Incompressible Inviscid Research Articles

A regularity result for incompressible elastodynamics equations in the ALE coordinates

Abstract We consider incompressible inviscid elastodynamics equations with a free surface and establish regularity of solutions for these equations. Compared with previous result on this free boundary problem [X. Gu and F. Wang, Well-posedness of the free boundary problem in incompressible elastodynamic under the mixed type stability condition, J. Math. Anal. Appl., 482, (2020), 123529] in space H 3 , we are able to establish regularity in space H 2.5+δ upon the Arbitrary Lagrangian-Eulerian (ALE) coordinates. It is achieved by reformulating the system into a new formulation with the ALE coordinates, presenting uniform estimates for the pressure, tangential estimates for the system, as well as curl and divergence estimates.

Global well-posedness and large time behavior for the inviscid Oldroyd-B model

In this paper, we consider the 3-dimensional incompressible inviscid Oldroyd-B model. Firstly, we establish the global existence of the solutions for the inviscid Oldroyd-B model with different coupling coefficient k>0. Then, we show the connection between the solution with the parameter k that is the reciprocal of Weissenberg number. On one hand, we prove that the solutions (u,τ) depend continuously on the parameter k>0, though (u,τ) corresponds to different decays rate for different k. On the other hand, when k→0 in large time, we prove that there is a gap between the L2 norm of solution uk(t,x) to the model of parameter k>0 with the L2 norm of solution u0(t,x) to the model of parameter k=0. In a word, we prove that the larger positive k induces the faster decay rate of the solution, but k cannot go to zero, or the dissipation will vanish instantly.

Stability Analysis of Two-Dimensional Ideal Flows With Applications to Viscous Fluids and Plasmas

Abstract We are interested in the stability analysis of two-dimensional incompressible inviscid fluids. Specifically, we revisit a known recent result on the stability of Yudovich’s solutions to the incompressible Euler equations in $L^{\infty }([0,T];H^{1})$ by providing a new approach to its proof based on the idea of compactness extrapolation and by extending it to the whole plane. This new method of proof is robust and, when applied to viscous models, leads to a remarkable logarithmic improvement on the rate of convergence in the vanishing viscosity limit of two-dimensional fluids. Loosely speaking, this logarithmic gain is the result of the fact that, in appropriate high-regularity settings, the smoothness of solutions to the Euler equations at times $t\in [0,T)$ is strictly higher than their regularity at time $t=T$. This “memory effect” seems to be a general principle that is not exclusive to fluid mechanics. It is therefore likely to be observed in other setting and deserves further investigation. Finally, we also apply the stability results on Euler systems to the study of two-dimensional ideal plasmas and establish their convergence, in strong topologies, to solutions of magnetohydrodynamic systems, when the speed of light tends to infinity. The crux of this asymptotic analysis relies on a fine understanding of Maxwell’s system.

Fluid Cohomology

The vorticity-streamfunction formulation for incompressible inviscid fluids is the basis for many fluid simulation methods in computer graphics, including vortex methods, streamfunction solvers, spectral methods, and Monte Carlo methods. We point out that current setups in the vorticity-streamfunction formulation are insufficient at simulating fluids on general non-simply-connected domains. This issue is critical in practice, as obstacles, periodic boundaries, and nonzero genus can all make the fluid domain multiply connected. These scenarios introduce nontrivial cohomology components to the flow in the form of harmonic fields. The dynamics of these harmonic fields have been previously overlooked. In this paper, we derive the missing equations of motion for the fluid cohomology components. We elucidate the physical laws associated with the new equations, and show their importance in reproducing physically correct behaviors of fluid flows on domains with general topology.

A regularity result for the incompressible inviscid magnetohydrodynamics equations in the Arbitrary Lagrangian-Eulerian coordinates

Consider herein the incompressible inviscid magnetohydrodynamics equations in a moving domain with free surface boundary conditions. Without smallness assumption in the fluid volume, a priori estimates for solutions of this model are established when the initial data in H2.5+δ (δ>0) upon the Arbitrary Lagrangian-Eulerian coordinates under the boundary condition p+12|B|2=0. Indeed, this is achieved by reformulating the system into a new formulation with the Arbitrary Lagrangian-Eulerian variables, presenting the uniform estimates for the pressure, the tangential estimates for the system, as well as the curl and divergence estimates.

Quantum complexity as hydrodynamics

As a new step towards defining complexity for quantum field theories, we map Nielsen operator complexity for $SU(N)$ gates to two-dimensional hydrodynamics. We develop a tractable large $N$ limit that leads to regular geometries on the manifold of unitaries as $N$ is taken to infinity. To achieve this, we introduce a basis of non-commutative plane waves for the $\mathfrak{su}(N)$ algebra and define a metric with polynomial penalty factors. Through the Euler-Arnold approach we identify incompressible inviscid hydrodynamics on the two-torus as a novel effective theory of large-qudit operator complexity. For large $N$, our cost function captures two essential properties of holographic complexity measures: ergodicity and conjugate points.

The 2D inviscid Boussinesq equations with fractional diffusion in bounded domain

The incompressible Boussinesq equations serve as an important model in geophysics especially in the study of Rayleigh–Bénard convection. This paper focuses on the 2D incompressible inviscid Boussinesq equations with fractional diffusion (−Δ)βθ in bounded domain, equipping with the slip boundary condition for velocity vector field and Dirichlet boundary condition for temperature. We obtain the global existence and uniqueness of classical solutions in the range of β∈[3/4,1) and also show the local corresponding result for β∈[0,1). To the best of our knowledge, this is the first paper considering the Boussinesq equation with fractional diffusion in bounded domain. This result extends the work by K. Zhao (2010) to a fractional dissipation for temperature in bounded domain.

Filling and emptying a tank of liquid

A right cylindrical tank open to the atmosphere is being filled by a laminar jet of incompressible inviscid liquid falling onto its free surface. At the same time, fluid is escaping through a hole centered in the bottom of the tank. Newton’s second law for variable mass and the unsteady Bernoulli equation are combined to find the time dependence of the liquid height in the tank. The level of analysis is suitable for an introductory undergraduate course in fluid dynamics.

Relaxed solutions for incompressible inviscid flows: a variational and gravitational approximation to the initial value problem.

Following Arnold's geometric interpretation, the Euler equations of an incompressible fluid moving in a domain [Formula: see text] are known to be the optimality equation of the minimizing geodesic problem along the group of orientation and volume preserving diffeomorphisms of D. This problem admits a well-established convex relaxation that generates a set of 'relaxed', 'multi-stream', version of the Euler equations. However, it is unclear that such relaxed equations are appropriate for the initial value problem and the theory of turbulence, due to their lack of well-posedness for most initial data. As an attempt to get a more relevant set of relaxed Euler equations, we address the multi-stream pressure-less gravitational Euler-Poisson system as an approximate model, for which we show that the initial value problem can be stated as a concave maximization problem from which we can at least recover a large class of smooth solutions for short enough times. This article is part of the theme issue 'Scaling the turbulence edifice (part 2)'.

Волновые движения жидкого топлива в тороидальных сосудах с учетом капиллярного эффекта

A numerical approach is proposed to solve the linear sloshing problem of an incompressible inviscid liquid with account for surface tension effects, which are predominant in the low-gravity environment. A variational formulation is derived by the linearization of motion equations for the liquid near its initial equilibrium state with consideration of a pressure drop on the free surface and a free-end boundary condition on the contact line. The continuous problem domain is discretized by the finite element method. After discretization, the classical generalized eigenvalue problem is obtained, whose solutions are the natural frequencies and mode shapes. Several examples show the effect of the Bond number and the fluid-filled volume on the liquid behavior in toroidal tanks. A comparison of numerical results with experimental measurements under ground conditions reveals that under microgravity condition, the surface tension force and the boundary condition on the contact line play an important role when determining the natural frequencies and mode shapes of the liquid sloshing. Each fluid-filled volume has its own characteristic Bond number, above which the natural frequencies approximate to the experimental values obtained under ground conditions. The presented results can be used in the coupling dynamic analysis of a spacecraft with propellant tanks. The author is grateful to the supervisor associate professor A.N. Tem-nov for help in formulating the problem and discussion of the results of the work.

Inviscid incompressible limit for compressible micro-polar fluids

In this paper we study the incompressible inviscid limit for a compressible micro-polar model. We prove that the weak solution of the compressible micro-polar system converges to the solution of the Navier–Stokes equations (Euler equations) in the limit of small Mach number (and vanishing viscosity).

Global Well-Posedness of Free Interface Problems for the Incompressible Inviscid Resistive MHD

We consider the plasma-vacuum interface problem in a horizontally periodic slab impressed by a uniform non-horizontal magnetic field. The lower plasma region is governed by the incompressible inviscid and resistive MHD, the upper vacuum region is governed by the pre-Maxwell equations, and the effect of surface tension is taken into account on the free interface. The global well-posedness of the problem, supplemented with physical boundary conditions, around the equilibrium is established, and the solution is shown to decay to the equilibrium almost exponentially. Our results reveal the strong stabilizing effect of the magnetic field as the global well-posedness of the free-boundary incompressible Euler equations, without the irrotational assumption, around the equilibrium is unknown. One of the key observations here is an induced damping structure for the fluid vorticity due to the resistivity and transversal magnetic field. A similar global well-posedness for the plasma-plasma interface problem is obtained, where the vacuum is replaced by another plasma.

Hydrodynamic model of a collision of a spherical plastic ice particle with a perfectly rigid substrate

Impact of a single particle onto a rigid substrate leads to its deformation and subsequent fragmentation. In this theoretical study a simplified model for the plastic flow in a deforming particle is developed which accounts for the dependence of the yield strength on the effective local strain rate. The flow in the particle is approximated by an incompressible inviscid flow which satisfies the conditions of impenetrability of the substrate. The expression for the pressure field in the particle is obtained from the momentum balance equations. This expression includes the plastic stresses, the inertial effects associated with the typical particle velocity and the effects, associated with the particle deceleration. The theoretical predictions for the maximum force, for the force evolution and for the evolution of the particle height agree well with the experimental data found in the literature. No adjustable parameters used in the model.

Global well-posedness of a model on 2D Boussinesq–Bénard equations

In this paper, we consider the classical solutions to a model of two-dimensional incompressible inviscid Boussinesq–Benard equations. Notice that, in the case when the source term of temperature equation in this model is the second component of velocity $$u_2$$ or no source term, there is no global-in-time existence result for the general initial data. Here, if the source term is only chosen as $$\Delta u_2$$ , then we can obtain the global well-posedness, inviscid limit and some exponential decay estimates. Our key observation is the nice symmetrical structure hidden in the corresponding system, which plays an extremely important role in the global well-posedness studied here.

Stability Criteria to the Incompressible Inviscid Linear Fluid between Two Rotation Coaxial Cylinders

The stability and instability phenomenon coupled with the rotation effect and the thermal convection between two concentric cylinders was studied. By means of the Normal-Modes method, the stability or instability criteria for the linearized system in terms of the oscillation frequency, the axial wavelength and the background thermal gradient are proved. Besides, some numerical simulation for the axisymmetric perturbations is presented.

Oblique Injection of Incompressible Ideal Fluid from a Slot into a Free Stream

This paper deals with a two-phase fluid free boundary problem in a slot-film cooling. We give two well-posedness results on the existence and uniqueness of the incompressible inviscid two-phase fluid with a jump relation on free interface. The problem formulates the oblique injection of an incompressible ideal fluid from a slot into a free stream. From the mathematical point of view, this work is motivated by the pioneer work [13] by A. Friedman, in which some well-posedness results are obtained in some special case. Furthermore, A. Friedman proposed an open problem in [14] on the existence and uniqueness of the injection flow problem for more general case. The main results in this paper solve the open problem and establish the well-posedness results on the physical problem.

A Transition-Loss Theory for Waves Reflected and Transmitted by an Overwashed Body

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 18 December 2020Accepted: 25 February 2021Published online: 10 May 2021Keywordsfree-surface potential flows for incompressible inviscid fluids, water waves, gravity waves, dispersion and scattering, nonlinear interaction, basic methods in fluid mechanicsAMS Subject Headings76B07, 76B15, 76M99Publication DataISSN (print): 0036-1399ISSN (online): 1095-712XPublisher: Society for Industrial and Applied MathematicsCODEN: smjmap

Geometrically nonlinear dynamic analysis of functionally graded porous partially fluid-filled cylindrical shells subjected to exponential loads

This article investigates the influence of graphene platelet reinforcements and nonlinear elastic foundations on geometrically nonlinear dynamic response of a partially fluid-filled functionally graded porous cylindrical shell under exponential loading. Material properties are assumed to be varied continuously in the thickness in terms of porosity and graphene platelet reinforcement. In this study, three different distributions for porosity and three different dispersions for graphene platelets have been considered in the direction of the shell thickness. The Halpin–Tsai equations are used to find the effective material properties of the graphene platelet–reinforced materials. The equations of motion are derived based on the higher-order shear deformation theory and Sanders’s theory. Displacements and rotations of the shell middle surface are approximated by combining polynomial functions in the meridian direction and truncated Fourier series with an appropriate number of harmonic terms in the circumferential direction. An incremental–iterative approach is used to solve the nonlinear equations of motion of partially fluid-filled cylindrical shells based on the Newmark direct integration and Newton–Raphson methods. The governing equations of liquid motion are derived using a finite strip formulation of incompressible inviscid potential flow. The effects of various parameters on dynamic responses are investigated. A detailed numerical study is carried out to bring out the effects of some influential parameters, such as fluid depth, porosity distribution, and graphene platelet dispersion parameters on nonlinear dynamic behavior of functionally graded porous nanocomposite partially fluid-filled cylindrical shells reinforced with graphene platelets.

RBF liquids

We introduce a novel liquid simulation approach that combines a spatially adaptive pressure projection solver with the Particle-in-Cell (PIC) method. The solver relies on a generalized version of the Finite Difference (FD) method to approximate the pressure field and its gradients in tree-based grid discretizations, possibly non-graded. In our approach, FD stencils are computed by using meshfree interpolations provided by a variant of Radial Basis Function (RBF), known as RBF-Finite-Difference (RBF-FD). This meshfree version of the FD produces differentiation weights on scattered nodes with high-order accuracy. Our method adapts a quadtree/octree dynamically in a narrow-band around the liquid interface, providing an adaptive particle sampling for the PIC advection step. Furthermore, RBF affords an accurate scheme for velocity transfer between the grid and particles, keeping the system's stability and avoiding numerical dissipation. We also present a data structure that connects the spatial subdivision of a quadtree/octree with the topology of its corresponding dual-graph. Our data structure makes the setup of stencils straightforward, allowing its updating without the need to rebuild it from scratch at each time-step. We show the effectiveness and accuracy of our solver by simulating incompressible inviscid fluids and comparing results with regular PIC-based solvers available in the literature.

Blow-up criterion for the density dependent inviscid Boussinesq equations

In this work, we consider the density-dependent incompressible inviscid Boussinesq equations in mathbb{R}^{N} (Ngeq 2). By using the basic energy method, we first give the a priori estimates of smooth solutions and then get a blow-up criterion. This shows that the maximum norm of the gradient velocity field controls the breakdown of smooth solutions of the density-dependent inviscid Boussinesq equations. Our result extends the known blow-up criteria.