Euler–Vlasov Model for Thick Sprays

Updated 10 September 2025

The Euler–Vlasov model for thick sprays is a rigorous kinetic–fluid framework that captures the dynamics of dense dispersed phases with significant volume fractions.
It incorporates gyroscopic forces and volume-exclusion effects to modify drag and pressure interactions beyond classical dilute approximations.
The model underpins applications in engine atomization and aerosol dispersion, with its well-posedness and stability supported by rigorous mathematical analyses.

The Euler–Vlasov model for thick sprays is a class of coupled kinetic–fluid systems describing the evolution of a dense dispersed phase—such as liquid drops or solid particles—immersed in a carrier fluid. The distinctive feature of the thick spray regime is that the volume fraction occupied by the dispersed phase is significant, fundamentally modifying the momentum, mass, and energy dynamics of both phases and requiring kinetic–fluid coupling beyond classical dilute approximations. These models have rigorous mathematical foundations and engineering relevance in combustion, aerosol, multiphase flows, and environmental dispersion.

1. Mathematical Formulation and Coupling

The Euler–Vlasov model for thick sprays is built from a set of partial differential equations that couple a kinetic equation for the dispersed phase (Vlasov-type) to Euler-type fluid equations for the carrier @@@@1@@@@. The standard incompressible form in two dimensions (Moussa et al., 2011) reads:

Fluid vorticity equation:

$\partial_t \omega + \operatorname{div}_x(\omega u) = 0$

Kinetic (Vlasov-type) equation:

$\partial_t f + \operatorname{div}_x(f) + \operatorname{div}_\theta\left[ f\, (\theta - u)^\perp \right] = 0$

with the fluid velocity field determined by a Biot–Savart integral involving both fluid vorticity and integrated particle density:

$u = K[\omega + p],\quad p(x) = \int_{\mathbb{R}^2} f(x,\theta)\, d\theta$

$K[g](x) = \int_{\mathbb{R}^2} \frac{(x-y)^\perp}{|x-y|^2} g(y)\, dy$

where $(\theta - u)^\perp$ denotes the rotation of the discrepancy vector between dispersed phase velocity and fluid velocity by 90°, introducing gyroscopic (lift) effects.

For compressible flows and finite volume fraction (Chen, 5 Sep 2025), the system generalizes to:

Gas mass and momentum:

$\partial_t (\alpha \rho) + \nabla_x \cdot (\alpha \rho u) = 0$

$\partial_t (\alpha \rho u) + \nabla_x \cdot (\alpha \rho u \otimes u) + \nabla_x p = m_g \int F(v) D(v-u)\, dv + \text{corrections}$

Particle phase (Vlasov):

$\partial_t F + v \cdot \nabla_x F + \nabla_v\left[ F \left( D(v-u) + \frac{4\pi}{3} a^3 \nabla_x p \right) \right] = (\text{remainder})$

with the void function $\alpha(x,t) = 1 - \frac{4\pi}{3} a^3 \int F(x,v)\, dv$ reflecting the volume excluded by the particles.

2. Derivation from Kinetic Theory

Modern derivations employ rigorous multiscale analysis starting from coupled Boltzmann or Enskog–Boltzmann equations for binary mixtures of particles and gas (Bernard et al., 2016, Chen, 5 Sep 2025). The derivation requires:

Proper nondimensionalization of lengths, times, and velocities,
Selection of scaling regimes such that particle mass and thermal velocities are comparable to the gas (a departure from dilute limits),
Expansion of collision operators to capture finite-volume corrections, i.e., terms $\mathcal{O}(a^3)$ due to particle radii.

The limiting process yields closure relations that couple the fluid and kinetic phases, including:

Modified drag forces,
Pressure–gradient forces accounting for the volume fraction,
Higher-order (viscous-like) corrections.

The mathematical passage ensures that the coarse-grained (mean-field) spray dynamics retain both momentum and volume exclusion effects essential for the thick regime.

3. Gyroscopic and Non-Dissipative Forces

Unlike classic drag-coupled models, the Euler–Vlasov formulation includes gyroscopic interactions, particularly in the incompressible case (Moussa et al., 2011). The term $\operatorname{div}_\theta\left[ f\, (\theta - u)^\perp \right]$ models the lift force acting orthogonally to the relative velocity, analogous to the Kutta–Joukowski force in aerodynamics.

Such coupling captures spray phenomena where the inertial and rotational characteristics of particles cannot be neglected. The spray not only experiences altered forces but also acts as a vorticity source for the carrier fluid, fundamentally modifying the structure and evolution of vortices.

4. Volume Fraction and Higher-Order Corrections

In dense sprays, the void function $\alpha(x,t)$ (or equivalently, the local gas volume fraction $\theta_f = 1 - \theta_p$ ) plays a pivotal role (Chen, 5 Sep 2025, Pakseresht et al., 2019, Pakseresht et al., 2020):

Fluid equations become variable-density, even in the zero-Mach limit;
Corrections to drag forces involve pressure gradients and tensorial contributions of order $\mathcal{O}(a^3)$ ;
Continuity and momentum equations reflect the loss of available fluid volume, modifying pressure fields and introducing non-divergence-free velocity.

Such terms enhance the mean and r.m.s. velocities of the gas phase where the particle volume fraction is high, as validated by DNS and LES studies in particle-laden jets; for volume loadings above 5%, these effects are significant and observable experimentally (Pakseresht et al., 2019, Pakseresht et al., 2020).

5. Analytical Properties: Well-Posedness, Stability, and Energy Conservation

Well-posedness is established under broad conditions for both weak and regular solutions (Moussa et al., 2011, Yao et al., 2016, Gamba et al., 2018):

Existence of global weak solutions via regularization and compactness;
Uniqueness in regimes of bounded vorticity and density (Yudovich-type);
Picard–Banach fixed-point arguments when using mollified kernels;
Stability estimates in Wasserstein metrics quantify the robustness of solutions to initial perturbations.

The system possesses a Hamiltonian structure:

$\frac{d}{dt} F = \{ F, H \}$

with a Poisson bracket defined on the phase space of $(\omega, f)$ (Moussa et al., 2011). The Hamiltonian incorporates both kinetic and potential (interaction) energies, ensuring conservation laws and geometric (symplectic) properties that can guide analytic and numerical studies.

6. Limiting Regimes and Singular Phenomena

Important limiting cases and behaviors include:

Massless particle limit: As particle inertia vanishes, their velocity relaxes to that of the fluid, rendering $f$ monokinetic and recovering classical Euler equations for the fluid (Moussa et al., 2011).
Finite-time blow-up: Under suitable initial data and coupling forms, classical solutions to Euler–Vlasov or Navier–Stokes–Vlasov equations may develop singularities in finite time (Choi, 2016). The energy–inertia functional grows unbounded, marking breakdowns manifesting as droplet clustering, shocks, or abrupt concentration changes.
Landau damping and instability: Coupling through pressure gradients induces a Landau-like damping and amplification mechanism, with the sign of the derivative of the distribution function at the sound speed governing whether acoustic waves are damped or amplified (Bian et al., 4 May 2025). Positive $f_0'(c)$ yields ill-posedness due to explosive growth of high-frequency modes, invalidating Sobolev stability and implying potential turbulence or noise amplification.

7. Practical Significance and Applications

The Euler–Vlasov thick spray model underpins simulations and analysis in:

Engine fuel injection and atomization;
Environmental aerosol dispersion;
Chemical process engineering;
Climate modeling with sea spray–air interaction (Veron et al., 2019).

Accurately modeling volumetric displacement, drag/pressure corrections, and nontrivial fluid–particle interactions is essential for capturing dense spray phenomena and their impact on macroscopic observables (e.g., pressure, mixing, heat/mass transfer).

Numerical methods employing Euler–Lagrange coupling, moment closure, or reduced kinetic approaches require careful incorporation of volume fraction effects, particularly in high-loading regimes above 5–10%, to avoid underprediction of velocity and turbulence levels (Pakseresht et al., 2019, Pakseresht et al., 2020).

Summary Table: Core Components of Euler–Vlasov Thick Spray Models

Mathematical Feature	Description	Reference
Kinetic–Fluid Coupling	Vlasov eq. + Euler/Navier-Stokes eqs., drag/lift, volume displacement	(Moussa et al., 2011, Chen, 5 Sep 2025)
Volume Fraction Effects	Modified continuity, pressure correction, drag enhancement	(Pakseresht et al., 2019, Pakseresht et al., 2020)
Gyroscopic Force	$(\theta - u)^\perp$ in kinetic equation	(Moussa et al., 2011)
Well-Posedness	Existence, uniqueness, stability, Hamiltonian structure	(Moussa et al., 2011, Yao et al., 2016)
Finite-Time Blow-Up	Analytical breakdown of classical solutions	(Choi, 2016)
Landau Damping	Damping/amplification of sound waves via kinetic coupling	(Bian et al., 4 May 2025)

In conclusion, the Euler–Vlasov model for thick sprays forms a mathematically rigorous and physically accurate framework for understanding and simulating high-volume loading dispersed phase flows. Its capacity to derive mesoscale equations from kinetic theory, capture nonclassical force interactions, and encompass complex limiting phenomena marks it as foundational in modern spray modeling and multiphase fluid mechanics.