Related papers: Towards a Fluid computer
Consider a fluid flowing through a junction between two pipes with different sections. Its evolution is described by the 2D or 3D Euler equations, whose analytical theory is far from complete and whose numerical treatment may be rather…
The concept of a fluid algebra was introduced by Sullivan over a decade ago as an algebraic construct which contains everything necessary in order to write down a form of the Euler equation, as an ODE whose solutions have invariant…
The interaction of electrical fields and liquids can lead to phenomena that defies intuition. Some famous examples can be found in Electrohydrodynamics as Taylor cones, whipping jets or non-coalescing drops. A less famous example is the…
We demonstrate the existence of smooth three-dimensional vector fields where the cross product between the vector field and its curl is balanced by the gradient of a smooth function, with toroidal level sets that are not invariant under…
The phase diagram of water harbours many mysteries: some of the phase boundaries are fuzzy, and the set of known stable phases may not be complete. Starting from liquid water and a comprehensive set of 50 ice structures, we compute the…
Tichler proved that a manifold admitting a smooth closed one-form fibers over a circle. More generally a manifold admitting $k$ independent closed one-forms fibers over a torus $T^k$. In this article we explain a version of this…
Geometric Hydrodynamics has flourished ever since the celebrated 1966 paper of V. Arnold. In this paper we present a collection of open problems along with several new constructions in fluid dynamics and a concise survey of recent…
Computational fluid dynamics and aerodynamics, which complement more expensive empirical approaches, are critical for developing aerospace vehicles. During the past three decades, computational aerodynamics capability has improved…
Understanding complexity in fluid mechanics is a major problem that has attracted the attention of physicists and mathematicians during the last decades. Using the concept of renormalization in dynamics, we show the existence of a locally…
Arnold pointed out that the Euler equation of incompressible ideal hydrodynamics describes geodesics on the group of volume-preserving diffeomorphisms. A simple analogue is the Euler equation for a rigid body, which is the geodesic equation…
We employ the relationship between contact structures and Beltrami fields derived in part I of this series to construct steady nonsingular solutions to the Euler equations on a Riemannian $S^3$ whose flowlines trace out closed curves of all…
Recently, the Nash-style convex integration has been becoming the main scheme for the mathematical study of turbulence, and the main building block of it has been either Beltrami flow (finite mode) or Mikado flow (compactly supported in the…
Two-dimensional (2-D) incompressible, inviscid fluids produce fascinating patterns of swirling motion. How and why the patterns emerge are long-standing questions, first addressed in the 19th century by Helmholtz, Kirchhoff, and Kelvin.…
After an introductory chapter on the quantum supersymmetric string, in which particular attention will be devoted to the techniques via which phenomenologically viable models can be obtained from the ultraviolet microscopic degrees of…
The relationship between computational models and dynamics has captivated mathematicians and computer scientists since the earliest conceptualizations of computation. Recently, this connection has gained renewed attention, fueled by T.…
Eulerian hydrodynamical simulations are a powerful and popular tool for modeling fluids in astrophysical systems. In this work, we critically examine recent claims that these methods violate Galilean invariance of the Euler equations. We…
The geometric nature of Euler fluids has been clearly identified and extensively studied over the years, culminating with Lagrangian and Hamiltonian descriptions of fluid dynamics where the configuration space is defined as the…
The article describes various aspects of mathematical modeling of fluid flows, both in general and with reference to hydraulic machinery. The article reviews historical development of corresponding methods of mathematical modeling.…
We prove that the dynamical system defined by the hydrodynamical Euler equation on any closed Riemannian 3-manifold $M$ is not mixing in the $C^k$ topology ($k > 4$ and non-integer) for any prescribed value of helicity and sufficiently…
In this note we first set up an analogy between spin and vorticity of a perfect 2d-fluid flow, based on the Borel-Weil contruction of the irreducible unitary representations of SU(2), and looking at the Madelung-Bohm velocity attached to…