Search

Exact radial-coordinate dependent Beltrami coefficients for unsteady axially symmetric viscous Beltrami vortices under external forces are sought in cylindrical coordinate system. Two types of solutions are presented. One breaks the scaling invariance and decays temporally exponentially. The other one keeps the scaling invariance and asymptotically exhibits a power-law decay. As a byproduct, the exact generalized Beltrami vortex solutions are found and are shown to have lower energy densities. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes a mixed variational formulation for the Navier-Stokes equations with variable viscosity that depends nonlinearly on the velocity gradient. Differently from previous works in which augmented terms are added to the formulation, here we employ a technique that had been previously applied to the stationary Boussinesq problem and the Navier-Stokes equations with constant viscosity. Firstly, a modified pseudostress tensor is introduced involving the diffusive and convective terms and the pressure. Secondly, by using the incompressibility condition, the pressure is eliminated, and the gradient of velocity is incorporated as an auxiliary unknown to handle the aforementioned nonlinearity. As a consequence, a Banach spaces-based formulation is obtained, which can be written as a perturbed twofold saddle point operator equation. We address the continuous and discrete solvabilities of this problem by linearizing the perturbation and employing a fixed-point approach along with a particular case of a known abstract theory. Given an integer ℓ ≥ 0 , feasible choices of finite element subspaces include discontinuous piecewise polynomials of degree ≤ ℓ for each entry of the velocity gradient, Raviart-Thomas spaces of order ℓ for the pseudostress, and discontinuous piecewise polynomials of degree ≤ ℓ for the velocity as well. Finally, optimal a priori error estimates are derived, and several numerical results confirming in general the theoretical rates of convergence, and illustrating the good performance of the scheme, are reported. [ABSTRACT FROM AUTHOR]

We frame the mechanical stress tensor decomposition in a general procedure which involves the Helmholtz–Hodge decomposition. We highlight the impact of the mechanical stress tensor decomposition on the Navier–Stokes equation, with emphasis on the dissipation function. For fluids with low compressibility, we draw some insights on the Reynolds Averaged Navier–Stokes equations, and on the Reynolds stress tensor decomposition. We derive a turbulent potential flow model, and investigate the transition from viscous potential flow to turbulent potential flow. Under low Mach number approximation, we apply the turbulent potential flow model to one-dimensional propagation of large amplitude pressure waves in liquid-filled pipe. [ABSTRACT FROM AUTHOR]

In this article, the authors use the special structure of helicity for the three-dimensional incompressible Navier-Stokes equations to construct a family of finite energy smooth solutions to the Navier-Stokes equations which critical norms can be arbitrarily large. [ABSTRACT FROM AUTHOR]

The Boltzmann equation of kinetic theory gives a statistical description of a gas of interacting particles. It is well known that the Boltzmann equation is related to the Euler and Navier–Stokes equations in the field of gas dynamics. In this paper we are concerned with the incompressible Navier–Stokes–Fourier limit of the Boltzmann equation. We prove the incompressible Navier–Stokes–Fourier limit globally in time and the time decay rate of the solution to the rescaled Boltzmann equation in a torus. For ɛ small, by using the truncated expansion and L x , v 2 – L x , v ∞ method, we prove such a limit for the general potentials γ ∈ (− 3 , 1 ] . [ABSTRACT FROM AUTHOR]

For the incompressible Navier-Stokes equations in R 3 with low viscosity ν > 0 , we consider the Cauchy problem with initial vorticity ω 0 that represents an infinitely thin vortex filament of arbitrary given strength Γ supported on a circle. The vorticity field ω (x , t) of the solution is smooth at any positive time and corresponds to a vortex ring of thickness ν t that is translated along its symmetry axis due to self-induction, an effect anticipated by Helmholtz in 1858 and quantified by Kelvin in 1867. For small viscosities, we show that ω (x , t) is well-approximated on a large time interval by ω lin (x − a (t) , t) , where ω lin (⋅ , t) = exp (ν t Δ) ω 0 is the solution of the heat equation with initial data ω 0 , and a ˙ (t) is the instantaneous velocity given by Kelvin's formula. This gives a rigorous justification of the binormal motion for circular vortex filaments in weakly viscous fluids. The proof relies on the construction of a precise approximate solution, using a perturbative expansion in self-similar variables. To verify the stability of this approximation, one needs to rule out potential instabilities coming from very large advection terms in the linearized operator. This is done by adapting V. I. Arnold's geometric stability methods developed in the inviscid case ν = 0 to the slightly viscous situation. It turns out that although the geometric structures behind Arnold's approach are no longer preserved by the equation for ν > 0 , the relevant quadratic forms behave well on larger subspaces than those originally used in Arnold's theory and interact favorably with the viscous terms. [ABSTRACT FROM AUTHOR]

Single‐crystal silicon carbide (SiC) is an important semiconductor material for the fabrication of power and radio frequency (RF) devices. The major technique for growing single‐crystal SiC is the so‐called physical vapor transport (PVT) method, in which not only the thermal field but also the fluid‐flow field and the distribution of gas species can be hardly measured directly. In this study, a multi‐component flow model is proposed that includes the inside and outside of a growth chamber and a joint between the seed crystal holder and crucible which allows exchanges of the gas species. The joint is simulated as a thin porous graphite sheet. The Hertz‐Knudsen equation is used to describe the sublimation and deposition. The convection and diffusion are described by the Navier–Stokes equations and mixture‐averaged diffusion model, in which the Stefan flow is taken into account. The numerical simulations are conducted by the finite element method (FEM) with a multi‐physics coupled model, which is able to predict the fluid flow field, species distribution field, crystal growth rate, and evolution of the molar concentration of dopant gas. Using this model, the effects of several experimental conditions on the transport of gas species and the growth rate of single‐crystal SiC are analyzed. [ABSTRACT FROM AUTHOR]

The coalbed methane (CBM) productivity is directly determined by the fracture permeability during hydraulic fracturing, which is regulated by the distribution of proppants. The proppant may be unevenly distributed in the fracture because of variables like the architecture of the fracture and the characteristics of the sand‐carrying fluid. This study used two types of random functions to produce different ununiform distributions of proppant clusters in large‐scale fractures, with the aim of investigating the effect of these distributions on the overall permeability of the fracture. A model of fluid‐structure coupling is proposed. The closure of large‐scale fractures under in‐situ stress is analyzed using solid mechanics and the penalty function; the CBM flowing in proppant clusters and the high‐speed channel between them is simulated using Darcy's law and the Navier–Stokes equation, respectively; and the overall permeability of fractures is computed using the fluid pressure drop throughout the fracture and the fluid flowing velocity in the fracture's outlet. Since most CBM flows along high‐speed channels between the proppant clusters, the simulated findings show that the overall permeability of fractures with an uneven distribution of proppant clusters is significantly higher than that of the proppant cluster itself. As CBM becomes more discretely distributed, the proportion of CBM flowing within the proppant cluster continuously drops. As the permeability of the proppant cluster increases, the volume ratio of proppant clusters decreases, and the distribution of proppant clusters becomes more discrete, the overall permeability of the fracture continuously increases. [ABSTRACT FROM AUTHOR]

This work focuses on the study of natural convection in a flat porous microchannel with asymmetric heating. The novelty of the work lies in the fact that for the first time the method of discrete symmetries was used to analyze the complete system of Navier–Stokes and energy equations in a two-dimensional approximation. Analytical solutions for velocity and temperature profiles have been derived based on symmetry analysis, taking into account boundary conditions such as slip and temperature jump at the channel walls. The effect of Grashof, Knudsen, Darcy, and Prandtl numbers on the flow characteristics in the microchannel and heat transfer coefficients was elucidated. At high Grashof numbers, an ascending flow near the hot wall and a descending flow near the cold wall arise. Increasing the Knudsen number leads to an increase in the velocity, temperature jump at the walls and a decrease in heat transfer coefficients. As the Darcy number increases, velocities amplify in both ascending and descending flows. The temperature jump at the hot wall grows up, while it remains unchanged at the cold wall. In the same time, the heat transfer coefficient at the hot wall decreases. [ABSTRACT FROM AUTHOR]

It is known that examining turbulence effects on medical devices has an important effect in design and optimization of blood-contacting devices. CFD has been commonly used on prosthetic heart valves, stents, and Ventricular Assist Devices (VADs) in both the design process and also on hemodynamics of the flow characteristics. In this study, flows in the FDA nozzle were modeled to examine Reynolds stresses in the whole domain. The flow behavior was determined by applying the Reynolds-Averaged Navier-Stokes model of turbulence (k-ω SST) to simulate five distinctive experimental cases in the nozzle taken from the literature. The Reynolds stress frequencies are determined for the five different experimental conditions. Results showed that the highest velocity case (corresponding throat Reynolds number of 6500) has much higher Reynolds stresses with a high number of frequencies. However, the lowest velocity case has very small Reynolds numbers in a very high frequency. When different parts of the nozzle were examined, the Reynolds stress values showed more fluctuations for the higher velocities and more regular profiles for the lower velocity cases. [ABSTRACT FROM AUTHOR]

• This work aims at defining altitude limit for a proper use of a Navier-Stokes (NS) solver in Titan atmospheric entry. • Non-equilibrium and anisotropy are quantified in the fluid flow field. • Computations were carried out by the commercial CFD (Fluent) and DSMC (DS2V and DS3V) codes. • In Titan atmosphere entry, altitude of 182 km can be considered the altitude upper limit for a proper use of a CFD code. • Burnett eq. solver, slip velocity corrections, bridging formulae are necessary in the interval 182–295 km. Titan is the largest moon of Saturn. It is the only moon in our solar system provided with atmosphere. Therefore, exploration of Titan demands aerodynamic computations aimed at a proper design of space vehicles that will bring instruments to its surface. To this purpose, the aerodynamic computing methodologies are Computational Fluid Dynamics (CFD) for the solution of continuum flow fields and Direct Simulation Monte Carlo (DSMC) method for the solution of rarefied flow fields. The failure of the Navier-Stokes equations at high altitude is due to non-equilibrium (different translational, rotational and vibrational temperatures) and to anisotropy (different components in the x, y and z direction of the translational temperature). The limitations in using DSMC at low altitudes are due to technical limitations of the computer, i.e. memory capacity and processing speed. In the present work, non-equilibrium has been quantified by the relative differences between translational and rotational (θ t-r), translational and vibrational (θ t-v) temperatures. Similarly, the relative differences between the translational temperature component along x and those along y (θ x-y) and z (θ x-z) quantified anisotropy. For each test condition, the maximum values of θ are representative of non-equilibrium and anisotropy. Computations have been carried out considering the Huygens capsule in axial-symmetric flow, along Titan atmospheric entry in the altitude interval 100–470 km by means of the commercial codes: 1) Fluent v18.1 in the interval 100–295 km, 2) DS2V-4.5 64 bits in the interval 295–470 km. The present computations stated that in the Titan entry of a capsule of dimension comparable with that of Huygens, the altitude of about 182 km should be the limit altitude for a proper use of a CFD code. [ABSTRACT FROM AUTHOR]

In this article, a numerical study was carried out to study the dynamic adaptive grid method, based on the concept of the equidistribution method. The article explores a method for adapting the computational grid to solving two-dimensional Navier-Stokes differential equations, which describe the physical processes of gas dynamics specifically for the problem of a two-dimensional channel with an expansion coefficient (H/h) = 2. Different flow characteristics were calculated at different Reynolds numbers Re = from 100 to 1000, to get the actual thread behavior. Calculations are performed for laminar flow mode. The results of the longitudinal velocity profiles in different sections of the channel and the length of the primary and secondary vortices are obtained with a change in the Reynolds number after the ledge. For the numerical solution of this problem, a second-order accuracy McCormack scheme was used. To confirm the adequacy and reliability of the numerical results, a careful comparison was made with the experimental data of Armaly V.F. et al., taken from the literature. It is also shown that as a result of using this method of adaptive grids, it is possible to improve the numerical accuracy obtained for a given number of node points. It is shown that the existing method of multiple 2D adaptive meshes makes it easier to concentrate meshes in the required areas. This method should prove useful for many Navier-Stokes flow calculations. [ABSTRACT FROM AUTHOR]

• A stationary model for flow and oxygen transport to design optimal axisymmetric radial-flow packed-bed bioreactors (rPBBs) • A stationary model for flow and oxygen transport in rPBBs accounting for Michaelian cellular consumption and bulk medium-to-cell surface oxygen transport resistance. • The effect of axial radial flux distribution on pericellular and bulk dissolved oxygen concentrations for relevant dimensionless parameters determining rPBB behaviour. • Operating conditions under which varying inlet flow rates relieve non-uniform oxygen concentration distribution and decay at cell surface also starting from non-uniform medium radial flux distribution. • Model predictions and integration runtimes feasible to develop algorithms to control dissolved oxygen concentration at cell surface as tissue matures. Radial flow packed-bed bioreactors (rPBBs) overcome the transport limitations of static and axial-flow perfusion bioreactors and enable development of clinical-scale bioengineered tissues. We developed criteria to design rPBBs with uniform medium radial flux distribution along bioreactor length ensuring uniform construct perfusion. We report a model-based analysis of the effect of non-uniform axial distribution of medium radial flux on pericellular concentration of oxygen and glucose. Albeit pseudo-homogeneous, the model predicts how medium flux, solutes transport and cellular consumption interact and determine the pericellular oxygen and glucose concentrations in the presence of pore transport resistance to design optimal axisymmetric rPBBs and enable control of pericellular environment. Thus, oxygen and glucose supply may match cell requirements as tissue matures. Flow and solute transport in bioreactor empty spaces and construct was described with Navier-Stokes and Darcy-Brinkman equations, and with convection–diffusion and convection–diffusion-reaction equations, respectively. Solute transport in construct accounted for Michaelian cellular consumption and bulk medium-to-cell surface oxygen transport resistance in terms of a transport-equivalent bed of Raschig rings. The effect of relevant dimensionless groups on pericellular and bulk solute concentrations was predicted under typical tissue engineering operation and evaluated against literature data for bone tissue engineering. Axial distribution of medium radial flux influenced the distribution of pericellular solutes concentration, more so at high cell metabolic activity. Increasing medium feed flow rates relieved non-uniform solute concentration distribution and decayed at cell surface for metabolic consumption, also starting from axially non-uniform radial flux distribution. Model predictions were obtained in runtimes compatible with on-line control strategies. [ABSTRACT FROM AUTHOR]

The aim of this note is to study the Cauchy problem for the 2D Euler equations under very low regularity assumptions on the initial datum. We prove propagation of regularity of logarithmic order in the class of weak solutions with L p initial vorticity, provided that p ≥ 4. We also study the inviscid limit from the 2D Navier-Stokes equations for vorticity with logarithmic regularity in the Yudovich class, showing a rate of convergence of order | log ⁡ ν | − α / 2 with α > 0. [ABSTRACT FROM AUTHOR]

The study of laminar incompressible fluid flow in the boundary layer revealed, even earlier, that the condition of complete adhesion of fluid particles to the surface (non-slip condition) of the moving body (half-plane) is not met in the flow development (formation) region. The assumption of constancy of the fluid velocity on the surface of a moving body, hence non-slip, leads, in the flow development region, to the complete absence of the normal component of the velocity field. And this contradicts the very concept of the flow development region, where there should be two velocity components - longitudinal (primary) and normal (secondary) ones. In the previous works of the authors, analytical solutions were obtained for the velocity field in the region of development of incompressible fluid flow in the boundary layer. Since the use of the incompressible fluid flow model is restricted by the Mach number, to further expand the speed range, the problem of the of slightly compressible fluid flow development region in the boundary layer was considered. It is analytically proven that all considerations regarding the impossibility of complete non-slip in the flow development region can be applied to a slightly compressible flow. Slight compressibility at the same time means the subsonic nature of the flow and the neglect of temperature effects due to friction. On the basis of a critical analysis of the existing approaches, which consider the flow of a fluid around a immobile plate in the framework of non-gradient flow (which is just impossible due to the lack of a mechanism for creating the motion of the fluid), it is shown that the system of equations is actually non-closed. For the region of flow development, where the longitudinal pressure gradient is not a constant value, one equation is missing. This equation, as in previous works, is obtained from the necessary condition for the extreme of the fluid rate functional. And although the complete solution for the longitudinal component of the velocity contains four constants of integration, to obtain the asymptotics near the solid surface it is sufficient to know only two quantities - the velocity and its first derivative (gradient). These values, as it turns out from the asymptotic solution, coincide with the case of incompressible flow, which allows us to expand the scope of the previously obtained results for a wider domain of Mach numbers, for example . And such values already correspond to the speeds of modern civil aircraft. The dimensionless distribution of pressure in the slightly compressible flow development region is presented and its significant heterogeneity is shown, which, in turn, indicates the importance of the obtained results. [ABSTRACT FROM AUTHOR]

Creating a high-quality aircraft engine is closely connected to the problem of obtaining the jet flow characteristics that appear while an aircraft's engine is in operation. As natural experiments are costly, studying turbulent jets by numerical simulation appears practical and acute. Biconic nozzle supersonic jet flow is the research subject of this article. A compression and expansion train of waves called barrels were formed in the jet flow at preset conditions. The simulation was performed on an unstructured numerical grid. In order to enhance the calculation accuracy in the shock-wave domain, a hybrid gradient computation scheme and numerical grid static adaptation method were applied in the regions of gas-dynamic values' significant differential. This approach resulted in a description of nozzle supersonic gas flow structure. It was shown that building local refinement when using a static adaptation numerical grid contributed to improving the accuracy of determining shock waves' fronts. In addition, this approach facilitated the identification of the Mach disk in the flow when using an unstructured grid, allowing for calculation schemes not higher than a second-order of accuracy. [ABSTRACT FROM AUTHOR]

The global stability of solutions to the compressible Navier-Stokes equations for a viscous and heat-conducting fluids around an infinite long cylinder is investigated. The global in time solution is proved to exist uniquely and approach the stationary state as $ t\rightarrow \infty $, provided with the corresponding initial boundary values are perturbed sufficiently small. [ABSTRACT FROM AUTHOR]

This paper presents a lifting‐line implementation in the framework of a Lagrangian vortex particle method (LL‐VP). The novelty of the present implementation lies in the fluid particles properties definition and in the particles shedding process. In spite of mimicking a panel method, the LL‐VP needs some peculiar treatments described in the paper. The present implementation converges rapidly and efficiently during the shedding sub‐iteration process. This LL‐VP method shows good accuracy, even with moderate numbers of sections. Compared to its panel or vortex filaments counterparts, more frequently encountered in the literature, the present implementation inherently accounts for the diffusion term of the Navier‐Stokes equations, possibly with a turbulent viscosity model. Additionally, the present implementation can also account for more complex onset flows: upstream ambient turbulence and upstream turbine wakes. After validation on an analytical elliptic wing configuration, the model is tested on the Mexnext‐III wind turbine application, for three reduced velocities. Accurate results are obtained both on the analytical elliptic wing and on the New MEXICO rotor cases in comparison with other similar numerical models. A focus is made on the Mexnext‐III wake analysis. The numerical wake obtained with the present LL‐VP is close to other numerical and experimental results. Finally, a last configuration with three tidal turbines in interaction is considered based on an experimental campaign carried out at the IFREMER wave and current flume tank. Enhanced turbine‐wake interactions are highlighted, with favourable comparisons with the experiment. Hence, such turbine interactions in a farm are accessible with this LL‐VP implementation, be it wind or tidal energy field. [ABSTRACT FROM AUTHOR]

We here investigate a modification of the compressible barotropic Euler system with friction, involving a fuzzy nonlocal pressure term in place of the conventional one. This nonlocal term is parameterized by ε > 0 and formally tends to the classical pressure when ε approaches zero. The central challenge is to establish that this system is a reliable approximation of the classical compressible Euler system. We establish the global existence and uniqueness of regular solutions in the neighborhood of the static state with density 1 and null velocity. Our results are demonstrated independently of the parameter ε , which enable us to prove the convergence of solutions to those of the classical Euler system. Another consequence is the rigorous justification of the convergence of the mass equation to various versions of the porous media equation in the asymptotic limit where the friction tends to infinity. Note that our results are demonstrated in the whole space, which necessitates to use the L 1 (R + ; B ˙ 2 , 1 σ (R d)) spaces framework. [ABSTRACT FROM AUTHOR]

In this study, we present an hp-multigrid preconditioner for a divergence-conforming HDG scheme for the generalized Stokes and the Navier–Stokes equations using an augmented Lagrangian formulation. Our method relies on conforming simplicial meshes in two- and three-dimensions. The hp-multigrid algorithm is a multiplicative auxiliary space preconditioner that employs the lowest-order space as the auxiliary space, and we develop a geometric multigrid method as the auxiliary space solver. For the generalized Stokes problem, the crucial ingredient of the geometric multigrid method is the equivalence between the condensed lowest-order divergence-conforming HDG scheme and a Crouzeix–Raviart discretization with a pressure-robust treatment as introduced in Linke and Merdon (Comput Methods Appl Mech Engrg 311:304–326, 2022), which allows for the direct application of geometric multigrid theory on the Crouzeix–Raviart discretization. The numerical experiments demonstrate the robustness of the proposed hp-multigrid preconditioner with respect to mesh size and augmented Lagrangian parameter, with iteration counts insensitivity to polynomial order increase. Inspired by the works by Benzi and Olshanskii (SIAM J Sci Comput 28:2095–2113, 2006) and Farrell et al. (SIAM J Sci Comput 41:A3073–A3096, 2019), we further test the proposed preconditioner on the divergence-conforming HDG scheme for the Navier–Stokes equations. Numerical experiments show a mild increase in the iteration counts of the preconditioned GMRes solver with the rise in Reynolds number up to 10 3 . [ABSTRACT FROM AUTHOR]

In this paper, we discuss the stability of a semi-discrete implicit difference scheme of the time fractional Navier–Stokes equations which is applied in many physical processes, and the convergence of the difference approximate solution. First, we introduce the concept of the average characteristic of the sequence obtained by the difference scheme and the concept of partial stability of the scheme, and then obtain several stability results according to the normality of the external force term. We also prove the convergence of the difference approximation sequence to the unique solution of the equation. [ABSTRACT FROM AUTHOR]

In this article, we propose an adaptive mesh-refining based on the multi-level algorithm and derive a unified a posteriori error estimate for a class of nonlinear problems. We have shown that the multi-level algorithm on adaptive meshes retains quadratic convergence of Newton's method across different mesh levels, which is numerically validated. Our framework facilitates to use the general theory established for a linear problem associated with given nonlinear equations. In particular, existing a posteriori error estimates for the linear problem can be utilized to find reliable error estimators for the given nonlinear problem. As applications of our theory, we consider the pseudostress-velocity formulation of Navier–Stokes equations and the standard Galerkin formulation of semilinear elliptic equations. Reliable and efficient a posteriori error estimators for both approximations are derived. Finally, several numerical examples are presented to test the performance of the algorithm and validity of the theory developed. [ABSTRACT FROM AUTHOR]

We consider the 3D stochastic Navier–Stokes equation on the torus. Our main result concerns the temporal and spatio-temporal discretisation of a local strong pathwise solution. We prove optimal convergence rates for the energy error with respect to convergence in probability, that is convergence of order (up to) 1 in space and of order (up to) 1/2 in time. The result holds up to the possible blow-up of the (time-discrete) solution. Our approach is based on discrete stopping times for the (time-discrete) solution. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the stability analysis for a contaminant convection-reaction-diffusion model of the recovered fracturing fluid (RFFM, for short), which couples a nonlinear and non-smooth stationary incompressible Navier-Stokes equation with a multivalued frictional boundary condition, and a nonlinear reaction-diffusion equation with mixed Neumann boundary conditions. First, we introduce a family of perturbation problems corresponding to (RFFM), and present the variational formulation of perturbation problem which is a perturbation elliptic hemivariational inequality driven by a perturbation nonlinear variational equation. Then, the existence of solutions and the uniform bound of the solution set to the perturbation problem are obtained. Finally, it is established that, as the perturbation parameter tends to zero, the solution set of the perturbation problems converge to the solution set of (RFFM) in the sense of the Kuratowski upper limit. This shows that (RFFM) is stable with respect to the perturbation data. [ABSTRACT FROM AUTHOR]

This work reports a numerical study on the effect of three nondimensionalization approaches that are commonly used to solve the classic problem of the 2‐D differentially heated cavity. The governing equations were discretized using orthogonal collocation with Legendre polynomials, and the resulting algebraic system was solved via Newton–Raphson method with LU factorization. The simulations were performed for Rayleigh numbers between 103 and 108, considering the Prandtl number equal to 0.71 and a geometric aspect ratio equal to 1, analyzing the convergence and the computation time on the flow lines, isotherms and the Nusselt number. The mesh size that provides independent results was 51 × 51. Approach II was the most suitable for the nondimensionalization of the differentially heated cavity problem. [ABSTRACT FROM AUTHOR]

Numerical simulations were performed to study the effect of composition gradient and obstacle arrangement on detonation initiation in fuel-rich hydrogen-air mixtures. A third-order WENO method with adaptive mesh refinement (AMR) was used to solve the unsteady, fully-compressible, reactive Navier-Stokes equations coupled to a calibrated chemical-diffusive model (CDM). An obstructed channel at a blockage ratio of 0.3 filled with non-uniform hydrogen-air mixtures of an average H 2 concentration 50 vol% was considered. The results show that unilateral obstacle arrangement is more conducive to flame acceleration (FA) and DDT than bilateral obstacle arrangement for both homogeneous and inhomogeneous mixtures. For inhomogeneous mixtures, the flame accelerates faster, and the detonation onset time is shorter when obstacles are placed on the sidewall with a hydrogen concentration closer to the stoichiometric ratio than the those when obstacles are placed on the other sidewall. However, the placement of unilateral obstacles on either the upper or lower sidewall does not significantly affect the DDT run-up distance. Besides, for fuel-rich inhomogeneous hydrogen-air mixtures, detonation tends to be initiated in the regions of high H 2 concentration. Idealized models were introduced to simplify the intricate interactions of the reaction waves and flow fields to better understand the connection between H 2 concentration distribution and detonation initiation. The analysis suggests that the minimum shock strength required for detonation initiation decreases with increasing equivalence ratio. This is supported by a kinetics analysis with detailed reaction model, which shows that ignition delay increases with decreasing equivalence ratio when detonation is about to be initiated. • Simulating flame acceleration (FA) and DDT using high-order numerical method. • Studying effects of obstacle arrangement and inhomogeneity of H 2 -air on FA and DDT. • Analyzing through simplified detonation onset models and detail chemical kinetics. • Providing valuable insights on detonation initiation physics in non-uniform mixtures. [ABSTRACT FROM AUTHOR]

This paper studies the zero-viscosity limit for the Navier-Stokes equations with rotation, an additive white noise and nonslip boundary condition in dimension three. By establishing the well-posedness of the limit problems and constructing approximate solutions, we obtain the convergence in the mean square for the Oseen type equation with nonlinear source term, and the convergence in probability of the martingale solution of the stochastic Navier-Stokes equations with rotation. [ABSTRACT FROM AUTHOR]

AbstractThis study concerns the numerical simulation of a system of swirling turbulent jets impacting a flat plate, by two numerical models of the closure of the Navier Stokes equations (k-ω sst). The jet impact technique is widely applied in the field of heating, drying and cooling of industrial objects. The experimental test bench comprising a diffuser of diameter D, impacting the perpendicular plate, such that the impact distance H = 4D. The swirl is obtained by a swirl generator, made up of 12 fins arranged at 60° from the vertical, placed just at the outlet of the diffuser. The temperature of the swirling jets blowing on the plate is measured with a VELOCICALC PLUS portable thermo-anemometer device, in different stations. It is noted that this diffuser configuration having balanced inlet temperatures (T, T, T), gives a uniform distribution of temperature and velocity on the surface of the plate. This temperature distribution results from a 95% spread of homogenization of plate heat transfer. The (k-ω sst) model gave temperature and velocity profiles that coincide with those of the experimental results. By way of comparison, the (k-ω sst) model better predicts impacting jets, particularly in fluctuating zones. The latter remains a relatively better numerical simulation tool in terms of quality. [ABSTRACT FROM AUTHOR]

We study the stationary Navier–Stokes equations in the region between two rotating concentric cylinders. We first prove that, for a small Reynolds number, if the fluid flow is axisymmetric and if its velocity is sufficiently small in the $L^\infty$ -norm, then it is necessarily the Taylor–Couette–Poiseuille flow. If, in addition, the associated pressure is bounded or periodic in the $z$ axis, then it coincides with the well-known canonical Taylor–Couette flow. We discuss the relation between uniqueness and stability of such a flow in terms of the Taylor number in the case of narrow gap of two cylinders. The investigation in comparison with two Reynolds numbers based on inner and outer cylinder rotational velocities is also conducted. Next, we give a certain bound of the Reynolds number and the $L^\infty$ -norm of the velocity such that the fluid is, indeed, necessarily axisymmetric. As a result, it is clarified that smallness of Reynolds number of the fluid in the two rotating concentric cylinders governs both axisymmetry and the Taylor–Couette–Poiseuille flow with the exact form of the pressure. [ABSTRACT FROM AUTHOR]

The bioconvective flows are truly connected to real-life and engineering development. So, the models of biomicrosystems and biocells are considered for the technical analysis in this paper. Our intention for the current analysis was to theoretically examine electrical conduction flow into mass and heat transfer by an extensive gyrotactic microorganism into an inclined magnetic field toward a vertical stretching sheet with nonlinear solar radiation with different solar thermal appliances. The effect on velocity slip and Joule heating was again studied in detail. This classical problem on Navier Stokes equations to the current imitation was decreased to ordinary differential equations by applying the comparison method. The numerical solutions were changed by boundary value problem (BVP) to clarify the subject into finite difference numerical scheme by applying MATLAB. The important results show that the density of motile microorganisms reduces to the bioconvection Lewis number and Peclet number, although reverse performance was noticed for the bioconvection Rayleigh number. Further, solar radiation fosters heat transport. In this paper, complete analysis is provided for potential functions of solar thermoelectric cells, solar ponds, solar thermal power fabrication, etc. [ABSTRACT FROM AUTHOR]

The oblique detonation waves have been studied extensively, but the wave characteristics influenced by the geometrical restriction have not been fully addressed. In this study, we examine the wave stability and the thrust performance in a contraction-expansion channel with varying velocity. A supersonic stoichiometric inflow of hydrogen-oxygen inflow is established in the computational domain, and the compressible reactive Navier-Stokes equations are solved using a comprehensive chemical model. As the velocity increases, the detonation wave inside the channel exhibits two successive unsteady states: the half normal detonation wave (half NDW) and the re-ignition oblique detonation wave (re-ignition ODW). In the half NDW state, the upper part of the wave surface is a basically stable NDW, while the lower part oscillates regularly as the reaction front. In the re-ignition ODW state, the explosion of the reaction front and the retreat of the detonation surface occur in proper order. Furthermore, the thrust associated with these newly discovered oscillation wave systems exhibits unstable behavior, with an important observation that it does not consistently decrease with increasing velocity. Notably, there is a significant increase in thrust during the transition from the half NDW state to the ODW state, as well as when the position of the oblique detonation wave shifts downstream. [Display omitted] • The oblique detonation wave structure and thrust performance in a contraction–expansion channel are revealed. • With increasing velocity, thermal choking unstarting, half normal detonation, and oblique detonation states emerge. • Two unsteady wave states were identified for the first time, and the primary flow characteristics were discerned. • Throughout the entire super-detonation process, the effect of V in / V cj on thrust is non-monotonic decreasing. [ABSTRACT FROM AUTHOR]

AbstractThe paper evaluates the application of several multi-physical fields reconstruction approaches based on the Tomographic Particle Image Velocimetry (PIV) in three-dimensional supersonic flows with shock waves, including the pressure, temperature, and density. The MacCormack method and the Flux Vector Splitting method with outstanding performance in two-dimensional supersonic flow are extended into three-dimensional forms. The conventional Poisson method is also considered because of its widespread application and high accuracy in subsonic and transonic flow. All of these approaches are established by the conservative Navier-Stokes equations and solved by the time-marching iteration process, combined with the perfect gas law and adiabatic flow assumption. The performances are evaluated by numerical velocimetry data. To simulate the typical three-dimensional features, the cases of uniform freestream at Mach 0.78 and 3 past a cone with a 20° half-angle are selected to obtain a sufficient spanwise velocity component. The results confirm the feasibility of the PIV-based reconstruction methods in the conical flow field. The conventional Poisson method performs well only in the subsonic case while the Flux Vector Splitting method has a better performance in supersonic flow, including higher accuracy, stability, and efficiency, with a low-level root-mean-square error of 0.449% and local maximum relative error of 1%. [ABSTRACT FROM AUTHOR]

This paper develops the mathematical apparatus of studying control problems for the stationary model of magnetic hydrodynamics of viscous heat-conducting fluid in the Boussinesq approximation. These problems are formulated as problems of conditional minimization of special cost functionals by weak solutions of the original boundary value problem. The model under consideration consists of the Navier–Stokes equations, the Maxwell equations without displacement currents, the generalized Ohm's law for a moving medium and the convection-diffusion equation for temperature. These relations are nonlinearly connected via the Lorentz force, buoyancy force in the Boussinesq approximation and convective heat transfer. Results concerning the existence and uniqueness of the solution of the original boundary value problem and of its generalized linear analog are presented. The global solvability of the control problem under study is proved and the optimality system is derived. Sufficient conditions on the data are established which ensure local uniqueness and stability of solutions of the control problems under study with respect to small perturbations of the cost functional to be minimized and one of the given functions. We stress that the unique stability estimates obtained in the paper have a clear mathematical structure and intrinsic beauty. [ABSTRACT FROM AUTHOR]

We consider evolution (non-stationary) spatially-periodic solutions to the n-dimensional non-linear Navier–Stokes equations of anisotropic fluids with the viscosity coefficient tensor variable in spatial coordinates and time and satisfying the relaxed ellipticity condition. Employing the Galerkin algorithm with the basis constituted by the eigenfunctions of the periodic Bessel-potential operator, we prove the existence of a global weak solution. [ABSTRACT FROM AUTHOR]

This work deals with gliding motion of bacteria over a non-Newtonian slime layer attached to a solid substrate. The surface of a glider is approximated with a simple wavy two-dimensional sheet while the sticky slime is modeled as Carreau–Yasuda fluid. The classical Navier–Stokes equations are transformed by using Galilean transformation and dimensionless variables. Flow beneath the organism is creeping and the lubrication assumption is also valid in this scenario, hence the equation is reduced to fourth-order BVP in terms of stream function. The basic purpose is to compute the gliding speed and flow rate of the slime which are present in the boundary conditions. MATLAB built-in bvp5c solver is utilized to calculate the numerical solution of the stream function. Further, unknowns (flow rate and cell speed) are refined by using modified Newton–Raphson method (MNRT). Further, these pairs are employed in the formula of energy expended. Velocity and stream function is also plotted for these computed pairs. This study is motivated by scientific interest and the desire to understand the dynamics of gliding bacteria. The findings of this study are thought to be beneficial in the development of artificial crawlers. [ABSTRACT FROM AUTHOR]

Bioprocess scale-up and technology transfer can be challenging due to multiple variables that need to be optimized during process development from laboratory scale to commercial manufacturing. Cell cultures are highly sensitive to key factors during process transfer across scales, including geometric variability in bioreactors, shear stress from impeller and sparging activity, and nutrient gradients that occur due to increasing blend times. To improve the scale-up and scale-down of these processes, it is important to fully characterize bioreactors to better understand the differences that will occur within the culture environment, especially the hydrodynamic profiles that will vary in vessel designs across scales. In this study, a comprehensive hydrodynamic characterization of the Ambr® 250 mammalian single-use bioreactor was performed using time-accurate computational fluid dynamics simulations conducted with M-Star computational fluid dynamics software, which employs lattice-Boltzmann techniques to solve the Navier-Stokes transport equations at a mesoscopic scale. The single-phase and two-phase fluid properties within this small-scale vessel were analyzed in the context of agitation hydrodynamics and mass transfer (both within the bulk fluid and the free surface) to effectively characterize and understand the differences that scale-down models possess when compared to their large-scale counterparts. The model results validate the use of computational fluid dynamics as an in-silico tool to characterize bioreactor hydrodynamics and additionally identify important free-surface transfer mechanics that need to be considered during the qualification of a scale-down model in the development of mammalian bioprocesses. • Multiphase hydrodynamic characterization of Ambr250 mammalian vessel using CFD. • Key scale-up parameters (mixing time, bubble size, and kLa) assessed. • Validated CFD characterization of small-scale model using experimental results. • During scale-up, free surface mass transfer is important in small-scale vessels. • CFD assessment of small-scale bioreactor helps to understand cell damage. [ABSTRACT FROM AUTHOR]

The initial flow past an impulsively started rotating and translating circular cylinder is asymptotically analysed using a Brinkman penalization method on the Navier-Stokes equation. In our previous study (J. Fluid Mech., vol. 929, 2021, A31), the asymptotic solution was obtained within the second approximation with respect to the small parameter, ϵ, that is of the order of 1/λ. Here, λ is the penalization parameter. In addition, the Reynolds number based on the cylinder radius and the translating velocity is assumed to be of the order of ϵ. The previous study asymptotically analysed the initial flow past an impulsively started translating circular cylinder and investigated the influence of the penalization parameter λ on the drag coefficient. It was concluded that the drag coefficient calculated from the integration of the penalization term exhibits a half-value of the results of Bar-Lev & Yang (J. Fluid Mech., vol. 72, 1975, pp. 625-647) as λ→∞. Furthermore, the derivative of vorticity in the normal direction was found to be discontinuous on the cylinder surface, which is caused by the tangential gradient of the pressure on the cylinder surface. The present study hence aims to investigate the variance on the drag coefficient against the result of Bar-Lev & Yang (1975). First, we investigate the problem of an impulsively started rotating circular cylinder. In this situation, the moment coefficient is independent of the pressure on the cylinder surface so that we can elucidate the role of the pressure to the hydrodynamic coefficients. Then, the problem of an impulsively started rotating and translating circular cylinder is investigated. In this situation, the pressure force induced by the unsteady flow far from the cylinder is found to play a key role on the drag force for the agreement with the result of Bar-Lev & Yang (1975), whereas the variance still exists on the lift force. To resolve the variance, an alternative formula to calculate the hydrodynamic force is derived, assuming that there is the pressure jump between the outside and inside of the cylinder surface. The pressure jump is obtained in this analysis asymptotically. Of particular interest is the fact that this pressure jump can cause the variance on the lift force calculated by the integration of the penalization term. [ABSTRACT FROM AUTHOR]

In 1945-1949, Lars Onsager made an exact analysis of the high-Reynolds-number limit for individual turbulent flow realisations modelled by incompressible Navier-Stokes equations, motivated by experimental observations that dissipation of kinetic energy does not vanish. I review here developments spurred by his key idea that such flows are well described by distributional or 'weak' solutions of ideal Euler equations. 1/3 Hölder singularities of the velocity field were predicted by Onsager and since observed. His theory describes turbulent energy cascade without probabilistic assumptions and yields a local, deterministic version of the Kolmogorov 4/5th law. The approach is closely related to renormalisation group methods in physics and envisages 'conservation-law anomalies', as discovered later in quantum field theory. There are also deep connections with large-eddy simulation modelling. More recently, dissipative Euler solutions of the type conjectured by Onsager have been constructed and his 1/3 Hölder singularity proved to be the sharp threshold for anomalous dissipation. This progress has been achieved by an unexpected connection with work of John Nash on isometric embeddings of low regularity or 'convex integration' techniques. The dissipative Euler solutions yielded by this method are wildly non-unique for fixed initial data, suggesting 'spontaneously stochastic' behaviour of high-Reynolds-number solutions. I focus in particular on applications to wall-bounded turbulence, leading to novel concepts of spatial cascades of momentum, energy and vorticity to or from the wall as deterministic, space-time local phenomena. This theory thus makes testable predictions and offers new perspectives on large-eddy simulation in the presence of solid walls. [ABSTRACT FROM AUTHOR]

Data assimilation (DA) integrates experimental measurements into computational models to enable high-fidelity predictions of dynamical systems. However, the cost associated with solving this inverse problem, from measurements to the state, can be prohibitive for complex systems such as transitional hypersonic flows. We introduce an accurate and efficient deep-learning approach that alleviates this computational burden, and that enables approximately three orders of magnitude computational acceleration relative to variational techniques. Our method pivots on the deployment of a deep operator network (DeepONet) as an accurate, parsimonious and efficient meta-model of the compressible Navier–Stokes equations. The approach involves two main steps, each addressing specific challenges. Firstly, we reduce the computational load by minimizing the number of costly direct numerical simulations to construct a comprehensive dataset for effective supervised learning. This is achieved by optimally sampling the space of possible solutions. Secondly, we expedite the computation of high-dimensional assimilated solutions by deploying the DeepONet. This entails efficiently navigating the DeepONet's approximation of the cost landscape using a gradient-free technique. We demonstrate the successful application of this method for data assimilation of wind-tunnel measurements of a Mach 6, transitional, boundary-layer flow over a 7-degree half-angle cone. [ABSTRACT FROM AUTHOR]

Once there is a velocity gradient in a viscous fluid-flow, such as that across a shock wave, a viscous force and viscous energy loss exist inside the flow according to the Navier-Stokes equation, which may confuse the relative contribution of compressibility and viscosity. In this paper, a viscous shear vector is defined as the component of gradient vector of local velocity magnitude perpendicular to the velocity vector. Then, a local viscous energy flux vector is defined from the viscous shear vector after being multiplied by the viscosity and the velocity magnitude. The divergence of the viscous energy flux vector results in new expressions for viscous force and loss of viscous energy, in which all the square terms of derivative of velocity components correspond to irreversible energy loss. The rest part can be taken as a kind of mechanical energy transfer done by the viscous force, from which the viscous force components can be got based on the assumption that the viscous force vector is parallel to the velocity vector. The new equations are different from and more complex than those in the traditional Navier-Stokes equation. By the new theory, it is shown that there is no shear viscous force and shear viscous energy loss in the flow across a normal shock wave without velocity gradient perpendicular to the flow direction. [ABSTRACT FROM AUTHOR]

The tasks of analyzing and visualizing the dynamics of viscous incompressible flows of complex geometry based on traditional grid and projection methods are associated with significant requirements for computer performance necessary to achieve the set goals. To reduce the computational load in solving this class of problems, it is possible to apply algorithms for constructing artificial neural networks (ANNs) using exact solutions of the Navier–Stokes equations on a given set of spatial regions as training sets. An ANN is implemented to construct flows in regions that are complexes made up of training sets of standard axisymmetric domains (cylinders, balls, etc.). To reduce the amount of calculations in the case of 3D problems, invariant flow manifolds of lower dimensions are used. This makes it possible to identify the structure of solutions in detail. It is established that typical invariant regions of such flows are figures of rotation, in particular, ones homeomorphic to the torus, which form the structure of a topological bundle, for example, in a ball, cylinder, and general complexes composed of such figures. The structures of flows obtained by approximation based on the simplest 3D unsteady vortex flows are investigated. Classes of exact solutions of the incompressible Navier–Stokes system in bounded regions of are distinguished based on the superposition of the above-mentioned topological bundles. Comparative numerical experiments suggest that the application of the proposed class of ANNs can significantly speed up the computations, which allows the use of low-performance computers. [ABSTRACT FROM AUTHOR]

This paper presents a study on the magnetic and thermal behaviors of a planar toroidal transformer, comprising two planar toroidal coils. In our configuration, the primary coil consists of twenty turns, while the secondary coil consists of ten turns. This design combines the advantages of both toroidal and planar transformers: it employs flat coils, akin to those utilized in planar transformers, while retaining a toroidal shape for its magnetic core. This combination enables leveraging the distinctive characteristics of both transformer types. This study delves into electromagnetic and thermal behaviors. Electromagnetic behavior is elucidated through Maxwell's equations, offering insights into the distribution of magnetic fields, potentials, and electric current densities. Fluid flow is modeled via the Navier–Stokes equations. By coupling these equation sets, a more comprehensive and accurate portrayal of the thermal phenomena surrounding electrical equipment is attained. Such research is invaluable in the design and optimization of electrical systems, empowering engineers to forecast and manage thermal effects more efficiently. Consequently, this aids in enhancing the reliability, durability, and performance optimization of electrical equipment. The mathematical model was solved using the finite element method integrated into the COMSOL Multiphysics software v. 6.0. The COMSOL Multiphysics simulation showed correct behavior of potential, electric field, current density, and uniformly distributed temperature. In addition, this planar toroidal coil transformer model offers many advantages, such as small dimensions, high resonance frequency, and high operating reliability. This study made it possible to identify the range of its optimal functioning. [ABSTRACT FROM AUTHOR]

A discretization method with non-matching grids is proposed for the coupled Stokes-Darcy problem that uses a mortar variable at the interface to couple the marker and cell (MAC) method in the Stokes domain with the Raviart-Thomas mixed finite element pair in the Darcy domain. Due to this choice, the method conserves linear momentum and mass locally in the Stokes domain and exhibits local mass conservation in the Darcy domain. The MAC scheme is reformulated as a mixed finite element method on a staggered grid, which allows for the proposed scheme to be analyzed as a mortar mixed finite element method. We show that the discrete system is well-posed and derive a priori error estimates that indicate first order convergence in all variables. The system can be reduced to an interface problem concerning only the mortar variables, leading to a non-overlapping domain decomposition method. Numerical examples are presented to illustrate the theoretical results and the applicability of the method. [ABSTRACT FROM AUTHOR]

The governing equations for a viscous, incompressible fluid, in the thin-shell approximation, written in a coordinate frame which ensures validity at the North Pole, is simplified by invoking the f-plane approximation. Together with suitable stress conditions at the surface, describing the interaction of wind over ice and ice over water, a solution is developed for the Transpolar Drift current. This involves prescribing the near-surface geostrophic ocean flow and also, for ease of calculation, a simple version of the Ekman flow. Then, via the stress conditions at the surface, the near-surface wind consistent with these flows is obtained. An example, which exhibits the reduction in ice thickness as the flow moves over the North Pole towards the Fram Strait, and also describes the relative directions and speeds of the wind, ice and water, is presented in graphical form; this solution recovers what is observed. The considerable freedom available in this approach, allowing choices to be made for the geostrophic flow and the wind (both variable), and the ice profile at some initial position, is explained, opening the way to further, extensive investigations. [ABSTRACT FROM AUTHOR]

Poiseuille flows are considered as flows of Newtonian fluids through stationary pipes, with applications ranging from the pharmaceutical industries to manufacturing companies. These flows have been extensively studied in several works of literature due to their relevance in many spheres of life. However, the Poiseuille flow for non-Newtonian flows has not gained much attention, even though most fluids are non-Newtonian. Based on this, this study investigates the Poiseuille flow of the second-grade fluid. Second-grade fluids are viscoelastic non-Newtonian fluids that exhibit both shear-thinning and shear-thickening with applications found in several industrial applications such as pharmaceutical, cosmetics and polymer processing. The Poiseuille flow of second-grade fluid is formulated from the underlying Navier-Stokes' equations and the general assumptions of Poiseuille flow are invoked to reduce the equations to the regular ordinary differential equations. An analytical solution for the flow problem is sought using the method of separation of variables and the results are graphed to show the response of velocity and flow rate to the parameters of the flow. The outcomes show that velocity distribution reduces as the pipe radius increases and second-grade fluid has lower velocity than the Newtonian fluid. [ABSTRACT FROM AUTHOR]

In this paper, we construct a class of spherically symmetric and self-similar analytical solutions to the pressureless Navier–Stokes equations with density-dependent viscosity coefficients satisfying h (ρ) = ρ , g (ρ) = (− 1) ρ for all > 0. Under the continuous density free boundary conditions imposed on the free surface, we investigate the large-time behavior of the solutions according to various > 1 and 0 < < 1. When the time grows up, such solutions exhibit interesting information: Case (i) If the free surface initially moves inward, then the free surface infinitely approaches to the symmetry center and the fluid density blows up at the symmetry center, or the free surface tends to an equilibrium state; Case (ii) If the free surface initially moves outward, then the free surface infinitely expands outward and the fluid density decays and tends to zero almost everywhere away from the symmetry center, or the free surface tends to an equilibrium state. We also study the large-time behavior of the solutions for = 1 without any boundary conditions. [ABSTRACT FROM AUTHOR]

This is the first research on the smart control and frequency analysis of a cylindrical sandwich nanoshell in the framework of the numerical-based generalized differential quadrature method (2D-GDQM). The current sandwich smart nanostructure is made of a honeycomb core and piezoelectric face sheets as sensor and actuator (PFSA). For modeling the size-dependent nanoshell, nonlocal stress-strain gradient theory (NSGT) is presented. Also, this nanostructure is under conveying viscous fluid, and the related force is calculated by the modified formulation of Navier–Stokes. Also, the current structure rotates around its axial direction. The stresses and strains are obtained using the higher-order shear deformation theory (HSDT). The external voltage is applied to the sensor layer, and a Proportional-Derivative (PD) controller is used for sensor output control. Governing equations and boundary conditions of the cylindrical sandwich nanoshell are obtained by implementing Hamilton's principle. The results show that the geometry of honeycomb core, PD controller, velocity of fluid flow, length to radius ratio (L/R), and applied voltage and have a significant influence on the frequency characteristics of a cylindrical sandwich nanoshell. Another important consequence is that applying the PD controller leads to an increase in the critical velocity of fluid flow in the smart nanostructure. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the conservation of energy and magnetic helicity of weak solutions for the three-dimensional electron magnetohydrodynamic (EMHD) equations. Firstly, we establish sufficient conditions to guarantee the energy (magnetic helicity) balance of weak solutions for the EMHD equations based on the magnetic field, which can be viewed as an analogue of famous Lions' energy balance criterion of the Navier–Stokes equations for the EMHD equations. Secondly, in the spirit of recent works due to Berselli and Chiodaroli (Nonlinear Anal 192: 111704, 2020), as reported by Berselli (Three-Dimensional Navier–Stokes Equations for Turbulence. Academic Press, London, 2021), Berselli (Mathematics 11(4): 1–16, 2023), Berselli (J Differ Equ 368: 350–375, 2023), Berselli and Georgiadis (Nonlinear Differ Equ Appl 31(33): 1–14, 2024), we present energy (magnetic helicity) preservation criteria in terms of the current density in this system for both the whole space and the torus cases. [ABSTRACT FROM AUTHOR]

We introduce a vorticity Leray- α model with eddy viscosity depending on d (x , ∂ Ω) η where ∂ Ω is the boundary of the domain and η ∈ ] 0 ; 1 [ . We prove that this system admits fairly regular weak solutions converging when α goes to 0 to the solution of a reference system [ABSTRACT FROM AUTHOR]