Stochastic Fluid Research Articles

Invariant Measures for a Class of Stochastic Third-Grade Fluid Equations in 2D and 3D Bounded Domains

Invariant Measures for a Class of Stochastic Third-Grade Fluid Equations in 2D and 3D Bounded Domains

Bridging scales in multiscale bubble growth dynamics with correlated fluctuations using neural operator learning

The intricate process of bubble growth dynamics involves a broad spectrum of physical phenomena from microscale mechanics of bubble formation to macroscale interplay between bubbles and surrounding thermo-hydrodynamics. Traditional bubble dynamics models including atomistic approaches and continuum-based methods segment the bubble dynamics into distinct scale-specific models. To bridge the gap between microscale stochastic fluid models and continuum-based fluid models for bubble dynamics, we develop a composite neural operator model to unify the analysis of nonlinear bubble dynamics across microscale and macroscale regimes by integrating a many-body dissipative particle dynamics (mDPD) model with a continuum-based Rayleigh–Plesset (RP) model through a novel neural network architecture, which consists of a deep operator network for learning the mean behavior of bubble growth subject to pressure variations and a long short-term memory network for learning the statistical features of correlated fluctuations in microscale bubble dynamics. Training and testing data are generated by conducting mDPD and RP simulations for nonlinear bubble dynamics with initial bubble radii ranging from 0.1 to 1.5 micrometers. The results show that the trained composite neural operator model can accurately predict bubble dynamics across scales, with a 99% predictive accuracy for the time evolution of the bubble radius under varying external pressure while containing correct size-dependent stochastic fluctuations in microscale bubble growth dynamics. The composite neural operator is the first deep learning surrogate for multiscale bubble growth dynamics that can capture correct stochastic fluctuations in microscopic fluid phenomena, which sets a new direction for future research in multiscale fluid dynamics modeling.

A stochastic fluid model approach to the stationary distribution of the maximal priority process

We consider a two-class priority queue in which the priority of a customer increases linearly at some constant, class-dependent rate. We describe the related maximum priority process { ( M 1 ( t ) , M 2 ( t ) ) : t ≥ 0 } , which is of interest since the stationary distribution of the maximum priority process at the times of the commencement of service gives information about the waiting time distributions of the two classes of customers. In the case where service times are exponential, we map a two-class maximum priority process { ( M 1 ( t ) , M 2 ( t ) ) : t ≥ 0 } to a tandem fluid queue, and use this mapping in order to derive the stationary distribution of { ( M 1 ( t ) , M 2 ( t ) ) : t ≥ 0 } . Further, we extended these results to the maximum priority process in which service times are phase-type distributed. We illustrate the theory with numerical examples.

Simulations of Stochastic Fluid Dynamics near a Critical Point in the Phase Diagram.

We present simulations of stochastic fluid dynamics in the vicinity of a critical endpoint belonging to the universality class of the Ising model. This study is motivated by the challenge of modeling the dynamics of critical fluctuations near a conjectured critical endpoint in the phase diagram of quantum chromodynamics (QCD). We focus on the interaction of shear modes with a conserved scalar density, which is known as model H. We show that the observed dynamical scaling behavior depends on the correlation length and the shear viscosity of the fluid. As the correlation length is increased or the viscosity is decreased we observe a crossover from the dynamical exponent of critical diffusion, z≃4, to the expected scaling exponent of model H, z≃3. We use our method to investigate the time-dependent correlation function of non-Gaussian moments M^{n}(t) of the order parameter. We find that the relaxation time depends in a nontrivial manner on the power n.

Lagrangian Perspectives on the Small-scale Structure of Alfvénic Turbulence and Stochastic Models for the Dispersion of Fluid Particles and Magnetic Field Lines in the Solar Wind

Lagrangian perspectives on the small-scale structure of anisotropic Alfvénic turbulence are adopted. We are interested in relating the statistical properties of the Eulerian field increments evaluated along the fluid particle trajectories, in the direction perpendicular to the guiding magnetic field and along the magnetic field lines. We establish the basis for a unified multifractal phenomenology of Eulerian and Lagrangian Alfvénic turbulence. The critical balance condition is generalized to structure functions of an order different than 2. A Lagrangian perspective is not only useful for investigating the small-scale structure of Alfvénic turbulence, it is also tailored to the modeling of large-scale turbulent transport. Therefore, we develop Lagrangian stochastic models for the dispersion of fluid particles and magnetic field lines in the solar wind. The transport models are based on the integrated Ornstein–Uhlenbeck process that is not Markov, yielding smooth stochastic fluid particle trajectories and magnetic field lines. Brownian diffusion is recovered by tending the integral scale parameter to zero while keeping the diffusivity finite.

Numerical analysis of finite element method for a stochastic active fluids model

In this paper, we first investigate the well-posedness and regularity of mild solution to a stochastic active fluids model driven by the additive noise. A fully-discrete scheme is proposed for solving the given model, which is based on the finite element method for spatial discretization and the backward Euler method for temporal discretization. By overcoming the difficulty of error analysis caused by the discrete Laplacian operator, we obtain the convergence results of the developed scheme. Finally, some numerical examples are provided to validate the theoretical results and we also simulate the motion states of the active fluids.

A geometric extension of the Itô-Wentzell and Kunita’s formulas

We extend the Itô-Wentzell formula for the evolution along a continuous semimartingale of a time-dependent stochastic field driven by a continuous semimartingale to tensor field-valued stochastic processes on manifolds. More concretely, we investigate how the pull-back (respectively, the push-forward) by a stochastic flow of diffeomorphisms of a time-dependent stochastic tensor field driven by a continuous semimartingale evolves with time, deriving it under suitable regularity conditions. We call this result the Kunita-Itô-Wentzell (KIW) formula for the advection of tensor-valued stochastic processes. Equations of this nature bear significance in stochastic fluid dynamics and well-posedness by noise problems, facilitating the development of certain geometric extensions within existing theories.

Innovative stochastic finite difference approach for modelling unsteady non-Newtonian mixed convective fluid flow with variable thermal conductivity and mass diffusivity

A novel stochastic numerical scheme is introduced to solve stochastic differential equations. The development of the scheme is based on two different parts. One part finds the solution for the deterministic equation, and the second part is the numerical approximation for the integral part of the Wiener process term in the stochastic partial differential equation. The scheme’s stability and consistency in the mean square sense are also ensured. Additionally, a respective mathematical model of the boundary layer flow of Casson fluid on a flat and oscillatory plate is formulated. Wiener process terms perturb the model to be studied. This scheme will be solved in contexts including deterministic and stochastic. The influence of different parameters on velocity, temperature, and concentration profiles is demonstrated in various graphical representations. The main objective of this study is to present a reliable numerical approach that surpasses the limitations of traditional numerical methods to analyze non-Newtonian mixed convective fluid flows with varying transport parameters. Our objective is to demonstrate the capabilities of the new stochastic finite difference scheme in enhancing our comprehension of stochastic fluid flow phenomena. This will be achieved by comprehensively examining its mathematical foundations and computer execution. Our objective is to develop a revolutionary method that will serve as a valuable resource for scientists and engineers studying the modeling and understanding of stochastic unstable non-Newtonian mixed convective fluid flow. This method will address the challenges posed by the fluid’s changing thermal conductivity and mass diffusivity.

Toward plasma drifts in EMC3: Implementation of gradient, divergence, and particle tracing schemes

AbstractThis paper presents a first implementation of gradient, divergence, and particle tracing schemes for the EMC3 code, a stochastic 3D plasma fluid code widely employed for edge plasma and impurity transport modeling in tokamaks and stellarators. These schemes are essential to accommodate plasma drift computations, which are currently absent in the code. Plasma drifts have been recognized to significantly influence transport of particles and energy, and their inclusion in future code upgrades will substantially enhance the code's predictive capabilities. For gradient and divergence calculations, we introduce a second‐order least‐squares gradient scheme. We confirm the second‐order convergence properties and assess the accuracy of several analytical test cases in the presence of synthetic noise. In the second part of this paper, we employ the validated gradient scheme in a fourth‐order Runge–Kutta particle tracing scheme to trace a particle through a generic drift velocity field. The impact of synthetic noise on the scheme's performance is investigated by evaluating various error metrics. We find that the implemented schemes function as intended and exhibit sufficient accuracy to enable drift computations.

A two‐stage reliable computational scheme for stochastic unsteady mixed convection flow of Casson nanofluid

AbstractResearchers can incorporate uncertainties in computational fluid dynamics (CFD) that go beyond the inaccuracies caused by numerical discretization thanks to stochastic simulations. This study confirms the validity of current stochastic modeling tools by providing examples of stochastic simulations in conjunction with numerical solutions for incompressible flows. A numerical technique for solving deterministic and stochastic models is developed in this work. Our approach employs the Euler‐Maruyama method for stochastic modeling, representing a stochastic version of the third‐order explicit‐implicit scheme. For the deterministic model, the scheme is third‐order accurate. The consistency and stability of the constructed scheme are provided in the mean square sense. The scheme is the predictor–corrector type that is built on two time levels. Moreover, a mathematical model of the Casson nanofluid flow with variable thermal conductivity is given with the effect of the chemical reaction. The appropriate transformations are used to condense the set of partial differential equations (PDEs) down to one that is dimensionless. The scheme is applied for the deterministic and stochastic models of dimensionless flow problems. The velocity profile's deterministic and stochastic behavior are shown using contour plots. Results show that growing values of the thermal mixed convection parameter enhance the velocity profile. This article presents the progress made in stochastic computational fluid dynamics (SCFD) and highlights the energy‐related aspects of our discoveries. Our computational approach and stochastic modeling techniques provide new insights into the energy properties of Casson nanofluid flow, specifically regarding the variability of thermal conductivity and chemical processes. Our objective is to clarify the complex interaction of these factors on energy dynamics. This article presents a contemporary summary of the latest SCFD advancements. Additionally, it highlights potential directions for future research and unresolved issues that require attention from the members of the field of computational mathematics.

Distribution-Path Dependent Nonlinear SPDEs with Application to Stochastic Transport Type Equations

By using a regularity approximation argument, the global existence and uniqueness are derived for a class of nonlinear SPDEs depending on both the whole history and the distribution under strong enough noise. As applications, the global existence and uniqueness are proved for distribution-path dependent stochastic transport type equations, which are arising from stochastic fluid mechanics with forces depending on the history and the environment. In particular, the distribution-path dependent stochastic Camassa-Holm equation with or without Coriolis effect has a unique global solution when the noise is strong enough, whereas for the deterministic model wave-breaking may occur. This indicates that the noise may prevent blow-up almost surely.

A new approach to stochastic relativistic fluid dynamics from information flow

We present a new general formalism for introducing thermal fluctuations in relativistic hydrodynamics, which incorporates recent developments on the causality and stability of relativistic hydrodynamic theories. Our approach is based on the information current, which measures the net amount of information carried by perturbations around equilibrium in a relativistic many-body system. The resulting noise correlators are guaranteed to be observer-independent for thermodynamically stable models. We obtain an effective action within our formalism and discuss its properties.

Well-posedness and invariant measures for 2D stochastic Oldroyd model of order one with pure jumps

In this paper, we consider the stochastic two-dimensional viscoelastic fluid flow equations driven by a multiplicative Lévy noise, arising from the Oldroyd model for the non-Newtonian fluid flows. The system carries the natural Lipschitz condition on balls and linear growth assumptions on the jump coefficient. We first prove the existence and uniqueness of a global strong solution by using stopping time technique and Banach fixed point theorem. Meanwhile, we overcome the difficulties posed by the jump and memory term, the latter makes some convergence findings more natural. By demonstrating the exponential stability of strong solutions, we establish the existence of a unique invariant measure for the stochastic system. Numerical results provide further support for these theoretical findings, the dissipative velocity of solution is more rapid and rigorous when compared with the stochastic Navier-Stokes equation.

Relativistic fluctuations in stochastic fluid dynamics

The state-of-the-art theoretical formalism for a covariant description of non-Gaussian fluctuation dynamics in relativistic fluids is discussed.

Well-Posedness of Solutions to Stochastic Fluid–Structure Interaction

Well-Posedness of Solutions to Stochastic Fluid–Structure Interaction

A field theory approach to the statistical kinematic dynamo

Variations in the geomagnetic field occur on a vast range of time scales, from milliseconds to millions of years. The advent of satellite measurements has allowed for detailed studies of short timescale geomagnetic field behaviour, but understanding its long timescale evolution remains challenging due to the sparsity of the paleomagnetic record. This paper introduces a field theory framework for studying magnetic field generation as a result of stochastic fluid motions. Starting from a stochastic kinematic dynamo model (the Kazantsev kinematic model), we derive statistical properties of the magnetic field that may be compared to observations from the paleomagnetic record. The fluid velocity is taken to be a Kraichnan field with general covariance, which acts as a random forcing obeying Gaussian statistics. Using the Martin-Siggia-Rose-Janssen-de Dominicis formalism, we compute the average magnetic field response function for fluid velocities with short correlation time. From this we obtain an estimate for the turbulent contribution to the magnetic diffusivity, and find that it is consistent with results from mean-field dynamo theory. This framework presents much promise for studying the geomagnetic field in a stochastic context.

Local strong solutions to the stochastic third grade fluid equations with Navier boundary conditions

Local strong solutions to the stochastic third grade fluid equations with Navier boundary conditions

Large deviations and long-time behavior of stochastic fluid queues with generalized fractional Brownian motion input

Large deviations and long-time behavior of stochastic fluid queues with generalized fractional Brownian motion input

High Value of Information Guided Data Enhancement for Heterogeneous Underwater Wireless Sensor Networks

Ensuring the freshness of high Value of Information (VoI) data has a significant practice meaning for marine observations and emergencies. The traditional forward method with an auv-aid is used to ensure the freshness of high VoI data. However, the methods suffer from two issues: an insufficient high VoI data throughput and random forwarding for cluster heads (CHs). The AUV (Autonomous Underwater Vehicle) with limited energy cannot meet the demand for the random generation of high VoI data. Low VoI data packets compete with high VoI data packets for channels, resulting in an insufficient high VoI data throughput and a low freshness. To address the above issues, we propose the Data Access Channel Scheme based on High Value of Information (DACS-HVOI), which is suitable for prioritizing the transmission packets with a high VoI. First, according to the level of VoI, the packets are divided into K classes, and the packets that are collected and forwarded by the AUV are defined as the highest K+1 class. Second, based on prior knowledge in the network, a Markov chain algorithm-based method is employed to predict which nodes should preferentially use the channel, to avoid conflict between a low and high VoI. Third, based on the stochastic fluid theory, a multilevel queueing system for CHs are constructed to avoid random forwarding. Last, compared with state-of-art protocols, experimental simulation shows that the proposed scheme has a low latency and high network throughput, while improving the throughput of high-VoI packets and ensuring the priority transmission of high-VoI packets.

Theoretical analysis and numerical approximation for the stochastic thermal quasi-geostrophic model

This paper investigates the mathematical properties of a stochastic version of the balanced 2D thermal quasigeostrophic (TQG) model of potential vorticity dynamics. This stochastic TQG model is intended as a basis for parametrization of the dynamical creation of unresolved degrees of freedom in computational simulations of upper ocean dynamics when horizontal buoyancy gradients and bathymetry affect the dynamics, particularly at the submesoscale (250 m–10 km). Specifically, we have chosen the Stochastic Advection by Lie Transport (SALT) algorithm introduced in [D. D. Holm, Variational principles for stochastic fluid dynamics, Proc. Roy. Soc. A: Math. Phys. Eng. Sci. 471 (2015) 20140963, http://dx.doi.org/10.1098/rspa.2014.0963 ] and applied in [C. Cotter, D. Crisan, D. Holm, W. Pan and I. Shevchenko, Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model, Found. Data Sci. 2 (2020) 173, https://doi.org/10.3934/fods.2020010 ; Numerically modeling stochastic lie transport in fluid dynamics, SIAM Multiscale Model. Simul. 17 (2019) 192–232, https://doi.org/10.1137/18M1167929 ] as our modeling approach. The SALT approach preserves the Kelvin circulation theorem and an infinite family of integral conservation laws for TQG. The goal of the SALT algorithm is to quantify the uncertainty in the process of up-scaling, or coarse-graining of either observed or synthetic data at fine scales, for use in computational simulations at coarser scales. The present work provides a rigorous mathematical analysis of the solution properties of the thermal quasigeostrophic (TQG) equations with SALT [D. D. Holm and E. Luesink, Stochastic wave-current interaction in thermal shallow water dynamics, J. Nonlinear Sci. 31 (2021), https://doi.org/10.1007/s00332-021-09682-9 ; D. D. Holm, E. Luesink and W. Pan, Stochastic mesoscale circulation dynamics in the thermal ocean, Phys. Fluids 33 (2021) 046603, https://doi.org/10.1063/5.0040026 ].