Perfect Incompressible Fluid Research Articles

The quantum perfect fluid in 2D

We consider the field theory that defines a perfect incompressible 2D fluid. One distinctive property of this system is that the quadratic action for fluctuations around the ground state features neither mass nor gradient term. Quantum mechanically this poses a technical puzzle, as it implies the Hilbert space of fluctuations is not a Fock space and perturbation theory is useless. As we show, the proper treatment must instead use that the configuration space is the area preserving Lie group {S\mathrm{Diff}}SDiff. Quantum mechanics on Lie groups is basically a group theory problem, but a harder one in our case, since {S\mathrm{Diff}}SDiff is infinite dimensional. Focusing on a fluid on the 2-torus T^2T2, we could however exploit the well known result {S\mathrm{Diff}}(T^2)\sim SU(N)SDiff(T2)∼SU(N) for N\to ∞N→∞, reducing for finite NN to a tractable case. SU(N)SU(N) offers a UV-regulation, but physical quantities can be robustly defined in the continuum limit N\to∞N→∞. The main result of our study is the existence of ungapped localized excitations, the vortons, satisfying a dispersion \omega \propto k^2ω∝k2 and carrying a vorticity dipole. The vortons are also characterized by very distinctive derivative interactions whose structure is fixed by symmetry. Departing from the original incompressible fluid, we constructed a class of field theories where the vortons appear, right from the start, as the quanta of either bosonic or fermionic local fields.

Existence of weak solutions to the two-dimensional incompressible Euler equations in the presence of sources and sinks

A classical model for sources and sinks in a two-dimensional perfect incompressible fluid occupying a bounded domain dates back to Yudovich's paper [44] in 1966. In this model, on the one hand, the normal component of the fluid velocity is prescribed on the boundary and is nonzero on an open subset of the boundary, corresponding either to sources (where the flow is incoming) or to sinks (where the flow is outgoing). On the other hand the vorticity of the fluid which is entering into the domain from the sources is prescribed. In this paper, we investigate the existence of weak solutions to this system by relying on {\it a priori} bounds of the vorticity, which satisfies a transport equation associated with the fluid velocity vector field. Our results cover the case where the vorticity has a $L^p$ integrability in space, with $p $ in $[1,+\infty]$, and prove the existence of solutions obtained by compactness methods from viscous approximations. More precisely we prove the existence of solutions which satisfy the vorticity equation in the distributional sense in the case where $p > \frac43$, in the renormalized sense in the case where $p > 1$, and in a symmetrized sense in the case where $p =1$.

Remote trajectory tracking of rigid bodies immersed in a two-dimensional perfect incompressible fluid

Remote trajectory tracking of rigid bodies immersed in a two-dimensional perfect incompressible fluid

Vortex dynamics on a Möbius strip

We consider the dynamics of a two-dimensional incompressible perfect fluid on a Möbius strip embedded in $\mathbb {R}^{3}$ . The vorticity–stream function formulation of the Euler equations is derived from an exterior-calculus form of the momentum equation. The non-orientability of the Möbius strip and the distinction between forms and pseudo-forms this introduces lead to unusual properties: a boundary condition is provided by the conservation of circulation along the single boundary of the strip, and there is no integral conservation for the vorticity density or for any odd function thereof. A finite-difference numerical implementation is used to illustrate the Möbius-strip realisation of familiar phenomena: translation of vortices along boundaries, shear instability and decaying turbulence.

Visualization of Exact Invariant Solutions Associated with Atmospheric Waves in a Thin Circular Layer

The purpose of the research was to investigate the exact solutions of nonlinear shallow water equations associated with planetary equatorial waves corresponding to the Cauchy-Poisson free boundary problem describing the nonstationary motion of an incompressible perfect fluid propagating around a solid circle. We consider water waves for which the ratio of the depth of fluid above the circular bottom to the radius of the circle is small (shallow water).

Алгоритм и программа расчета параметров вибрации трубопровода с учетом протекания жидкости

This paper discusses the development of calculation complex (model, algorithm and software) needed to investigate vibration parameters (amplitudes of displacements, internal forces and support responses) of a constant cross-section pipeline with a perfect incompressible fluid flowing inside it. This paper presents a pipeline model as quasi-monomeric finite-element system. Presently, the study discusses vibration of a straight constant cross-section pipeline resting on two elastic supports. Calculation algorithm is based on the discrete variant of partial-response method. The effect of fluid flow is taken into account as an additional inertial load incorporated, in its turn, by means of corrections and modifications of inertia & stiffness parameters of pipeline model. The study gives calculation expressions for partial responses and partial parameters, needed to implement the algorithm suggested by the authors. The problem formulated in this paper was solved as per specially developed mathematical model taking into account the forces due to the flow in the pipe. The paper also suggests calculation algorithm for vibration parameters of the adopted model. These vibration parameters were obtained in specially developed Koriolis software. The study also investigated the effect of additional inertial load components upon vibration parameters and natural frequencies of the structure at question. All these activities made it possible to accomplish the task of the whole study, i.e. to develop the calculation complex for determination of pipeline vibration parameters.

On Schwarzschild’s Interior Solution and Perfect Fluid Star Model

We solve the boundary value problem for Einstein’s gravitational field equations in the presence of matter in the form of an incompressible perfect fluid of density ρ and pressure field p(r) located in a ball r≤r0. We find a 1-parameter family of time-independent and radially symmetric solutions ga,ρa,pa:−2m<a<a1 satisfying the boundary conditions g=gS and p=0 on r=r0, where gS is the exterior Schwarzschild solution (solving the gravitational field equations for a point mass M concentrated at r=0) and containing (for a=0) the interior Schwarzschild solution, i.e., the classical perfect fluid star model. We show that Schwarzschild’s requirement r0>9κM/(4c2) identifies the “physical” (i.e., such that pa(r)≥0 and pa(r) is bounded in 0≤r≤r0) solutions {pa:a∈U0} for some neighbourhood U0⊂(−2m,+∞) of a=0. For every star model {ga:a0<a<a1}, we compute the volume V(a) of the region r≤r0 in terms of abelian integrals of the first, second, and third kind in Legendre form.

External boundary control of the motion of a rigid body immersed in a perfect two-dimensional fluid

We consider the motion of a rigid body due to the pressure of a surrounded two-dimensional irrotational perfect incompressible fluid, the whole system being confined in a bounded domain with an impermeable condition on a part of the external boundary. Thanks to an impulsive control strategy we prove that there exists an appropriate boundary condition on the remaining part of the external boundary (allowing some fluid going in and out the domain) such that the immersed rigid body is driven from some given initial position and velocity to some final position and velocity in a given positive time, without touching the external boundary. The controlled part of the external boundary is assumed to have a nonvoid interior and the final position is assumed to be in the same connected component of the set of possible positions as the initial position.

Dynamics of rigid bodies in a two dimensional incompressible perfect fluid

We consider the motion of several rigid bodies immersed in a two-dimensional incompressible perfect fluid, the whole system being bounded by an external impermeable fixed boundary. The fluid motion is described by the incompressible Euler equations and the motion of the rigid bodies is given by Newton's laws with forces due to the fluid pressure. We prove that, for smooth solutions, Newton's equations can be recast as a second-order ODE for the degrees of freedom of the rigid bodies with coefficients depending on the fluid vorticity and on the circulations around the bodies, but not anymore on the fluid pressure. This reformulation highlights geodesic aspects linked to the added mass effect, gyroscopic features generalizing the Kutta-Joukowski-type lift force, including body-body interactions through the potential flows induced by the bodies' motions, body-body interactions through the irrotational flows induced by the bodies' circulations, and interactions between the bodies and the fluid vorticity.

Dynamics of a rigid body in a two-dimensional incompressible perfect fluid and the zero-radius limit

In this survey we report some recent results on the dynamics of a rigid body immersed in a two-dimensional incompressible perfect fluid, with an emphasis on the zero-radius limit.

Static energetics in gravity

A stress-energy tensor for linear gravity adapted to the harmonic gauge was recently proposed by Butcher, Hobson and Lasenby. By removing gauge constraints and imposing full metrical GR, we find a natural generalisation to the pseudotensor of Einstein. M{\o}ller's pseudotensor is an alternative to that of Einstein formulated using tetrads. We obtain the pseudotensor of M{\o}ller for gauge theory gravity (GTG) using a variational approach, identifying a potentially interesting recipe for constructing conserved currents in that theory. We show that in static, spherical spacetimes with a central gravitational mass M{\o}ller's pseudotensor describes gravitational stress-energy as if the gravitational potential were a scalar (i.e. Klein-Gordon) field coupled to a gravitational mass density on the Minkowski background. The old Newtonian formula successfully describes the potential of even strong fields in this picture. The Newtonian limit of this effect was previously observed in the tensor of Butcher; we recover a local virial theorem in this limit. We demonstrate using the `Schwarzschild star' solution for an incompressible perfect fluid ball.

Numerical Simulation of Unsteady Capillary-Gravity Waves

In this work, plane problems of numerical simulation of wave motion are studied. Potential flows of a perfect incompressible fluid are considered. A numerical algorithm for calculating the shape of a free boundary is proposed. The algorithm is based on the boundary element method with the use of quadrature formulas with no saturation. The algorithm is used for studying the breaking of capillary gravity waves and calculating thin cumulative jets. The stability of the scheme and high accuracy in calculations of sharp cumulative jets are achieved due to special control for the distribution of grid points and a decrease in the grid step in the neighborhood of the forward end of the cumulative jet with an ultimately rapid growth of the curvature.

Point vortex dynamics as zero-radius limit of the motion of a rigid body in an irrotational fluid

The vortex point system is usually considered as an idealized model where the vorticity of an ideal incompressible two-dimensional fluid is concentrated in a finite number of moving points. In the case of a single vortex in an otherwise irrotational ideal fluid occupying a bounded and simply-connected two-dimensional domain the motion is given by the so-called Kirchhoff-Routh velocity which depends only on the domain. The main result of this paper establishes that this dynamics can also be obtained as the limit of the motion of a rigid body immersed in such a fluid when the body shrinks to a pointwise massless particle with fixed circulation. The rigid body is assumed to be only accelerated by the force exerted by the fluid pressure on its boundary, the fluid velocity and pressure being given by the incompressible Euler equations, with vanishing vorticity. The circulation of the fluid velocity around the particle is conserved as time proceeds according to Kelvin's theorem and gives the strength of the limit point vortex. In order to understand the solid dynamics one has to evaluate the pressure field on the boundary of the solid, that is, where the singularity is concentrated at the limit. Moreover the Newton equations driving the particle's dynamics involve a singular perturbation problem in time whereas the fluid state may be seen as solving an auxiliary problem. The fluid velocity can indeed be recovered by an elliptic-type problem where time appears only as a parameter. Since the analysis involves singular perturbation problem both in space and in time and that nothing excludes a priori some sharp energy transfer from the fluid toward the rigid immersed particle it is crucial to recast the solid dynamics under normal forms. This form makes appear the gyroscopic structure of these singularities and allows one to obtain uniform estimates on the dynamics thanks to some energy-type quantities modulated by the limit dynamics, and therefore to describe the transition of the dynamics in the limit. In order to get such normal forms we first establish that the Newton equations can be seen as a geodesic equation, with a metric associated with the well-known ``added inertia'' phenomenon, under an applied force similar to the Lorentz force which can be seen as an extension of the celebrated Kutta-Joukowski lift force. Then, in the limit, surprising relations between leading and subprincipal orders of various terms and the modulation variables show up and allow us to establish a normal form with a more gyroscopic structure. As a byproduct of our analysis we also prove that in the different regime where the body shrinks with a fixed mass the limit equation is a second-order differential equation involving a Kutta-Joukowski-type lift force, which extends the result of [Glass O., Lacave C., Sueur F., On the motion of a small body immersed in a two dimensional incompressible perfect fluid, Bull. Soc. Math. France Volume 142, fascicule 3, 489-536, 2014. ] to the case where the domain occupied by the solid-fluid system is bounded.

Invariant solutions and shock atmospheric waves in a thin circular layer

The objective of this paper is to investigate the nonlinear mathematical model describing equatorial waves from Lie group analysis point of view in order to understand the nature of shallow water model theory, which is associated to planetary equatorial waves. Such waves correspond to the Cauchy–Poisson free boundary problem on the nonstationary motion of a perfect incompressible fluid circulating around a solid circle of a large radius.

Corrigendum to: A variational $\boldsymbol{H}({\rm div})$ finite-element discretization approach for perfect incompressible fluids

Corrigendum to: A variational $\boldsymbol{H}({\rm div})$ finite-element discretization approach for perfect incompressible fluids

Cumulative jet generation in a plane parallel potential flow of the perfect incompressible fluid

This manuscript deals with the plane parallel unsteady potential flow of the perfect incompressible fluid limited by a free boundary. As the result of the free boundary deformation, cumulative jets may form. A cumulative jet generation in a problem of cylindrical cavity deformation is considered in this manuscript. A numerical algorithm for this problem is proposed. Based on the boundary elements method with no saturation, the algorithm has a high order of approximation and reasonable computation time. The distribution of points along the length of the cavity boundary is required to be constant at computation time. This requirement provides the numerical stability of the algorithm. To verify the correctness of the proposed algorithm, conservation laws are used.

Formation of a cumulative jet in the plane-parallel flow of a perfect fluid

The system of equations and numerical results for the problem of cylindrical cavity evolution in the plane-parallel potential flow of a perfect incompressible fluid are given.

Interior solution for the Kerr metric

A, recently presented, general procedure to find static and axially symmetric, interior solutions to the Einstein equations, is extended to the stationary case, and applied to find an interior solution for the Kerr metric. The solution, which is generated by an anisotropic fluid, verifies the energy conditions for a wide range of values of the parameters, and matches smoothly to the Kerr solution, thereby representing a globally regular model describing a non spherical and rotating source of gravitational field. In the spherically symmetric limit, our model converges to the well known incompressible perfect fluid solution.The key stone of our approach is based on an ansatz allowing to define the interior metric in terms of the exterior metric functions evaluated at the boundary source. The physical variables of the energy-momentum tensor are calculated explicitly, as well as the geometry of the source in terms of the relativistic multipole moments.

Axially symmetric static sources of gravitational field

A general procedure to find static and axially symmetric, interior solutions to the Einstein equations is presented. All the so obtained solutions, verify the energy conditions for a wide range of values of the parameters, and match smoothly to some exterior solution of the Weyl family, thereby representing globally regular models describing non-spherical sources of gravitational field. In the spherically symmetric limit, all our models converge to the well known incompressible perfect fluid solution. The key stone of our approach is based on an ansatz allowing to define the interior metric in terms of the exterior metric functions evaluated at the boundary source. Some particular sources are obtained, and the physical variables of the energy–momentum tensor are calculated explicitly, as well as the geometry of the source in terms of the relativistic multipole moments. The total mass of different configurations is also calculated, it is shown to be equal to the monopole of the exterior solution.

On the short-wave nature of Richtmyer–Meshkov instability

In the case of a variable period (wavelength) of a perturbed interface, the instability and stability of Richtmyer–Meshkov vortices in perfect gas and incompressible perfect fluid, respectively, are investigated numerically and analytically. Taking into account available experiments, the instability of the interface between the argon and xenon in the case of a relatively small period is modeled. An estimate of the magnitude of the critical period is given. The nonlinear (for arbitrary initial conditions) stability of the corresponding steady-state vortex flow of perfect fluid in a strip (vertical periodic channel) in the case of a fairly large period is shown.