Fluid Equations Research Articles

Electron acceleration by wakefield generated by the propagation of chirped laser pulse in plasma

A study of generation of wakefield and particle acceleration via propagation of linearly polarised, chirped, Gaussian laser pulse in preformed plasma channel is presented. Perturbation technique is used to separate slow and fast varying plasma electron velocities and density. Considering the laser pulse length and amplitude to be evolving along the propagation distance, nonlinear fluid equations are used to derive longitudinal electric wakefields at varying propagation distances for chirped as well as unchirped pulses. It has been seen that longitudinal wakefield amplitude generated by positively (negatively) chirped laser pulses is higher (lower) than the amplitude obtained by unchirped laser pulses. The wakefield amplitudes are optimized with respect to the propagation distance. Further, trapping and acceleration of electrons by the generated wakes, is analysed. Comparing the energy of an accelerated test electron using chirped and unchirped laser pulses, it is shown that positively chirped laser pulses are capable of accelerating test electron to maximum energy using minimum injection energy. Hence, highest energy gain can be obtained by propagation of positively chirped laser pulse in plasma.

First‐order fractional step finite element method for the 2D/3D unstationary incompressible thermomicropolar fluid equations

AbstractIn this paper, the first‐order fractional step finite element method to solve the 2D/3D unstationary incompressible thermomicropolar fluid equations (ITME) is proposed. At each time step, the projection strategy about the linear velocity is adopted to get the linear velocity and pressure values in the first two sub‐steps, and then the angular velocity and temperature values are solved in the last two sub‐steps. Based on the proposed method, the variational problem is decomposed into four linear subproblems. By using the energy analysis technique, the unconditional stability and error estimates about the first‐order semi‐discrete method are derived, and the unconditional stability about the first‐order fully discrete method is analyzed. Finally, some numerical experiments are carried out to show the feasibility and effectiveness of the method.

Gravitational versus Magnetohydrodynamic Waves in Curved Spacetime in the Presence of Large-Scale Magnetic Fields

The general-relativistic (GR) magnetohydrodynamic (MHD) equations for a conductive plasma fluid are derived and discussed in the curved spacetime described by Thorne’s metric tensor, i.e., a family of cosmological models with inherent anisotropy due to the existence of an ambient, large-scale magnetic field. In this framework, it is examined whether the magnetized plasma fluid that drives the evolution of such a model can be subsequently excited by a transient, plane-polarized gravitational wave (GW) or not. To do so, we consider the associated set of the perturbed equations of motion and integrate them numerically in order to study the evolution of instabilities triggered by the GW propagation. In particular, we examine to what extend perturbations of the electric and/or the magnetic field can be amplified due to a potential energy transfer from the GW to the electromagnetic (EM) degrees of freedom. The evolution of the perturbed quantities depends on four free parameters, namely, the conductivity of the fluid, σ; the speed of sound square, 13<Csc2≡γ<1, which in this model may serve also as a measure of the inherent anisotropy; the GW frequency, ωg; and the associated angle of propagation with respect to the direction of the magnetic field, θ. We find that GW propagation in the anisotropic magnetized medium under consideration does excite several MHD modes; in other words, there is energy transfer from the gravitational to the EM degrees of freedom that can result in the acceleration of charged particles at the spot and in the subsequent damping of the GW.

A three-dimensional fictitious domain method for direct numerical simulations of particle-laden flows with heat transfer

In the present work, a three-dimensional fictitious domain method for particulate flows with heat transfer is proposed. For the case of fixed particle temperature, an iterative scheme for the temperature Lagrange multiplier is proposed, in order to determine its initial value and overcome the spurious oscillation of the explicit scheme at the initial time stage for different initial fluid and particle temperatures. Both explicit and implicit schemes are proposed for the solution of coupled fluid and solid temperature equations in the case of freely evolving particle temperature. The implicit scheme is suited to the case of large density ratios, specific heat ratios, or thermal conductivity ratios. Our method for the case of fixed particle temperature is verified via the test problems of a stationary hot sphere heating the surrounding quiescent fluid, a fixed sphere, and spheroid, respectively, in uniform flow, and sedimentation of a sphere and spheroid, respectively, in a vertical channel. We propose a new correlation of particle Nusselt number for an isolated sphere in a relatively small domain. Our code for the case of varying particle temperatures is verified via the effective thermal conductivity of a motionless sphere and the rising of a catalyst particle in an enclosure. Our method is applied to the sedimentation of a sphere at different Grashof numbers, specific heat ratios, and conductivity ratios. In addition, some preliminary results on heat transfer in turbulent channel flows laden with neutrally buoyant spherical and spheroidal particles, respectively, from fully resolved simulations with our method are reported.

Modulation of electromagnetic waves in a relativistic degenerate plasma at finite temperature

We study the modulational instability (MI) of a linearly polarized electromagnetic (EM) wave envelope in an intermediate regime of relativistic degenerate plasmas at a finite temperature (T≠0) where the thermal energy (KBT) and the rest-mass energy (mec2) of electrons do not differ significantly, i.e., βe≡KBT/mec2≲ (or ≳) 1, but the Fermi energy (KBTF) and the chemical potential energy (μe) of electrons are still a bit higher than the thermal energy, i.e., TF>T and ξe=μe/KBT≳1. Starting from a set of relativistic fluid equations for degenerate electrons at finite temperature, coupled to the EM wave equation and using the multiple scale perturbation expansion scheme, a one-dimensional nonlinear Schödinger (NLS) equation is derived, which describes the evolution of slowly varying amplitudes of EM wave envelopes. Then, we study the MI of the latter in two different regimes, namely, βe<1 and βe>1. Like unmagnetized classical cold plasmas, the modulated EM envelope is always unstable in the region βe>4. However, for βe≲1 and 1<βe<4, the wave can be stable or unstable depending on the values of the EM wave frequency, ω, and the parameter ξe. We also obtain the instability growth rate for the modulated wave and find a significant reduction by increasing the values of either βe or ξe. Finally, we present the profiles of the traveling EM waves in the form of bright (envelope pulses) and dark (voids) solitons, as well as the profiles (other than traveling waves) of the Kuznetsov–Ma breather, the Akhmediev breather, and the Peregrine solitons as EM rogue (freak) waves, and discuss their characteristics in the regimes of βe≲1 and βe>1.

Analysis of CO2 Migration in Horizontal Saline Aquifers during Carbon Capture and Storage Process

The storage of CO2 has become an important worldwide problem, considering that an excess of CO2 in the Earth’s atmosphere causes dramatic changes in its climate. One possible solution is to remove the excess of CO2 from the atmosphere, capture it in the process of creation, and store it safely, negating the possibility of its return into the atmosphere. This is the process of Carbon Capture and Storage (CCS). In the following paper, the authors investigate horizontal saline aquifers and their ability to store CO2. The authors’ application of sensitivity analysis on horizontal migrations uncovered that CO2 permeability and aquifer porosity have a considerable impact on horizontal migrations. During the migration process, CO2 can reach tens of kilometers from its injection point. By introducing effective CO2 density to the conduction velocity term, the authors showcase that the convection-diffusion equation for compressible fluids can be replaced with the equation for incompressible fluids. The buoyancy factor in convective velocity is as density dependent as in conduction velocity. By means of introducing an effective density to the aforementioned term, the process of transport via variable convective velocity can be substituted for a process which is effective, constant, and not density dependent.

Stationary solutions of the Schrödinger-Poisson-Euler system and their stability

We present the construction of stationary boson-fermion spherically symmetric configurations governed by Newtonian gravity. Bosons are described in the Gross-Pitaevskii regime and fermions are assumed to obey Euler equations for an inviscid fluid with polytropic equation of state. The two components are coupled through the gravitational potential. The families of solutions are parametrized by the central value of the wave function describing the bosons and the central density of the fluid. We explore the stability of the solutions using numerical evolutions that solve the time dependent Schrödinger-Euler-Poisson system, using the truncation error of the numerical methods as the perturbation. We find that all configurations are stable as long as the polytropic equation of state (EoS) is enforced during the evolution. When the configurations are evolved using the ideal gas EoS they all are unstable that decay into a sort of twin solutions that approach a nearly stationary configuration. We expect these solutions and their evolution serve to test numerical codes that are currently being used in the study of Fuzzy Dark Matter plus baryons.

Global Existence and Optimal Estimation of the Cauchy Problem to the 3D Fluid Equations

Global Existence and Optimal Estimation of the Cauchy Problem to the 3D Fluid Equations

Mechanism of enhanced oil recovery of ultra-dry CO2-in-water foams in porous media: Experiments and numerical simulation

The application of ultra-dry CO2 foam in enhanced oil recovery (EOR) accelerates recovery and enables CO2 utilization and sequestration. The flow characteristics of foam in porous media play a crucial role in the foam EOR process. Herein, ultra-dry CO2-in-water (C/W) foam with 10% water was prepared at 80 °C and 12 MPa using cocamidopropyl betaine (CAPB) and nonionic polyacrylamide (NPAM). Additionally, the constitutive equation for the foam fluid was determined by flow experiments using tubes with different diameters. The CO2 foam was a shear-thinning, non-Newtonian fluid and conformed to the Herschel–Bulkley (H–B) model. The core displacement experiment with the permeability of 470 mD showed that the oil recovery of foam flooding increased from 56.01% of water flooding to 76.01% at the same temperature and pressure. The maximum pressure difference rose sharply from 0.48 MPa of water flooding to 1.2 MPa of foam flooding, increasing by 2.5 times; the pressure difference finally stabilized at 0.64 MPa, which was approximately 4.3 times that of water flooding. Moreover, a two-dimensional (2D) model of the porous media was constructed using digital core technology. Using the validated model, sensitivity analysis based on the phase-field method (PFM) was performed to investigate the effect of different power-law exponents on displacement. The results indicate that increasing the power-law exponent of the foam fluid slowed down breakthrough and improved the sweep efficiency, resulting in higher oil recovery.

Probabilistic Descriptions of Fluid Flow: A Survey

Fluids can behave in a highly irregular, turbulent way. It has long been realised that, therefore, some weak notion of solution is required when studying the fundamental partial differential equations of fluid dynamics, such as the compressible or incompressible Navier–Stokes or Euler equations. The standard concept of weak solution (in the sense of distributions) is still a deterministic one, as it gives exact values for the state variables (like velocity or density) for almost every point in time and space. However, observations and mathematical theory alike suggest that this deterministic viewpoint has certain limitations. Thus, there has been an increased recent interest in the mathematical fluids community in probabilistic concepts of solution. Due to the considerable number of such concepts, it has become challenging to navigate the corresponding literature, both classical and recent. We aim here to give a reasonably concise yet fairly detailed overview of probabilistic formulations of fluid equations, which can roughly be split into measure-valued and statistical frameworks. We discuss both approaches and their relationship, as well as the interrelations between various statistical formulations, focusing on the compressible and incompressible Euler equations.

Non-isothermal non-Newtonian fluids: The stationary case

The stationary Navier–Stokes equations for a non-Newtonian incompressible fluid are coupled with the stationary heat equation and subject to Dirichlet-type boundary conditions. The viscosity is supposed to depend on the temperature and the stress depends on the strain through a suitable power law depending on [Formula: see text] (shear thinning case). For this problem we establish the existence of a weak solution as well as we prove some regularity results both for the Navier–Stokes and the Stokes cases. Then, the latter case with the Carreau power law is approximated through a FEM scheme and some error estimates are obtained. Such estimates are then validated through some two-dimensional numerical experiments.

Vortex Model of Plane Couette Flow

We present the theoretical description of plane Couette flow based on the previously proposed equations of vortex fluid, which take into account both the longitudinal flow and the vortex tubes rotation. It is shown that the considered equations have several stationary solutions describing different types of laminar flow. We also discuss the simple model of turbulent flow consisting of vortex tubes, which are moving chaotically and simultaneously rotating with different phases. Using the Boussinesq approximation, we obtain an analytical expression for the stationary profile of mean velocity in turbulent Couette flow, which is in good agreement with experimental data and results of direct numerical simulations. Our model demonstrates that near-wall turbulence can be described by a coordinates-independent coefficient of eddy viscosity. In contrast to the viscosity of the fluid itself, this parameter characterizes the turbulent flow and depends on Reynolds number and roughness of the channel walls. Potentially, the proposed model can be considered as a theoretical basis for the experimental measurement of the eddy viscosity coefficient.

Numerical Investigations of Pseudo-Boiling and Multi-Component Mixing Under Trans-/supercritical Conditions for Engine Applications

ABSTRACT The fuel injection and mixing process are often carried out under trans-/supercritical conditions for engine applications; however, the process is still unclear. In this study, based on OpenFOAM, a multi-component trans-/supercritical spray model based on the KT/KNP scheme is developed, which covers the special real fluid equation of state, various mixing rules and modified thermodynamic properties. In addition, the PIMPLE algorithm is extended to deal with the high nonlinearity. First, a one-dimensional advection case is applied to evaluate the performance of the model. Then, the effect of various mixing rules and pseudo-boiling phenomenon on n-heptane/nitrogen jet are thoroughly analyzed under trans-/supercritical conditions. The results show that the jet is extremely sensitive to the changes of the initial jet temperature and chamber pressure, and the “heat shield” effect of pseudo-boiling delays the breakup and mixing in the transcritical jet. With the mixing of different components, unlike the single-component condition, the pseudo-boiling intensity will increase. The increase of pressure and occurrence of mixing will lead to the decrease of pseudo-boiling temperature, and the pseudo-boiling will end earlier, which is obviously different from the single-component fluid case. For the n-heptane/nitrogen mixture, when the mole fraction of n-heptane in the mixture changes from 1.0 to 0.7, and the pseudo-boiling temperature decreases from 571 K to 465 K. With increasing injection temperature or chamber pressure, the pseudo-boiling intensity gradually decreases, and the potential core shortens, the difference among various mixing rules decreases. When the injection temperature is higher than 571 K, the pseudo-boiling phenomenon no longer exists. When the chamber pressure reaches 9 MPa, the pseudo-boiling strength is very low. Therefore, the spray mixing under supercritical conditions will be more sufficient. With the occurrence of mixing, the critical parameter gradually transitions from the critical value of n-heptane to the critical value of nitrogen, which strongly depends on the mixing rule.

Stiffness matrix method for modelling wave propagation in arbitrary multilayers

Natural and engineered media usually involve combinations of solid, fluid and porous layers, and accurate and stable modelling of wave propagation in such complex multilayered media is fundamental to evaluating their properties with wave-based methods. Here we present a general stiffness matrix method for modelling waves in arbitrary multilayers. The method first formulates stiffness matrices for individual layers based on the governing wave equations for fluids and solids, and the Biot theory for porous materials. Then it utilises the boundary conditions considered at layer interfaces to assemble the layer matrices into a global system of equations, to obtain solutions for reflection and transmission coefficients at any incidence. Its advantage over existing methods is manifested by its unconditional computational stability, and its validity is proved by experimental validations on single solid sheets, porous layers, and porous-solid-porous battery electrodes. This establishes a powerful theoretical platform that allows us to develop advanced wave-based methods to quantitatively characterise properties of the layers, especially for layers of porous materials.

Analysis of the relationship between the L-box test and the yield stress of fresh concrete: an analytical approach

The objective of this work was to study fresh concrete flow in the L-box test, predict the final deposit shape after flow stoppage and estimate the yield stress. This is very useful for controlling concrete casting in real structures, detecting problems and developing solutions on time. Concrete rheometers used for measuring the rheological parameters of concrete are expensive, not always available and can lead to measurement disparity. To overcome these issues, an effective method is to analyse concrete flow phenomena physically and solve the problem analytically based on L-box measurements. In the first part of this work, fresh concrete was considered as a homogeneous yield stress fluid moving under the effect of its self-weight. By developing momentum equations for an incompressible fluid, a theoretical approach was used to study three probable final shapes depending on the concrete rheology. Expressions were established for predicting the final shape and the yield stress. In the second part, the results obtained were compared with those of previous models and good agreement was found, thus validating the proposed approach. This study provides a consolidation of rheometer results, allowing evaluation and perhaps a calibration to reinforce their validity.

Viscoelastic effects on the deformation and breakup of a droplet on a solid wall in Couette flow

The deformation, movement and breakup of a wall-attached droplet subject to Couette flow are systematically investigated using an enhanced lattice Boltzmann colour-gradient model, which accounts for not only the viscoelasticity (described by the Oldroyd-B constitutive equation) of either droplet (V/N) or matrix fluid (N/V) but also the surface wettability. We first focus on the steady-state deformation of a sliding droplet for varying values of capillary number ( $Ca$ ), Weissenberg number ( $Wi$ ) and solvent viscosity ratio ( $\beta$ ). Results show that the relative wetting area $A_r$ in the N/V system is increased by either increasing $Ca$ , or by increasing $Wi$ or decreasing $\beta$ , where the former is attributed to the increased viscous force and the latter to the enhanced elastic effects. In the V/N system, however, $A_r$ is restrained by the droplet elasticity, especially at higher $Wi$ or lower $\beta$ , and the inhibiting effect strengthens with an increase of $Ca$ . Decreasing $\beta$ always reduces droplet deformation when either fluid is viscoelastic. The steady-state droplet motion is quantified by the contact-line capillary number $Ca_{cl}$ , and a force balance is established to successfully predict the variations of $Ca_{cl}/Ca$ with $\beta$ for each two-phase viscosity ratio in both N/V and V/N systems. The droplet breakup is then studied for varying $Wi$ . The critical capillary number of droplet breakup monotonically increases with $Wi$ in the N/V system, while it first increases, then decreases and finally reaches a plateau in the V/N system.

Breakdown of smooth solutions to the Müller–Israel–Stewart equations of relativistic viscous fluids

Breakdown of smooth solutions to the Müller–Israel–Stewart equations of relativistic viscous fluids

Scalar fluid in scalar-tensor gravity from an equivalent picture of thermodynamic and fluid descriptions of gravitational dynamics

We revisit the thermodynamic description of fluid, represented by scalar field in scalar-tensor gravity theory through a general approach to study the thermodynamics of relativistic fluids. In order to identify the fluid energy-momentum tensor, contrary to the existing way, we use equivalent description of gravitational dynamics both in Jordan and Einstein frames as thermodynamical identity and fluid equation on a generic null surface. Such an approach provides the energy-momentum tensors for scalar fluid as that of an ideal fluid in both the frames. We then mention few issues in the existing way of using Eckart's formalism for an ideal fluid. Our investigation suggests that the Eckart's frame may not be suitable to consider ideal fluid. Consequently a possible alternative description, valid in other than Eckart's frame, is being suggested for a general ideal fluid. Based on this we obtain a unified thermodynamic description in both Jordan and Einstein frames from our identified energy-momentum tensors. Finally, the relations between thermodynamic entities on different frames are being put forwarded. Thus, the description in either of the frames is enough to provide the complete picture.

Gravity waves in strong magnetic fields

ABSTRACT Strong magnetic fields in the cores of stars are expected to significantly modify the behaviour of gravity waves: this is likely the origin of suppressed dipole modes observed in many red giants. However, a detailed understanding of how such fields alter the spectrum and spatial structure of magnetogravity waves has been elusive. For a dipole field, we analytically characterize the horizontal eigenfunctions of magnetogravity modes, assuming that the wavevector is primarily radial. For axisymmetric modes (m = 0), the magnetogravity wave eigenfunctions become Hough functions, and they have a radial turning point for sufficiently strong magnetic fields. For non-axisymmetric modes (m ≠ 0), the interaction between the discrete g-mode spectrum and a continuum of Alfvén waves produces nearly discontinuous features in the fluid displacements at critical latitudes associated with a singularity in the fluid equations. We find that magnetogravity modes cannot propagate in regions with sufficiently strong magnetic fields, instead becoming evanescent. When encountering strong magnetic fields, ingoing gravity waves are likely refracted into outgoing slow magnetic waves. These outgoing waves approach infinite radial wavenumbers, which are likely to be damped efficiently. However, it may be possible for a small fraction of the wave power to escape the stellar core as pure Alfvén waves or magnetogravity waves confined to a very narrow equatorial band. The artificially sharp features in the Wentzel–Kramers–Brillouin-separated solutions suggest the need for global mode solutions which include small terms neglected in our analysis.

Fourth-order accurate numerical modeling of the multi-fluid plasma equations with adaptive mesh refinement

This work develops a multi-fluid plasma modeling capability in extension of a parallel, adaptive, fourth-order, finite-volume method, computational fluid dynamics algorithm. The governing equations of the multi-fluid plasma model are derived from moments of the Boltzmann equation, under the assumption of local thermodynamic equilibrium. Maxwell's equations describe the electromagnetic field evolution and are coupled to the fluid equations of charged species. The present work concentrates on two, charged, collisionless fluids as the first step, although the algorithm is designed to accommodate more fluids, both charged and neutral, and collisions between fluids; the extension to multiple fluids is straightforward. The numerical algorithm featuring parallel, high-order (≥4 for smooth flows), adaptive mesh refinement is expected to help cope with the inherent stiffness of the physics involved. The standard, fourth-order Runge-Kutta scheme is used to evolve the solution in time. The current study is primarily focused on verification and testing of the new, adaptive algorithm; we demonstrate 4th-order accuracy using problems with exact solutions and benchmark our solutions for common plasma simulations against references from literature. Using the 4th-order scheme, our multi-fluid model is shown to provide accurate solutions to plasma problems using half the cell count as traditional, 2nd-order methods. Furthermore, adaptive mesh refinement is successfully applied to three, plasma test problems, providing a significant reduction in mesh size. In subsequent work, an optimized version of our multi-fluid plasma algorithm will be applied to plasma test cases with more practical configurations, with a rigorous assessment of the improvements to computational efficiency afforded by our adaptive, higher-order scheme.