Equations Of Incompressible Fluid Research Articles

Existence of energy-variational solutions to hyperbolic conservation laws

We introduce the concept of energy-variational solutions for hyperbolic conservation laws. Intrinsically, these energy-variational solutions fulfill the weak-strong uniqueness principle and the semi-flow property, and the set of solutions is convex and weakly-star closed. The existence of energy-variational solutions is proven via a suitable time-discretization scheme under certain assumptions. This general result yields existence of energy-variational solutions to the magnetohydrodynamical equations for ideal incompressible fluids and to the Euler equations in both the incompressible and the compressible case. Moreover, we show that energy-variational solutions to the Euler equations coincide with dissipative weak solutions.

NUMERICAL SIMULATION OF TURBULENT FLOW AROUND DARRIEUS AND SAVONIUS ROTORS

The study of unsteady processes during the flow around the rotors of vertical-axis wind turbines was carried out based on the Navier-Stokes equations. General definition of the coupled problem on vertical-axis wind turbines rotors aerodynamics and dynamics is formulated. The solution to the system of initial equations is obtained using an implicit finite-volume numerical algorithm based on the method of artificial compressibility and multi-block structured grids. A code for the calculation of vertical-axis wind turbines rotors aerodynamic and power characteristics is developed on the base of Reynolds averaged Navier-Stokes equations for incompressible fluids. Two types of Darrieus and Savonius rotors with different geometric parameters and number of blades were calculated. The influence of Reynolds numbers, tip-speed-ratios and solidity on the power characteristics of the rotors has been established.

Stability for a system of the 2D incompressible magneto-micropolar fluid equations with partial mixed dissipation

This paper focuses on the 2D incompressible anisotropic magneto-micropolar fluid equations with vertical dissipation, horizontal magnetic diffusion, and horizontal vortex viscosity. The goal is to investigate the stability of perturbations near a background magnetic field in the 2D magneto-micropolar fluid equations. Two main results are obtained. The first result is based on the linear system. Global existence for any large initial data and asymptotic linear stability are established. The second result explores stability for the nonlinear system. It is proven that if the initial data are sufficiently small, then the solution for some perturbations near a background magnetic field remains small. Additionally, the long-time behaviour of the solution is presented. The most challenging terms in the proof are the linear terms in the velocity equation and the micro-rotation equation that will grow with respect to time t. We are able to find some background fields to control the growth of the linear terms. Our results reveal that some background fields can stabilise electrically conducting fluids.

Structure and role of the pressure Hessian in regions of strong vorticity in turbulence

Amplification of velocity gradients, a key feature of turbulent flows, is affected by the non-local character of the incompressible fluid equations expressed by the second derivative (Hessian) of the pressure field. By analysing the structure of the flow in regions where the vorticity is the highest, we propose an approximate expression for the pressure Hessian in terms of the local vorticity, consistent with the existence of intense vortex tubes. Contrary to the often used simplification of an isotropic form for the pressure Hessian, which in effect inhibits vortex stretching, the proposed approximate form of the pressure Hessian enables much stronger vortex stretching. The prediction of the approximation proposed here is validated with results of direct numerical simulations of turbulent flows.

Three results on the energy conservation for the 3D Euler equations

We consider the 3D Euler equations for incompressible homogeneous fluids and we study the problem of energy conservation for weak solutions in the space-periodic case. First, we prove the energy conservation for a full scale of Besov spaces, by extending some classical results to a wider range of exponents. Next, we consider the energy conservation in the case of conditions on the gradient, recovering some results which were known, up to now, only for the Navier–Stokes equations and for weak solutions of the Leray-Hopf type. Finally, we make some remarks on the Onsager singularity problem, identifying conditions which allow to pass to the limit from solutions of the Navier–Stokes equations to solution of the Euler ones, producing weak solutions which are energy conserving.

Incompressible Schrödinger Flow with Heat-Transfer: An Introduction to the Analysis of Isotropic Fluid Dynamics in Sobolev Spaces for an Immersed Arbitrary Isothermal Geometry

This research investigates the intricate interplay of incompressible Schrödinger flow, heat-transfer, and the presence of an immersed isothermal body. The mathematical framework encompasses the Schrödinger equation for incompressible fluids, the heat transfer equation, and introduces a term that represents the thermal influence of an immersed isothermal geometry. Emphasizing the modeling and analysis of isotropic fluid dynamics, the study seeks to unravel the subtle relationship between the principles of quantum mechanics and the classical behavior of fluids. The initial discoveries produce an important theorem that leads the name of the authors, allowing new and valuable insights into the effects of the isothermal body immersed in a fluid medium. As a result, it was found that the term temperature source offers a unique perspective at the intersection of quantum mechanics and fluid dynamics.

Dissipation closure of fluid turbulence from energy spectrum

Direct method of the turbulence closure consists of obtaining expression for small-scale velocity and inserting this expression in the Reynolds tensor and in the transport equations for turbulent energy, dissipation, eddy and so on, implementing some averaging. Previously such approaches were used for Burgers-Lundgren vortex. Here we used ‘naïve’ perturbation method for the transport equation for small-scale velocity incompressible fluid - the equation suggested by one of the authors earlier. It is assumed that the energy spectrum of turbulence is given.

Boundedness of solutions for parabolic-elliptic predator-prey chemotaxis-fluid system with logistic source term

This paper considers the 2-species chemotaxis-Stokes system with competitive kinetics{(n1)t+u⋅∇n1=Δn1−χ∇⋅(n1∇w)+n1(λ1−μ1n1+an2),x∈Ω,t>0,(n2)t+u⋅∇n2=Δn2+ξ∇⋅(n2∇z)+n2(λ2−μ2n2−bn1),x∈Ω,t>0,wt+u⋅∇w=Δw−w+n2,x∈Ω,t>0,u⋅∇z=Δz−z+n1,x∈Ω,t>0,ut+∇P=Δu+(n1+n2)∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 under no-flux boundary conditions for n1, n2, w and z in three-dimensional bounded domains and no-slip boundary conditions for u, this is∂n1∂ν=∂n2∂ν=∂w∂ν=∂z∂ν=0,u=0,x∈∂Ω,t>0, where χ>0, ξ>0, μ1≥0, μ2≥0, λ1≥0, λ2≥0, a≥0, b≥0 and ϕ∈W2,∞(Ω). This system is a coupled system of the chemotaxis equations and viscous incompressible fluid equations. Under appropriate assumptions, this problem exhibits a global classical bounded solution.

Effect of vertical porous baffle on sloshing mitigation of two-layered liquid in a swaying tank

The sloshing behavior of two-layered liquid in a rectangular sloshing tank with a vertical porous baffle under sway excitation has been investigated using both analytical and numerical methods. The analytical model is developed using the matched eigenfunction expansion method in conjunction with linear potential flow theory. The existence of vertical porous baffle results in energy dissipation by applying the boundary condition that pressure drop across the baffle is proportional only to the quadratic drag term. For comparison, numerical simulations are conducted using computational fluid dynamics (CFD) with unsteady, implicit, and incompressible fluid equations. The study explores the effects of the liquid density ratio, the baffle's porosity, and its submergence depth, on various parameters including the free surface and interfacial elevations, hydrodynamic force on the tank surface, and horizontal force on the baffle. The trends observed in the analytical model align well with the CFD results and experiments. It is found that a fully submerged vertical porous baffle with a porosity close to P = 0.1 leads to stable and less dynamically active sloshing behavior in a two-layered liquid tank.

Nearly Toroidal, Periodic and Quasi-periodic Motions of Fluid Particles Driven by the Gavrilov Solutions of the Euler Equations

We consider the smooth, compactly supported solutions of the steady 3D Euler equations of incompressible fluids constructed by Gavrilov (Geom Funct Anal (GAFA) 29(1):190–197, 2019), and we study the corresponding fluid particle dynamics. This is an ode analysis, which contributes to the description of Gavrilov’s vector field.

Численное исследование двухмерного симметричного течения несжимаемой жидкости с зависящей от координат вязкостью между частично неровными плоскостями

The structure of a symmetric two-dimensional unsteady flow between two planes having a periodic step-like irregularity is investigated by numerical methods using the finite element package Freefem++. The kinematic viscosity of the fluid is chosen as typical for turbulent flows, depending on the coordinates as 𝜇 = 𝜇0(𝜂2(𝑥) − 𝑧2), 𝜂(𝑥) = ℎ(𝑥) + 𝛿, where 𝜇0 is the characteristic viscosity, 𝛿 is the roughness, ℎ(𝑥) is the function defining the boundary 𝑧 = 𝜂(𝑥) (profiles of planes mirror-symmetric with respect to 𝑧 = 0), 𝑥, 𝑧 are longitudinal and transverse coordinates along the main flow. In contrast to the usual parabolic profile of the Poiseuille flow between the planes, with the specified choice of viscosity, the flow profile is logarithmic and has a singularity at the boundary of the region, to prevent which the roughness of the boundary is introduced. A distinctive feature of the problem under study is the setting of boundary conditions on an irregular part of the boundary. On plane sections of the boundary, we maintain the usual non-slip conditions for a viscous liquid, but on the irregular part of the boundary we set only the impermeability conditions of the boundary for the liquid (no sticking!). Computational experiment - numerical solution of the Navier-Stokes equations for a viscous incompressible fluid using a modified penalty method, showed that for a wide set of parameters, the structure of the flow described by the initial non-stationary complete problem and the quasi-stationary flow described by the asymptotic model constructed in the presented paper, for which there is an exact solution, are in good agreement - in the non-stationary flow, a stable regular system of vortices is established concentrated near those surface irregularities for which the boundary has a negative curvature (`deepening’of boundary).

On Rayleigh–Taylor instability in Navier–Stokes–Korteweg equations

This paper focuses on the Rayleigh–Taylor instability in the two-dimensional system of equations of nonhomogeneous incompressible viscous fluids with capillarity effects in a horizontal periodic domain with infinite height. First, we use the modified variational method to construct (linear) unstable solutions for the linearized capillary Rayleigh–Taylor problem. Then, motivated by the Grenier’s idea in (Grenier in Commun. Pure Appl. Math. 53(9):1067–1091, 2000), we further construct approximate solutions with higher-order growing modes to the capillary Rayleigh–Taylor problem and derive the error estimates between both the approximate solutions and nonlinear solutions of the capillary Rayleigh–Taylor problem. Finally, we prove the existence of escape points based on the bootstrap instability method of Hwang–Guo in (Hwang and Guo in Arch. Ration. Mech. Anal. 167(3):235–253, 2003), and thus obtain the nonlinear Rayleigh–Taylor instability result. Our instability result presents that the Rayleigh–Taylor instability can occur in the fluids with capillarity effects for any capillary coefficient kappa >0 if the critical capillary coefficient is infinite. In particular, it improves the previous Zhang’s result in (Zhang in J. Math. Fluid Mech. 24(3):70–23, 2022) with the assumption of smallness of the capillary coefficient.

Extended Korteweg-de Vries equation for long gravity waves in incompressible fluid without strong limitation to surface deviation

We have derived the extended Korteweg-de Vries equation describing the long gravity waves without limitation to surface deviation. The only restriction to the surface deviation is connected with the stability condition for appropriate solutions. The derivation of extended KdV equation is based on the Euler equations for inviscid irrotational and incompressible fluid. It is shown that the extended KdV equation reduces to standard KdV equation for small amplitude of the waves. We have also generalized the extended KdV equation for describing the decaying effect of the waves. Quasi-periodic and solitary wave solutions for extended KdV equation with decaying effect are found as well. We also demonstrate that the fundamental approach based on the inverse scattering method is applicable for solving the extended KdV equation in the case when decaying effect is negligibly small. Such case always occur for restricted propagation distances of the waves.

An evolving space framework for Oseen equations on a moving domain

This article considers non-stationary incompressible linear fluid equations in a moving domain. We demonstrate the existence and uniqueness of an appropriate weak formulation of the problem by making use of the theory of time-dependent Bochner spaces. It is not possible to directly apply established evolving Hilbert space theory due to the incompressibility constraint. After we have established the well-posedness, we derive and analyse a first order time discretisation of the system.

Spatial Asymptotic Expansions in the Navier–Stokes Equation

Abstract We prove that the Navier–Stokes equation for a viscous incompressible fluid in ${\mathbb {R}}^{d}$ is locally well-posed in spaces of functions allowing spatial asymptotic expansions with log terms as $|x|\to \infty $ of any a priori given order. The solution depends analytically on the initial data and time so that for any $0<\vartheta <\pi /2$ it can be holomorphically extended in time to a conic sector in ${\mathbb {C}}$ with angle $2\vartheta $ at zero. We discuss the approximation of solutions by their asymptotic parts.

Role of preferential hydration on diffusiophoresis of globular proteins

Diffusiophoresis is the migration of a colloidal particle through a fluid, caused by a cosolute concentration gradient. Preferential hydration is a thermodynamic phenomenon responsible for particle diffusiophoresis from high to low cosolute concentration in water. Here, steady-state Navier-Stokes equation for incompressible fluids, with particle modeled as an inpenetrable rigid sphere, is employed for deriving a mathematical expression of the diffusiophoresis coefficient. Mean-force potentials with arbitrary interaction range but with interaction energies small or comparable with thermal energy are employed to describe water-mediated particle-cosolute interactions. This choice of potential is movitated by previous work showing that only fluid particles weakly interacting with the colloildal particle can cause diffusiophoresis. Preferential hydration has been extensively investigated for proteins. Thus, the proposed model is applied to available experimental diffusiophoresis coefficients of lysozyme in aqueous NaCl and KCl at pH 4.5 and 25 °C. This analysis also includes data on salt osmotic diffusion, describing salt diffusion caused by protein concentration gradients. The preferential-hydration contribution of experimental lysozyme diffusiophoresis is obtained after removing electrophoretic diffusiophoresis. Examination of these experimental data shows that the mean-force potential describing preferential hydration is indeed weak, with an interaction range that is an appreciable fraction of particle size.

Computational fluid dynamics simulations of silicon μ-structure formation during excimer laser modification of silicon-nanoparticle layers

The laser modification of silicon-nanoparticle layers with a nanosecond pulsed excimer laser leads to the self-organized formation of crystalline, μ-cone-shaped silicon structures with good electronic properties, which have allowed the demonstration of their potential for printed flexible electronics. With the current nanosecond laser process, silicon exhibits only short melting times, resulting in a method-defined substrate contact angle, instead of this parameter being defined by the substrate surface energy as expected for equilibrium conditions. This substrate material-independent non-equilibrium contact angle of the silicon melt was experimentally determined in this study to be Θ = 68 ± 10°. To gain deeper insight into the process of the sequential melting and the formation of the silicon μ-cone structures during laser modification, a two-dimensional computational fluid dynamics simulation was performed in COMSOL Multiphysics® solving the Navier–Stokes equation for incompressible fluids. The simulation uses an effective medium approach by applying the conservative level set method to describe the porous silicon-nanoparticle layer. Its sequential melting during the pulsed laser modification is modeled using a newly developed simulation methodology, which uses a time- and depth-dependent dynamic viscosity of the molten silicon. The simulation was carried out for different laser energy densities and verified using scanning electron microscopy images of corresponding laser-modified samples. The simulation results agree well with the experiment and were subsequently used to optimize the laser process.

Energy conservation for weak solutions of incompressible fluid equations: The Hölder case and connections with Onsager's conjecture

In this paper we give elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes equations in the class of Hölder continuous functions, relaxing some of the assumptions on the time variable (both integrability and regularity at initial time) and presenting them in a unified way.Then, in the final section we prove (for the Navier-Stokes equations) a result of energy conservation in presence of a solid boundary and with Dirichlet boundary conditions. This result seems the first one –in the viscous case– with Hölder type hypotheses, but without additional assumptions on the pressure.

Self-similar solutions to the hypoviscous Burgers and SQG equations at criticality

After reviewing the source-type solution of the Burgers equation with standard dissipativity, we study the hypoviscous counterpart of the Burgers equation. (1) We determine an equation that governs the near-identity transformation underlying its self-similar solution. (2) We develop its approximation scheme and construct the first-order approximation. (3) We obtain the source-type solution numerically by the Newton–Raphson iteration scheme and find it to agree well with the first-order approximation. Implications of the source-type solution are given, regarding the possibility of linearisation of the hypoviscous Burgers equation. Finally we address the problems of the incompressible fluid equations in two dimensions, centering on the surface quasi-geostrophic equation with standard and hypoviscous dissipativity.

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

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