Fluid Equations Research Articles

Soliton, lump and hybrid solutions of a generalized (2+1)-dimensional Benjamin-Ono equation in fluids

Soliton, lump and hybrid solutions of a generalized (2+1)-dimensional Benjamin-Ono equation in fluids

Impact of relativistic positron beam on ion-acoustic solitary, periodic and breather waves in Earths’ ionospheric region through the framework of KdV and modified KdV equation

Abstract This study explores the propagation characteristics of ion-acoustic periodic, soliton, and breather waves in electron-positron-ion (EPI) plasma with a relativistic positron beam. The Korteweg–de Vries (KdV) equation is obtained by applying the traditional reductive perturbation method (RPM) to the fundamental set of fluid equations. When the KdV model is unable to accurately represent the nonlinear system’s evolution, a modified Korteweg–de Vries (mKdV) equation is constructed. In both models, Jacobi elliptic functions are used to derive periodic solutions, and a connection between periodic waves and soliton solutions is established. Hirota’s bilinear method is used to generate breathers directly from the KdV type framework without utilizing the modified Schrödinger framework inferred from the KdV type framework, which is a prevalent method in studies of nonlinear waves. Numerical knowledge of various physical factors in the ionospheric region is incorporated into the model to elucidate wave propagation in the Earth’s upper atmosphere.

New a Priori estimate for stochastic 2D Navier–Stokes equations with applications to invariant measure

AbstractThe paper deals with the stochastic two-dimensional Navier–Stokes equations for homogeneous and incompressible fluids, set in a bounded domain with Dirichlet boundary conditions. We consider additive noise in the form $$G\,\textrm{d}W$$ G d W , where W is a cylindrical Wiener process and G a bounded linear operator with range dense in the domain of $$A^\gamma $$ A γ , A being the Stokes operator. While it is known that existence of invariant measure holds for $$\gamma >1/4$$ γ > 1 / 4 , previous results show its uniqueness only for $$\gamma > 3/8$$ γ > 3 / 8 . We fill this gap and prove uniqueness and strong mixing property in the range $$\gamma \in (1/4, 3/8]$$ γ ∈ ( 1 / 4 , 3 / 8 ] by adapting the so-called Sobolevskiĭ-Kato-Fujita approach to the stochastic N–S equations. This method provides new a priori estimates, which entail both better regularity in space for the solution and strong Feller and irreducibility properties for the associated Markov semigroup.

An open-source, adaptive solver for particle-resolved simulations with both subcycling and non-subcycling methods

We present the IAMReX (incompressible flow with adaptive mesh refinement for the eXascale), an adaptive and parallel solver for particle-resolved simulations on the multi-level grid. The fluid equations are solved using a finite-volume scheme on the block-structured semi-staggered grids with both subcycling and non-subcycling methods. The particle-fluid interaction is resolved using the multidirect forcing immersed boundary method. The associated Lagrangian markers used to resolve fluid-particle interface only exist on the finest-level grid, which greatly reduces memory usage. The volume integrals are numerically calculated to capture the free motion of particles accurately, and the repulsive potential model is also included to account for the particle–particle collision. We demonstrate the versatility, accuracy, and efficiency of the present multi-level framework by simulating fluid-particle interaction problems with various types of kinematic constraints. The cluster of monodisperse particles case is presented at the end to show the capability of the current solver in handling multiple particles. It is demonstrated that the three-level AMR (Adaptive Mesh Refinement) simulation leads to a 72.46% grid reduction compared with the single-level simulation. The source code and testing cases used in this work can be accessed at https://github.com/ruohai0925/IAMR/tree/development. Input scripts and raw postprocessing data are also available for reproducing all results.

Nonlinear Perturbations and Weak Shock Waves in Isentropic Atmospheres

Acoustic perturbations to stellar envelopes can lead to the formation of weak shock waves via nonlinear wave steepening. Close to the stellar surface, the weak shock wave increases in strength and can potentially lead to the expulsion of part of the stellar envelope. While accurate analytic solutions to the fluid equations exist in the limits of low-amplitude waves or strong shocks, connecting these phases generally requires simulations. We address this problem using the fact that the plane-parallel Euler equations, in the presence of a constant gravitational field, admit exact Riemann invariants when the flow is isentropic. We obtain exact solutions for acoustic perturbations and show that after they steepen into shock waves, Whitham’s approximation can be used to solve for the shock’s dynamics in the weak to moderately strong regimes, using a simple ordinary differential equation. Numerical simulations show that our analytic shock approximation is accurate up to moderate (∼few–15) Mach numbers, where the accuracy increases with the adiabatic index.

Navier-Stokes: Singularities and Bifurcations

This article presents substantial advances in the analysis of the Navier-Stokes equations for both compressible and incompressible fluids, focusing on the formation of singularities, hypercomplex bifurcations, and regularity in Sobolev and Besov spaces. Through new theorems, we extend the theory of singularities in fluid dynamics and introduce quaternionic bifurcations, representing an innovative extension of classical bifurcation theory. Moreover, we delve into the investigation of the regularity of compressible fluids, exploring the conditions under which solutions remain smooth or develop singularities. These contributions are fundamental to the understanding of global regularity issues, directly linked to the renowned Millennium Prize problem, which seeks definitive answers on the existence and smoothness of solutions to the Navier-Stokes equations. Additionally, we discuss how these theoretical advancements offer new approaches to unresolved problems related to the formation of singularities in turbulent flows and the multiscale behavior of solutions, which are crucial for a comprehensive understanding of fluid dynamics. This work not only broadens the scope of traditional mathematical analysis of the Navier-Stokes equations but also establishes a robust theoretical framework for the investigation of bifurcations and regularity in advanced functional spaces, fostering a deeper understanding of global regularity phenomena and the complex dynamics governing fluid systems.

Electromagnetic Surface Waves in a Relativistic Magnetized Plasma Propagating in a Cylindrical Column

Dispersion relations of electromagnetic surface waves in a magnetized relativistic plasma propagating in a cylindrical column surrounded by vacuum are derived by means of the relativistic fluid equations. The zero order relativistic beam is assumed to stream along the direction of the axial magnetic field. The coupled equations of the E- and B-wave modes are solved analytically in terms of the cylindrical coordinates in weak or intermediate values of the magnetic field in which the cyclotron frequency is much smaller than the relativistic Doppler-shifted frequency. The entangled field components of the E- and B-waves are decoupled by assuming that the flow is incompressible. The surface wave dispersion relations of the E- and B-wave modes show a strong similarity and symmetry. The static magnetic field splits the E- and B-waves, thus doubling the modes in unmagnetized plasma. By taking the radius of the cylinder infinite, the dispersion relations in a semi-infinite plasma are derived.

Generalized Brenier Principle and the Closure Problem of Landgren–Monin–Novikov Hierarchy for Vorticity Field

Brenier’s concept – a representation of solutions to the equations of ideal incompressible fluids in terms of probability measures on the set of Lagrangian trajectories in the case of their stochasticity, is a generalization of Arnold’s principle of least action of finding smooth solutions of Euler’s equations. In this work, the variational generalized Brenier principle (Brenier, J. Am. Math. Soc. 1989) is used to close the infinite chain of Landgren–Monin–Novikov equations for the n-point probability density functions fn of the vortex field of two-dimensional turbulence. In addition, within the framework of the statistical approach, an approximation of the variational problem with conditions at the ends posed by Shnirelman (Mat. Sat. 1985) for the Euler equation is proposed.

Fast simulation strategy for capacitively-coupled plasmas based on fluid model

Fluid simulations are widely used in optimizing the reactor geometry and improving the performance of capacitively coupled plasma (CCP) sources in industry, so high computation speed is very important. In this work, a fast method for CCP fluid simulation based on the framework of Multi-physics Analysis of Plasma Sources (MAPS) is developed, which includes a multi-time-step explicit upwind scheme to solve electron fluid equations, a semi-implicit scheme and an iterative method with in-phase initial value to solve Poisson's equation, an explicit upwind scheme with limited artificial diffusion to solve heavy particle fluid equations, and an acceleration method based on fluid equation modification to reduce the periods required to reach equilibrium. In order to prove the validity and efficiency of the newly developed method, benchmarking against COMSOL and comparison with experimental data have been performed in argon discharges on the Gaseous Electronics Conference (GEC) reactor. Besides, the performance of each acceleration method is tested, and the results indicated that the multi-time-step explicit Euler scheme can effectively decline the computational burden in the bulk plasma and reduce the time cost on the electron fluid equations by half. The in-phase initial value method can greatly decrease the iteration times required to solve linear equations and lower the computational time of Poisson's equation by 77 %. The acceleration method based on equation modification can reduce the periods required to reach equilibrium by two-thirds.

Impact of temperature-dependent viscosity on linear and weakly nonlinear stability of double-diffusive convection in viscoelastic fluid

The impact of temperature-dependent viscosity on the double-diffusive viscoelastic convective fluids is investigated, with applications in heat transfer, polymer processing, food industry, biomedical engineering, etc. Both linear and weakly nonlinear analyses are carried out to determine the stability of fluids using the Oldroyd-B model. Analytical criteria for the onset of linear stationary and oscillatory convection are derived. The effects of rheological parameters, linearly and exponentially varying viscosity, salinity Rayleigh number and Prandtl number on the system’s stability are investigated. Linear stability analysis reveals that the interaction between thermal and solute diffusions with rheological parameters favours oscillatory convection over stationary convection. In weakly nonlinear theory, a power series expansion derives a Landau amplitude equation, allowing heat and mass transfer analysis in viscoelastic fluids with temperature-dependent viscosity. Numerical results highlight the effects of variable viscosity on heat and mass transfer rates, represented by Nusselt and Sherwood numbers. Further, the weakly non-linear stability analysis evaluates the impact of the Prandtl number, rheological parameters and salinity Rayleigh number on heat and mass transfer rates. These findings are compared with existing results to validate and enhance our understanding of fluid behaviour under different conditions.

Coupled Tanks Level Control Based on Volume Balance Method

Abstract The laboratory bench “Coupled tanks” represents a model of real technological plans where the levels and flow rates of liquids must be controlled. Such training benches are very common and there is a very large number of publications related to their control. All publications work with a non-linear model based on the Bernoulli equation for ideal fluids, which is non-linear – the square root of the fluid height is present. This equation or system of equations is linearized and attempts are made to control such systems by means of linear models. This article shows that this approach is not correct and proposes a completely different one based on the volume balance method.

Entropy Generation in Third-Grade Non-Newtonian Fluid Flow and Heat Transport Through Porous Medium in a Horizontal Channel under Heat Generation

In this study, entropy production in flow along with heat transport of non-Newtonian fluid via porous medium, is analyzed. For fluid flow via porous medium, a modified Darcy resistance term, is taken in the momentum equation for third-grade fluid. Temperature-dependent viscosity is considered using Vogel’s model viscosity. Using adequate transformations, the momentum and heat transport equation are reduced to non-dimensional form and solved analytically invoking homotopy analysis method on MATHEMATICA software. Effects of parameters arising in the study are depicted by graphs on velocity distribution, temperature distribution, and entropy production with Bejan number and discussed. For validity of current findings, the values of velocity and temperature are computed for particular values of the parameters and equated with previously published results, excellent agreement achieved. Furthermore, skin-friction coefficient and Nusselt number values are expressed in tabular form for various values of relevant parameters and discussed. It is noticed that slip parameters $\left( \gamma \right)$ and $\left( \beta \right)$ reduce the entropy generation number $\left( NS \right)$. Also noticed that skin friction coefficient upsurges with rising velocity slip parameter $\left( \gamma \right)$ value while, effect of temperature slip parameter $\left( \beta \right)$ is observed to lessen Nusselt number in the absolute sense. HIGHLIGHTS Entropy production in third grade non-Newtonian fluid flow and heat transfer in a channel via porous medium is investigated in the presence of a heat source. Slip boundary conditions are applied. The resulting governing equations are solved analytically using the homotopy analysis method (HAM). Maximum entropy production observed near the lower wall of channel and Bejan number attains maximum value in middle of the channel. GRAPHICAL ABSTRACT

Gyrotactic micro-organism dusty fluid flow over an extended surface

The conventional magnetohydrodynamics (MHD) equations are modeled for the dusty fluid flow past a stretching sheet embedded in a porous medium. These equations are coupled with motile microorganisms, temperature and concentration equations for both fluid and dust particles dynamical equations. The two-phase dusty fluid flow is subjected to magnetic effect, porosity factor, 2nd order chemical reaction, Brownian diffusion, thermophoretic effect and Arrhenius activation energy. An appropriate transformation is used to reset the system of partial differential equations (PDEs) into non-linear system of ODEs (ordinary differential equations). The flow equations are numerically solved by using the parametric continuation method (PCM). For validity of the applied methodology, the results are compared using numerical and graphical results of Matlab built-in package “BVP4c”. Both results show accuracy up to four decimal places. Furthermore, from graphical results, it can be noticed that the fluid particles interaction parameter causes an attractive force among fluid particles which prevents the dust particles phase flow. This technique can be used for biological separations, filters, fluid dust separators and crude oil purification.

A nonequilibrium statistical mechanics derivation of the hydrodynamic equations of simple fluids using a noncanonical form of the Poisson bracket

The continuum level hydrodynamic equations of a simple fluid were derived by Irving and Kirkwood directly from discrete particle dynamics using statistical mechanics almost 75 years ago. Their elegant derivation demonstrated the fundamental molecular basis of macroscopic fluid flow and culminated in molecular expressions for the stress tensor and heat current density that have since been employed in countless molecular simulations to date. In this article, an alternative derivation is presented, which leads to more general expressions for the fundamental transport relationships and which arrives at them in a more straightforward chain of consistency that ensues directly from the Principle of Least Action. The main point of departure from the Irving–Kirkwood derivation is the application of a transformation mapping of the total momentum of each individual particle onto the sum of its peculiar momentum and its momentum relative to the local velocity field. This mapping provides a phase-space distribution function applicable in the space of particle positions and peculiar momentum, from which a noncanonical Poisson bracket can be derived in terms of the same set of microscopic variables. For a given dynamic variable, expressed in terms of particle positions and peculiar momenta, the expectation value of the noncanonical Poisson bracket of the dynamic variable is shown to correspond to the evolution equation of the expectation value of the dynamic variable. This allows for a direct derivation of all macroscopic density evolution equations (mass, momentum, and energy density fields) using a systematic procedure free of assumptions concerning the macroscopic state of the system. Furthermore, an explicit expression of the time evolution of the entropy density at the hydrodynamic level is derived following the same procedure. Finally, in the limit of short-range interparticle interactions, a molecular-based expression for the local stress tensor as properly defined from continuum mechanics is developed at the hydrodynamic level that elucidates the continuum mechanics connection of the general stress expression of Irving and Kirkwood.

The MOOSE fluid properties module

The Fluid Properties module within the Multiphysics Object-Oriented Simulation Environment (MOOSE) is used to compute fluid properties for numerous applications, ranging from nuclear reactor thermal hydraulics to geothermal energy. Those applications drove the development of the module to enable numerous different fluid equations of states, property lookups with primitive and conserved flow variable to cater to pressure and density-driven solvers, and an object-oriented design facilitating expansion and maintenance. Each fluid property is implemented in its own class but inherits capabilities such as automatic differentiation, automated out-of-bounds handling or variable conversion capabilities. This paper presents the module, its design, its user and developer interface, its content in terms of fluids and properties, and several of its applications showing its major role in the MOOSE simulation ecosystem. Program summaryProgram title: MOOSE Fluid Properties moduleCPC Library link to program files:https://doi.org/10.17632/cwzhsyp6pd.1Developer's repository link:https://github.com/idaholab/moose/tree/next/modules/fluid_propertiesLicensing provisions: LGPLProgramming language: C++, PythonNature of problem: The simulation of thermal hydraulics of advanced nuclear reactor systems, such as heat pipe micro-reactors or molten-salt cooled pebble bed reactors, requires a wide variety of discretizations of the fluid flow equations, from 1D thermal hydraulics to computational fluid dynamics at various levels of fidelity, with a wide variety of coolants. Applications are developed within the MOOSE C++ framework by Argonne and Idaho National Laboratories to simulate these reactors for research and design purposes. These applications (Sockeye, SAM, others) rely on MOOSE for the computation of fluid properties. The fluid properties module contains properties for most advanced nuclear reactor coolants, including an interface to the Molten Salt Thermodynamics Database (MSTDB) developed by Oak Ridge National Laboratory. Single phase, two phase, and gas mixtures fluid properties are computed by the module.Solution method: The fluid properties module includes numerous numerical methods to support the wide range of applications, notably forward automatic differentiation, conversion methods between pressure and density-driven variable sets, spline-based table lookups which are the current state of the art for the fast computation of fluid properties. The integration with MOOSE facilitates uncertainty quantification with regards to the fluid properties and optimization studies with regards to the fluid composition.

Features of MHD on Secant Curved Annular Circular Plate Lubricant as a Couple-Stress Fluid with Slip Velocity

The purpose of this study is to examine the magneto-hydrodynamic (MHD) and slip velocity effect on secant curved annular circular plates lubricated with couple stress fluid and to derive the modified Reynolds’s equation for non-Newtonian fluid under various operating conditions to obtain the optimum bearing parameters. Based upon the MHD theory and Stokes theory for couple stress fluid, the governing equations relevant to the problem under consideration are derived. This paper analyses the effect of slip velocity on secant curved annular circular plates with an electrically conducting fluid in the presence of a transverse magnetic field. It is found that there is an increase in squeeze film pressure, load carrying capacity and squeeze film time in secant curved annular circular plates due to the presence of magnetic effects with couple stress fluid and slip velocity.

Nonlinear dynamics of micropolar two-phase fluids: Multiple exact solutions

Dual and triple solutions induced by a flexible planar surface for a micropolar two-phase fluid model are studied. The two-phase behavior in the micropolar fluid model occurs due to phase transitions between the fluid phases, influenced by interfacial stresses and heat transfer. The physical implications of these transitions are significant in understanding flow behavior under different mechanical and thermal conditions. This study examines the critical parameters and conditions that lead to these phase transitions, resulting in dual or triple solutions in the flow dynamics. The flow and thermal fields are exact solutions of the steady, two-dimensional two-phase micropolar fluid equations in the form of similarity solution. It is shown that dual and triple exact solutions exist for a highly nonlinear system. Triple solutions exist for the skin friction and temperature gradient identified by the critical numbers ac and μc. It is noted that for sufficiently small values of stretching strength parameter the dual branches for two of the triple solutions exist only in the regions μ≥μc3, and μ≤μc4, where μc3=−5.23 and μc4=−7.72. Numerical results are also provided, validating the model and offering insights into its accuracy and behavior of the model.

Thermodynamic and Kinetic Behavior Study of Gas Bubbles in High‐Temperature Steel Melts

This study provides an exhaustive exploration of the thermal expansion of low‐temperature argon bubbles in high‐temperature molten steel and the resulting dynamic effects. The energy equation of compressible fluid simultaneously with the volume of fluid model under the Euler framework is solved. The generation and rising behavior of bubbles are validated through rigorous physical water model experiments. Findings reveal that, within a heating time of 0.2–0.45 ms, gas bubbles already complete 65–90% of their internal energy increase. The total heat transfer time and heat transfer coefficient of bubbles exhibit distinctive patterns influenced by the initial diameter and temperature of the bubbles. Specifically, under the conditions studied, bubbles exhibit a total heat transfer time of ≈20–60 ms, accompanied by a heat transfer coefficient ranging from 500 to 2600 W (m2 * K)−1. During the thermal expansion process, gas bubbles experience energy transfer and conversion within the two‐phase flow. This article establishes a dynamic model describing bubble thermal expansion in steel melts based on underwater bubble dynamics’ first principles. The model provides a quantitative depiction of instantaneous changes in bubble size, velocity, and kinetic energy under specific conditions.

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

Role of Cairns–Gurevich ion distribution on nonlinear wave propagation in negatively charged dusty plasma

The study of dust-acoustic (DA) nonlinear electrostatic waves in dusty plasma is crucial for understanding plasma behavior in both astrophysical and laboratory environments. Dusty plasmas, characterized by the presence of negatively charged dust particles, exhibit complex dynamics influenced by various factors, including nonthermal electron and ion populations following Cairns–Gurevich distributions. This study addresses the problem of understanding wave propagation under these specific conditions by employing a set of fluid hydrodynamic equations for dust fluid, alongside appropriate electron and Cairns–Gurevich ion distributions, to model the dynamics and propagation of DA nonlinear electrostatic waves under specific plasma conditions with negatively charged dust particles. We derived a nonlinear Zakharov–Kuznetsov (ZK) equation to model the evolution of these waves and further transformed it into the nonlinear Schrödinger (NLS) equation to analyze rogue wave formation and identify unstable and stable zones within the plasma. We focused our analysis on the effects of key parameters, such as the nonthermal index, magnetic field strength, negative dust temperature ratio, polarization force, and density ratio, on the behavior of dust-acoustic waves (DAWs). The results demonstrated that soliton waves, explosive waves, and kink waves exhibit distinct behaviors depending on these parameters and the spatial context of Earth's magnetosphere. These findings provide new insights compared to previous studies into the conditions under which these waveforms emerge and evolve, especially in magnetized environments where earlier models were less effective at accurately characterizing or predicting wave behavior. By applying two different analytical methods, a direct integration approach and a generalized Kudryashov method, we expanded our understanding of nonlinear wave phenomena in dusty plasmas. Our results not only advance theoretical knowledge but also have practical implications for interpreting space plasma observations and guiding experimental design in plasma physics research. This study thus contributes to a more comprehensive understanding of plasma dynamics, with potential applications to astrophysical observation and laboratory plasma experimentation.