. Relativistic third-order hydrodynamics and broader generalizations

In relativistic hydrodynamics, the third-grade analogue is a constitutive expansion of the stress–energy tensor and conserved currents through third order in spacetime gradients. For a neutral fluid in curved spacetime, the constitutive relation

$A_1,A_2,A_3,\dots$10

is organized as a gradient expansion in $A_1,A_2,A_3,\dots$11, $A_1,A_2,A_3,\dots$12 or $A_1,A_2,A_3,\dots$13, and the metric. The third-order classification in four dimensions yields $A_1,A_2,A_3,\dots$14 new transport coefficient candidates in the conformal case and $A_1,A_2,A_3,\dots$15 in the non-conformal case, before any entropy-current constraints are imposed. Those coefficients contribute to third-order corrections to the linear dispersion relations for diffusion and sound, to two-point functions of the stress tensor, and to the Bjorken-flow energy density (Grozdanov et al., 2015).

The charged relativistic theory in a general hydrodynamic frame is structurally richer. Using the Irreducible-Structure algorithm, the constitutive relations are expanded in scalars, transverse vectors, and transverse symmetric traceless tensors built from

$A_1,A_2,A_3,\dots$16

For a non-conformal charged fluid, the paper finds $A_1,A_2,A_3,\dots$17 first-order, $A_1,A_2,A_3,\dots$18 second-order, and $A_1,A_2,A_3,\dots$19 third-order transport coefficients in a general frame; after frame fixing and constraints, the counts become $A_1,A_2,A_3,\dots$20 at first order, $A_1,A_2,A_3,\dots$21 at second order, and $A_1,A_2,A_3,\dots$22 at third order. Linearized frame-invariant combinations determine the dispersion relations of shear, sound, and diffusive modes (Diles et al., 2023).

Multi-gradient fluid theory provides a further generalization in which the internal energy depends on density, volumetric entropy, and their gradients up to order $A_1,A_2,A_3,\dots$23,

$A_1,A_2,A_3,\dots$24

The corresponding generalized chemical potential and temperature are

$A_1,A_2,A_3,\dots$25

$A_1,A_2,A_3,\dots$26

and the conservative equation of motion retains the thermodynamic form

$A_1,A_2,A_3,\dots$27

This framework extends perfect compressible fluids, second-gradient capillary models, and higher-grade descriptions within a single variational structure (Gouin, 2018).

Across these formulations, third-grade fluid models share a common constitutive theme: stress depends on higher-order kinematic or thermodynamic structure rather than only on the instantaneous linear rate of deformation. The precise realization varies—from Rivlin–Ericksen tensors, to Korteweg density gradients, to relativistic derivative expansions—but in each case the point of third-grade modeling is to retain nonlinear, nonlocal, or higher-order contributions that are absent from Newtonian theory yet remain compatible with continuum mechanics and thermodynamics.