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The Lattice Boltzmann Method (LBM) is an approach for modeling mesoscopic fluid flow and heat transfer, based on modeling distributions of particles moving and colliding on a lattice. Using a perturbative formulation of the Boltzmann equation, it scales to the macroscopic Navier–Stokes equation. We simulate natural thermal convection via LBM in a 2D rectangular box being heated from below and cooled from above, and use the results as training, testing, and generalization datasets to build a deep learning model. GoogLeNet, a convolutional neural network, is used to classify the simulation results based on two parameters: Rayleigh (R a) and Prandtl (P r) numbers, from a single snapshot of either the entire modeling field of resolution 1024 × 1024 , or a 224 × 224 crop. For each fixed P r in a range from 1 to 128, increasing by a factor of 2, we estimate R a with an accuracy varying from 40% to 90%, depending on the chosen augmentation strategy. For each fixed R a in the range from 10 5 to 10 9 , increasing of a factor 10 , the method predicts P r with a systematically lower accuracy ranging from 30% to 80%. This approach has great potential for industrial applications like being able to control the industrial flow or scientific research on geophysical ones including the transport of heat in the earth's interiors, ocean, and atmosphere. [ABSTRACT FROM AUTHOR]

In this paper we consider the vanishing viscosity limit of solutions to the initial boundary value problem for the compressible viscoelastic equations in the half space. When the initial deformation gradient does not degenerate and there is no vacuum initially, we establish the uniform regularity estimates of solutions to the initial-boundary value problem for the three-dimensional compressible viscoelastic equations in the Sobolev spaces. Then we justify the vanishing viscosity limit of solutions of the compressible viscoelastic equations based on the uniform regularity estimates and the compactness arguments. Both the no-slip boundary condition and the Navier-slip type boundary condition on velocity are addressed in this paper. On the one hand, for the corresponding vanishing viscosity limit of the compressible Navier-Stokes equations with the no-slip boundary condition, it is impossible to derive such uniform energy estimates of solutions due to the appearance of strong boundary layers. Consequently, our results show that the deformation gradient can prevent the formation of strong boundary layers. On the other hand, our results also provide two different kinds of boundary conditions suitable for the well-posedness of the initial-boundary value problem of the elastodynamic equations via the method of vanishing viscosity. Finally, it is worth noting that we take advantage of the Lagrangian coordinates to study the vanishing viscosity limit for the fixed boundary problem in this paper. [ABSTRACT FROM AUTHOR]

The Kolmogorov flow represents the stationary sinusoidal solution (sin ⁡ y , 0) to a two-dimensional spatially periodic Navier-Stokes system, driven by an external force. This system admits the additional non-stationary solution (sin ⁡ y , 0) + e − ν t (sin ⁡ y , 0) , which tends exponentially to the Kolmogorov flow at the minimum decay rate determined by the viscosity ν. Enhanced damping or enhanced dissipation of the problem is obtained by presenting higher decay rate for the difference between a solution and the non-stationary basic solution. Moreover, for the understanding of the metastability problem in an explicit manner, a variety of exact solutions are presented to show enhanced and unenhanced dampings. [ABSTRACT FROM AUTHOR]

We study the dynamics of a coupled system, formed by a rigid body with a cavity entirely filled with magnetohydrodynamic compressible fluid. Our aim is to derive the global existence of the unique classical solutions and weak solutions to this system. Moreover, we show the weak-strong uniqueness principle which means that a weak solution coincides with a strong solution on the time existence of a strong solution, provided they emanate from the same initial data. [ABSTRACT FROM AUTHOR]

We present the hydrodynamic limit of the nonlinear Maxwell-Schrödinger equations. Under the a priori boundedness assumption on the density, we show that the Maxwell-Schrödinger system converges to the compressible Euler-Maxwell system as the Planck constant tends to 0. Our analysis is based on the modulated energy estimate. [ABSTRACT FROM AUTHOR]

In this note we investigate interior regularity criteria for suitable weak solutions to the 3D Naiver-Stokes equations, and obtain the solutions are regular in the interior if the L t p L x q (Q 1) norm of the velocity is sufficiently small, where 1 ≤ 2 p + 3 q < 2 and 2 ≤ p ≤ ∞. It improves the recent result of p , q > 2 by Kwon [15] , and also generalizes Chae-Wolf's L t ∞ L x 3 2 + criterion [3]. [ABSTRACT FROM AUTHOR]

In this work, we address parametric non–stationary fluid dynamics problems within a model order reduction setting based on domain decomposition. Starting from the optimisation–based domain decomposition approach, we derive an optimal control problem, for which we present a convergence analysis in the case of non–stationary incompressible Navier–Stokes equations. We discretise the problem with the finite element method and we compare different model order reduction techniques: POD–Galerkin and a non–intrusive neural network procedures. We show that the classical POD–Galerkin is more robust and accurate also in transient areas, while the neural network can obtain simulations very quickly though being less precise in the presence of discontinuities in time or parameter domain. We test the proposed methodologies on two fluid dynamics benchmarks with physical parameters and time dependency: the non–stationary backward–facing step and lid–driven cavity flow. • Optimisation-based domain-decomposition algorithm for non-stationary computational fluid dynamics. • Convergence analysis for the domain-decomposition optimal control problem. • Comparison of the intrusive POD-Galerkin and a non-intrusive Neural Networks based approaches. [ABSTRACT FROM AUTHOR]

This paper proposes a low-cost, penalty parameter-free, and pressure-robust Stokes solver based on the enriched Galerkin (EG) method with a discontinuous velocity enrichment function. The EG method employs the interior penalty discontinuous Galerkin (IPDG) formulation to weakly impose the continuity of the velocity function. However, despite its advantage of symmetry, the symmetric IPDG formulation requires a lot of computational effort to choose an optimal penalty parameter and compute different trace terms. To reduce such effort, we replace the derivatives of the velocity function with its weak derivatives computed by the geometric data of elements. Therefore, our modified EG (mEG) method is a penalty parameter-free numerical scheme that has reduced computational complexity and conserves the optimal convergence orders. Moreover, we achieve pressure robustness for the mEG method by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. The theoretical results are confirmed through numerical experiments with two- and three-dimensional examples. [ABSTRACT FROM AUTHOR]

The DPG method with optimal test functions for solving linear quadratic optimal control problems with control constraints is studied. We prove existence of a unique optimal solution of the nonlinear discrete problem and characterize it through first-order optimality conditions. Furthermore, we systematically develop a priori as well as a posteriori error estimates. Our proposed method can be applied to a wide range of constrained optimal control problems subject to, e.g., scalar second-order PDEs and the Stokes equations. Numerical experiments that illustrate our theoretical findings are presented. [ABSTRACT FROM AUTHOR]

The mixing and combustion processes are linked and challenging for a supersonic combustion engine. Supersonic air doesn't spend much time in the combustor, making it challenging to blend fuel with supersonic air. The expansion of shear-mixing layer with shock waves and their interactions are the primary characteristics of mixing augmentation. This paper investigates the same methodology for a model scramjet combustion chamber with wedge and wavy sides. The novel scramjet combustion models include wedge and way-wall sidewalls based on the basic scramjet model. By combining the finite volume approach with a second-order upwind discretization method, the Reynolds-averaged Navier-Stokes equations are numerically analyzed. The interplay between wedge and wavy-wall effects and free stream flows is taken into consideration by using the shear stress transport (SST) k−ω turbulence model. The effect of the wedge and wavy wall combustor has been evaluated by analysing the shock production and their interactions with the mixing layer and internal flow structure. The turbulence characteristics are also examined to analyse the degree of flow disturbance for wedge and wavy-wall combustion. The wavy wall combustor model results in a 14.5% increase in turbulence intensity, which raises air-fuel mixing and there by improves mixing and combustion efficiency by 12.59% and 11.23%, compared to the base wedge type combustor. [Display omitted] • Shock-induced mixing and combustion in the scramjet combustor with wavy and wedge walls. • The combustor's flow structure has been evaluated for shock wave production and interactions. • The wavy wall combustor promotes combustion in the radial direction. • The wavy wall combustor increases turbulent intensity by 14.5%, improving air-fuel mixing. • Wavy wall combustor improve mixing and combustion efficiency by 12.59% and 11.23%, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial-boundary value problems of the compressible isentropic Navier-Stokes equations with density-dependent viscosity on two dimensional solid balls which was first introduced by Kazhikhov where shear viscosity μ is assumed to be constant and the bulk viscosity λ is a polynomial of density up to power β. Under the condition of β > 1 , we prove the global existence of the radially symmetric strong solutions to the Kazhikhov models under Dirichlet boundary conditions for arbitrary large initial smooth data. Moreover, the density is shown to be uniformly bounded with respect to time when β ∈ (max ⁡ { 1 , γ + 2 4 } , γ ]. This improves the previous result of [14,15] for general 2D domains where they require β > 4 / 3 to ensure global existence and is the first result concerning the global existence of classical solutions to the radially symmetric compressible Navier-Stokes equations in 2D solid balls under Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

We consider the homogenisation of the quasi-stationary Stokes equations in a porous medium that evolves over time. The evolution is a priori given. At the interface of the pore space and the solid part, we prescribe an inhomogeneous Dirichlet boundary condition, which enables a no-slip boundary condition at the evolving boundary. We pass rigorously to the homogenisation limit employing the two-scale transformation method. In order to derive uniform a priori estimates, we show a Korn-type inequality for the two-scale transformation method. The homogenisation result is a new version of Darcy's law. It features a time- and space-dependent permeability tensor, which accounts for the local pore structure, and a macroscopic inhomogeneous divergence condition, which induces a new source term for the pressure. In the case of a no-slip boundary condition at the interface, this source term relates to the change of the local pore volume. [ABSTRACT FROM AUTHOR]

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]

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]

• 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 paper, we describe an upgrade of the Alya code with up-to-date parallel linear solvers capable of achieving reliability, efficiency and scalability in the computation of the pressure field at each time step of the numerical procedure for solving a Large Eddy Simulation formulation of the incompressible Navier–Stokes equations. We developed a software module in the Alya's kernel to interface the libraries included in the current version of PSCToolkit, a framework for the iterative solution of sparse linear systems, on parallel distributed-memory computers, by Krylov methods coupled to Algebraic MultiGrid preconditioners. The Toolkit has undergone various extensions within the EoCoE-II project with the primary goal of facing the exascale challenge. Results on a realistic benchmark for airflow simulations in wind farm applications show that the PSCToolkit solvers significantly outperform the original versions of the Conjugate Gradient method available in the Alya's kernel in terms of scalability and parallel efficiency and represent a very promising software layer to move the Alya code toward exascale. [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]

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]

In this paper, a second‐order time discretizing block‐centered finite difference method is introduced to solve the compressible wormhole propagation. The optimal second‐order error estimates for the porosity, pressure, velocity, concentration and its flux are established carefully in different discrete norms on non‐uniform grids. Then by introducing Lagrange multiplier, a novel bound‐preserving scheme for concentration is constructed. Finally, numerical experiments are carried out to demonstrate the correctness of theoretical analysis and capability for simulations of compressible wormhole propagation. [ABSTRACT FROM AUTHOR]

We show that the discrete velocity solution converges at the optimal order when solving the steady state Stokes equations by the Pk$$ {P}_k $$‐Pk−1disc$$ {P}_{k-1}^{\mathrm{disc}} $$ mixed finite element method for k≥4$$ k\ge 4 $$ on 2D triangular grids or k≥6$$ k\ge 6 $$ on tetrahedral grids, even in the case the inf‐sup condition fails. By a simple L2$$ {L}^2 $$‐projection of the discrete Pk−1$$ {P}_{k-1} $$ pressure to the space of continuous Pk−1$$ {P}_{k-1} $$ polynomials, we show this post‐processed pressure solution also converges at the optimal order. Both 2D and 3D numerical tests are presented, verifying the theory. [ABSTRACT FROM AUTHOR]

We introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for the dual-porosity-Stokes problem. This coupled problem describes the interaction between free flow in macrofractures/conduits, governed by the Stokes equations, and flow in microfractures/matrix, governed by a dual-porosity model. We prove that the HDG method is strongly conservative, well-posed, and give an a priori error analysis showing dependence on the problem parameters. Our theoretical findings are corroborated by numerical examples. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the three-dimensional globally modified Navier–Stokes equations driven by additive white noise on some unbounded domains 풪. By using the Ornstein–Uhlenbeck process, we transfer the original equation to a random dynamical system, and then we prove the existence of pullback attractors for the random dynamical system equations under suitable conditions. Due to the unboundedness of the domains, we get the asymptotic compactness of the solutions by Ball’s idea of energy equations. The periodicity of the attractors is also proved when the deterministic non-autonomous external terms are periodic in time. [ABSTRACT FROM AUTHOR]

In the diffuse interface theory, the motion of two incompressible viscous fluids and the evolution of their interface are described by the well-known model H. The model consists of the Navier–Stokes equations, nonlinearly coupled with a convective Cahn–Hilliard type equation. Here we consider a stochastic version of the model driven by a noise of Lévy type, and where the standard Cahn–Hilliard equation is replaced by its nonlocal version with a singular (e.g., logarithmic) potential. The case of smooth potentials with arbitrary polynomial growth has been already analyzed in [G. Deugoué, A. Ndongmo Ngana and T. Tachim Medjo, Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type, Phys. D 398 2019, 23–68]. Taking advantage of this previous result, we investigate this more challenging and physically relevant case. Global existence of a martingale solution is proved with no-slip and no-flux boundary conditions in both 2D and 3D bounded domains. In the two-dimensional case, we prove the uniqueness of weak solutions when the viscosity is constant. [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]

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]

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]

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]

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]

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]

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 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]

Purpose: This paper aims to investigate the tribological benefits of a biomimetic teardrop surface texture inspired by snakeskin compared to conventional surface textures with the help of geometrical and flow parameters using computational fluid dynamics techniques. Design/methodology/approach: The lubricant is assumed to be Newtonian, and the flow is laminar with constant viscosity and isothermal property. The governing equations, continuity and Navier–Stokes equation, are discretised by the finite volume method, and cavitation modelling is included. The discretisation for the momentum equations is carried out using the second-order difference method for the SIMPLEC algorithm of pressure–velocity coupling. Findings: The results indicate that biomimetic teardrop surface texturing performs better than conventional shapes surface textures in improving tribological performance. Furthermore, the parallel texture orientation along with the flow generates a high-pressure distribution relative to other orientations. Surface texture area density also highly influences the load-carrying capacity, which is optimum at 29%. Zigzag pattern arrangement performs better compared to linear pattern arrangement of texturing. Originality/value: The paper proposes that this unique biomimetic teardrop shape can give better tribological performance than conventional shapes. Peer review: The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-02-2024-0053/ [ABSTRACT FROM AUTHOR]

The dynamic behaviour of wildfires is mainly influenced by weather, fuel, and topography. Based on fundamental conservation laws involving numerous physical processes and large scales, atmospheric models require substantial computational resources. Therefore, coupling wildfire and atmospheric models is impractical for high resolutions. Instead, a static atmospheric wind field is typically input into the wildfire model, either as boundary conditions on the control surface or distributed over the control volume. Wildfire propagation models may be (i) data-driven ; (i i) theoretical ; or (i i i) mechanistic surrogates. Data-driven models are beyond the scope of this paper. Theoretical models are based on conservation laws (species, energy, mass, momentum) and are, therefore, computationally intensive; e.g. the Fire Dynamics Simulator (FDS). Mechanistic surrogate models do not closely follow fire dynamics laws, but related laws observed to make predictions more efficiently with sufficient accuracy; e.g. FARSITE, and FDS with the Level Set model (FDS-LS). Whether theoretical or mechanistic surrogate, these wildfire models may be coupled with or decoupled from wind models (e.g. Navier-Stokes equations). Only coupled models account for the effect of the fire on the wind field. In this paper, a series of simulations of wildfire propagation on grassland are performed using FDS-LS to study the impact of the fire-induced wind on the fire propagation dynamics. Results show that coupling leads to higher Rates of Spread (RoS), closer to those reported from field experiments, with increasing wind speeds and higher terrain slopes strengthening this trend. Aiming to capture the fire–wind interaction without the hefty cost of solving Navier-Stokes equations, a conceptual framework is proposed: 1) a swarm of unmanned aerial vehicles measure wind velocities at flight height; 2) the wind field is constructed with the acquired data; 3) the high-altitude wind field is mapped to near-surface, and 4) the near-surface wind field is fed into the wildfire model periodically. A series of simulations are performed using an in-house decoupled physics-based reduced-order fire propagation model (FireProM-F) enhanced by wind field "measurements". In this proof of concept, wind velocities are not measured but extracted from physics-based Large Eddy Simulations taken as ground truth. Unsurprisingly, higher measurement frequencies lead to more accurate and realistic predictions of the propagating fire front. An initial attempt is made to study the effect of wind measurement uncertainty on the model predictions by adding Gaussian noise. Preliminary results show that higher noise leads to the fire front displaying more irregular shapes and slower propagation. • Higher wildfire model prediction accuracy normally achieved by coupling it with wind model such as Navier-Stokes equations. • Coupled fire-wind models often too computationally expensive for real-time use in operational contexts. • Prediction accuracy of wildfire model uncoupled from wind model enhanced by periodic inputs of updated wind field. • Wind field to be constructed from UAV swarm measurements, surrogated here to velocities obtained from full CFD simulations. • Proposed method to enhance wildfire model predictions robust to Gaussian noise added to represent measurement uncertainty. [ABSTRACT FROM AUTHOR]

In thermal spraying, molten YSZ (Yttria Stabilized Zirconia) drop impact is crucial to the understanding of formation of splats and hence coatings. However, existing studies hardly cast attention to the possible rebounding dynamics when successive molten YSZ drops impinge onto newly solidified splats whose temperature is still high so that freezing may be delayed. This work uses a newly developed numerical model based on the conservative level set method to probe into the rebounding dynamics of YSZ drops in practical thermal spraying conditions. The free-energy-based surface tension model in the phase field method was introduced to avoid the computing of curvature, which is however necessary in the continuum surface tension model. The projection method was invoked to solve the Navier–Stokes equations. The model was validated against an experiment of drop impact with rebounding, showing reasonable agreement. Moreover, the model was applied to both dense and hollow YSZ drop impact, with the details of fluid flow being analyzed. The maximum spread factor was found in agreement with existing scaling laws. [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]

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]

This paper deals with the semilinear Rayleigh–Stokes equation with the fractional derivative in time of order α ∈ (0 , 1) , which can be used to model anomalous diffusion in viscoelastic fluids. An operator family related to this problem is defined, and its regularity properties are investigated. We firstly give the concept of the mild solutions in terms of the operator family and then obtain the existence of global mild solutions by means of fixed point technique. Moreover, the existence and regularity of classical solutions are given. [ABSTRACT FROM AUTHOR]

In this paper, we mainly consider global well-posedness and long time behavior of compressible Navier–Stokes equations without heat conduction in L p -framework. This is a generalization of Peng and Zhai (SIMA 55(2):1439–1463, 2023), where they obtained the corresponding result in L 2 -framework. Based on the key observation that we can release the regularity of non-dissipative entropy S in high frequency in Peng and Zhai (2023), we ultimately achieve the desired L p estimate in the high frequency via complicated calculations on the nonlinear terms. In addition, we get the L p -decay rate of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the optimal instability threshold of the Couette flow for Navier–Stokes equations with small viscosity ν>0$\nu >0$, when the perturbations are in the critical spaces Hx1Ly2$H^1_xL_y^2$. More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size ν12−δ0$\nu ^{\frac{1}{2}-\delta _0}$ with any small δ0>0$\delta _0>0$, which implies that ν12$\nu ^{\frac{1}{2}}$ is the sharp stability threshold. In our method, we prove a transient exponential growth without referring to eigenvalue or pseudo‐spectrum. As an application, for the linearized Euler equations around shear flows that are near the Couette flow, we provide a new tool to prove the existence of growing modes for the corresponding Rayleigh operator and give a precise location of the eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper, we show the global regularity and the optimal decay of weak solutions to the generalized Leray problem with critical dissipation. Our approach hinges on the maximal smoothing effect, L^{p}-type elliptic regularity of linearization, and the action of the heat semigroup generated by the fractional powers of Laplace operator on distributions with Fourier transforms supported in an annulus. As a by-product, we construct a self-similar solution to the three-dimensional incompressible Navier-Stokes equations. Most notably, we prove the global regularity and the optimal decay without the need for additional requirements found in existing literatures. [ABSTRACT FROM AUTHOR]

Inspired by some experimental (numerical) works on fractional diffusion PDEs, we develop a rigorous framework to prove that solutions to the fractional Navier–Stokes equations, which involve the fractional Laplacian operator (- Δ) α 2 with α < 2 , converge to a solution of the classical case, with - Δ , when α goes to 2. Precisely, in the setting of mild solutions, we prove uniform convergence in the L t , x ∞ -space and derive a precise convergence rate, revealing some phenomenological effects. As a bi-product, we prove strong convergence in the L t p L x q -space. Finally, our results are also generalized to the coupled setting of the Magnetic-hydrodynamic system. [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 Keller–Segel–Navier–Stokes system governs chemotaxis in liquid environments. This system is to be solved for the organism and chemoattractant densities and for the fluid velocity and pressure. It is known that if the total initial organism density mass is below 2π there exist globally defined generalised solutions, but what is less understood is whether there are blow-up solutions beyond such a threshold and its optimality. Motivated by this issue, a numerical blow-up scenario is investigated. Approximate solutions computed via a stabilised finite element method founded on a shock capturing technique are such that they satisfy a priori bounds as well as lower and L1(Ω) bounds for the organism and chemoattractant densities. In particular, these latter properties are essential in detecting numerical blow-up configurations, since the non-satisfaction of these two requirements might trigger numerical oscillations leading to non-realistic finite-time collapses into persistent Dirac-type measures. Our findings show that the existence threshold value 2π encountered for the organism density mass may not be optimal and hence it is conjectured that the critical threshold value 4π may be inherited from the fluid-free Keller–Segel equations. Additionally it is observed that the formation of singular points can be neglected if the fluid flow is intensified. [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]