Thick Compressible Fluid Models

• Thick compressible fluids are non-Newtonian flows characterized by a maximal admissible shear rate, providing a singular, constrained model.
• They are derived as the asymptotic limit of compressible power-law fluids, enforcing |D(u)| ≤ 1 via a Lagrange multiplier in the stress.
• The framework extends to stochastic and viscoelastic regimes, employing variational methods and measure-valued solutions to address complex flow behavior.

Searching arXiv for papers on thick compressible fluids and closely related compressible non-Newtonian/viscoelastic models. Thick compressible fluids are compressible non-Newtonian flows associated with shear-rate–dependent viscosity, and in a particularly sharp asymptotic formulation they are compressible flows with a hard cap on the admissible shear rate. In the deterministic singular-limit setting, the cap arises by sending the power-law exponent $p\to\infty$ in compressible power-law systems, producing a constrained model with $|D(u)|\le 1$ and a Lagrange multiplier in the stress. In a broader stochastic setting, the same topic includes barotropic compressible flows with an implicit convex constitutive law for the extra stress, treated through measure-valued solutions because strong compactness is generally unavailable (Bresch et al., 21 Jul 2025, Ludvík et al., 27 Apr 2025).

1. Definition and conceptual scope

In the asymptotic sense developed by Bresch, Burtea, and Szlenk, a thick compressible fluid is the limit of a compressible power-law fluid when the power-law exponent becomes infinite. The resulting model enforces a maximal admissible shear rate,

$|D(u)|\le \dot\gamma_{\max},$

usually scaled so that $\dot\gamma_{\max}=1$, and introduces a nonnegative Lagrange multiplier $\pi(x,t)$ satisfying

$$|D(u)|\le1,\qquad \pi\ge0,\qquad \pi\,(1-|D(u)|)=0,$$

with stress

$\tau=\pi\,D(u).$

Within this formulation, the defining feature is not merely large viscosity but a unilateral constraint on the symmetric velocity gradient $D(u)$ (Bresch et al., 21 Jul 2025).

A broader constitutive usage appears in the stochastic analysis of Ludvík and Macha, where the governing system is a generalized compressible Navier–Stokes model for shear-rate–dependent fluids. There the extra stress is described implicitly by a convex potential $F$, and the theory covers compressible, shear-rate-dependent “thick or thin” fluids with stochastic forcing. This suggests that the phrase “thick compressible fluids” is used at two scales: as a specific singular constrained model, and as part of a wider class of compressible non-Newtonian fluids with shear-dependent rheology (Ludvík et al., 27 Apr 2025).

2. Governing equations and constitutive structure

The power-law compressible system used as the starting point for the singular asymptotic is

$\partial_t \rho_p+\nabla\!\cdot(\rho_p u_p)=0,$

$|D(u)|\le 1$0

with barotropic pressure $|D(u)|\le 1$1, $|D(u)|\le 1$2, and

$|D(u)|\le 1$3

A prototypical interpretation is $|D(u)|\le 1$4, so that as $|D(u)|\le 1$5 the viscosity diverges whenever $|D(u)|\le 1$6. The limit therefore replaces a very steep constitutive law by a constrained one (Bresch et al., 21 Jul 2025).

In the stochastic barotropic framework, the equations are written as

$|D(u)|\le 1$7

where $|D(u)|\le 1$8 is a cylindrical Wiener process and

$|D(u)|\le 1$9

A prototypical constitutive potential is

$|D(u)|\le \dot\gamma_{\max},$0

More generally, $|D(u)|\le \dot\gamma_{\max},$1 may grow according to an $|D(u)|\le \dot\gamma_{\max},$2-function $|D(u)|\le \dot\gamma_{\max},$3, with

$|D(u)|\le \dot\gamma_{\max},$4

and the conjugacy relation

$|D(u)|\le \dot\gamma_{\max},$5

The pressure potential is

$|D(u)|\le \dot\gamma_{\max},$6

This constitutive structure places thick compressible fluids within the Orlicz and convex-analysis setting rather than exclusively within polynomial-growth Sobolev theory (Ludvík et al., 27 Apr 2025).

3. Singular asymptotic from power-law rheology

The asymptotic regime $|D(u)|\le \dot\gamma_{\max},$7 is the central derivation of the constrained thick-fluid model. Uniform energy estimates in $|D(u)|\le \dot\gamma_{\max},$8 show that $|D(u)|\le \dot\gamma_{\max},$9 remains bounded and that $\dot\gamma_{\max}=1$0 for any $\dot\gamma_{\max}=1$1. In the one-dimensional analysis, uniform $\dot\gamma_{\max}=1$2-bounds on the density are obtained by a maximum-principle argument applied to the Cauchy stress

$\dot\gamma_{\max}=1$3

The hyperbolic–parabolic coupling between density and velocity is a principal obstacle, and the proof uses an adaptation of Hoff’s method together with a favorable transport–Gronwall inequality for a modified quantity $\dot\gamma_{\max}=1$4 (Bresch et al., 21 Jul 2025).

A key structural step is a variational or Minty-monotonicity inequality: for any admissible test field $\dot\gamma_{\max}=1$5 with $\dot\gamma_{\max}=1$6,

$\dot\gamma_{\max}=1$7

As $\dot\gamma_{\max}=1$8, the middle term blows up unless $\dot\gamma_{\max}=1$9, and the limit becomes a variational inequality characterizing the constrained flow. The limiting stress tensor $\pi(x,t)$0 is identified through

$\pi(x,t)$1

which yields

$\pi(x,t)$2

The constraint is therefore not imposed ad hoc; it is the limit of a monotone constitutive sequence (Bresch et al., 21 Jul 2025).

The main deterministic results split into two settings. In one dimension, for periodic data with $\pi(x,t)$3, $\pi(x,t)$4, and $\pi(x,t)$5, there exists a global weak solution satisfying

$\pi(x,t)$6

Along a subsequence,

$\pi(x,t)$7

and the limit satisfies

$\pi(x,t)$8

In the semi-stationary multidimensional Stokes system, one freezes inertia and obtains a limit with $\pi(x,t)$9 almost everywhere together with a variational inequality involving the indicator functional

$$|D(u)|\le1,\qquad \pi\ge0,\qquad \pi\,(1-|D(u)|)=0,$$0

A one-dimensional singular-stress extension leads to the same limiting constraint (Bresch et al., 21 Jul 2025).

4. Solution concepts and analytical frameworks

For stochastic compressible non-Newtonian flows, strong convergence of the nonlinear terms is generally unavailable, so Ludvík and Macha introduce a probabilistic Young measure

$$|D(u)|\le1,\qquad \pi\ge0,\qquad \pi\,(1-|D(u)|)=0,$$1

encoding oscillation and concentration of $$|D(u)|\le1,\qquad \pi\ge0,\qquad \pi\,(1-|D(u)|)=0,$$2. The averaged fields are

$$|D(u)|\le1,\qquad \pi\ge0,\qquad \pi\,(1-|D(u)|)=0,$$3

and two defect measures $$|D(u)|\le1,\qquad \pi\ge0,\qquad \pi\,(1-|D(u)|)=0,$$4 capture concentrations in the convective flux and pressure. The measure-valued triple $$|D(u)|\le1,\qquad \pi\ge0,\qquad \pi\,(1-|D(u)|)=0,$$5 is required to satisfy the weak continuity equation, the weak momentum equation with defect contributions, a martingale structure, and an energy–dissipation inequality with a nonnegative defect energy $$|D(u)|\le1,\qquad \pi\ge0,\qquad \pi\,(1-|D(u)|)=0,$$6. The existence proof combines Galerkin approximation, stochastic energy estimates, Jakubowski–Skorokhod compactness, de la Vallée–Poussin, Dunford–Pettis, and weak lower semicontinuity. The final existence theorem yields a Young measure solution on a $$|D(u)|\le1,\qquad \pi\ge0,\qquad \pi\,(1-|D(u)|)=0,$$7 bounded domain, with stress and strain in the Orlicz spaces dictated by $$|D(u)|\le1,\qquad \pi\ge0,\qquad \pi\,(1-|D(u)|)=0,$$8 and $$|D(u)|\le1,\qquad \pi\ge0,\qquad \pi\,(1-|D(u)|)=0,$$9, and uniform moment bounds of arbitrary order $\tau=\pi\,D(u).$0 (Ludvík et al., 27 Apr 2025).

Closely related analytical machinery appears in the theory of compressible viscoelastic rate-type fluids with stress diffusion. Bulíček, Feireisl, and Málek study

$\tau=\pi\,D(u).$1

$\tau=\pi\,D(u).$2

$\tau=\pi\,D(u).$3

with

$\tau=\pi\,D(u).$4

Their energy inequality is

$\tau=\pi\,D(u).$5

and the existence proof hinges on a modified effective viscous flux identity that controls the extra stress contributions without changing the classical threshold $\tau=\pi\,D(u).$6. The result is global-in-time weak solvability for any finite-energy initial data with $\tau=\pi\,D(u).$7 (Bulíček et al., 2018).

A stronger, local theory is available near equilibrium for three-dimensional compressible viscoelastic fluids with deformation gradient $\tau=\pi\,D(u).$8. Hu and Wang consider

$\tau=\pi\,D(u).$9

$D(u)$0

$D(u)$1

with $D(u)$2, $D(u)$3, and equilibrium $D(u)$4. For small initial data in $D(u)$5, they obtain local existence and uniqueness of a strong solution and uniform estimates on density, velocity, and deformation gradient. A central device is the “good velocity”

$D(u)$6

which exposes dissipation of the elastic component and enables $D(u)$7-$D(u)$8 maximal-regularity estimates (Hu et al., 2010).

5. Relation to neighboring compressible regimes

A nearby but distinct regime is the semi-compressible fluid of Roubíček. There the fluid is a slightly compressible visco-elastic “liquid” for which density changes are everywhere small, for example $D(u)$9 with $F$0, but not strictly zero; instead, density variations are slaved to pressure through a large but finite bulk modulus $F$1. The Eulerian model uses the quasi-continuity law

$F$2

and the momentum balance

$F$3

with

$F$4

The model is designed to comply with energy conservation while capturing dispersive pressure-wave propagation (Roubíček, 2020).

Linearization yields the one-dimensional dispersion relation

$F$5

and hence

$F$6

The exact energy balance includes kinetic energy, volumetric-elastic energy, viscous dissipation, the $F$7 term, and mechanical work of gravity and boundary pressure. Classical quasi-incompressible variants are criticized for replacing the material derivative by $F$8, neglecting the quadratic internal-energy pressure $F$9, or omitting the thermodynamically required $\partial_t \rho_p+\nabla\!\cdot(\rho_p u_p)=0,$0 term. In the limit $\partial_t \rho_p+\nabla\!\cdot(\rho_p u_p)=0,$1, $\partial_t \rho_p+\nabla\!\cdot(\rho_p u_p)=0,$2, the model recovers incompressible Navier–Stokes equations (Roubíček, 2020).

This neighboring theory is not a theory of thick compressible fluids in the constrained-shear sense. It addresses small density variations and pressure-wave propagation rather than a maximal admissible shear rate. The comparison is nevertheless useful because it separates two distinct departures from incompressibility: one based on finite bulk modulus and dispersive pressure dynamics, and one based on asymptotic shear-rate saturation.

6. Physical interpretation, applications, and misconceptions

The physical motivation emphasized in the constrained theory is high-pressure squeezing flow. In deep-ocean lubrication or polymer compression molding, pressures may become so large that the flow compresses and develops very high local shear rates. The $\partial_t \rho_p+\nabla\!\cdot(\rho_p u_p)=0,$3 limit then yields a model in which the fluid behaves Newtonian up to a maximal strain rate, beyond which it becomes effectively solid-like, providing a macroscopic description of discontinuous shear-thickening under compression rigorously linked to power-law rheologies (Bresch et al., 21 Jul 2025).

A recurrent misconception is to treat thick compressible fluids as merely compressible versions of standard incompressible shear-thickening models. The deterministic asymptotic results show that compressibility changes the problem qualitatively because the mass equation and momentum equation are coupled in a mixed hyperbolic–parabolic way, and the density must be controlled uniformly, in one dimension through a maximum-principle argument on the Cauchy stress. Another misconception is terminological: in (Bresch et al., 21 Jul 2025), the phrase denotes a singular constrained model with $\partial_t \rho_p+\nabla\!\cdot(\rho_p u_p)=0,$4, whereas in (Ludvík et al., 27 Apr 2025) the analysis covers the broader class of compressible, shear-rate-dependent “thick or thin” fluids under stochastic forcing. This suggests that the term should be interpreted from context rather than assumed to have a single canonical meaning.

The current research picture is therefore stratified. At one level, there is a rigorous asymptotic derivation of constrained thick compressible-fluid models from compressible power-law systems, with one-dimensional evolutionary and multidimensional semi-stationary results. At another, there is a general existence theory for stochastic barotropic compressible flows with nonlinear stress laws based on convex potentials and Young measures. Around these lie adjacent theories of compressible viscoelasticity, stress diffusion, and semi-compressibility, which supply complementary techniques—effective viscous flux identities, maximal regularity, defect-measure formulations, and energy-based compactness—for understanding compressible non-Newtonian flow beyond the Newtonian regime (Ludvík et al., 27 Apr 2025).