Vortex dynamics in R4R^4

The vortex dynamics of Euler's equations for a constant density fluid flow in $R^4$ is studied. Most of the paper focuses on singular Dirac delta distributions of the vorticity two-form $\omega$ in $R^4$. These distributions are supported on two-dimensional surfaces termed {\it membranes} and are the analogs of vortex filaments in $R^3$ and point vortices in $R^2$. The self-induced velocity field of a membrane is shown to be unbounded and is regularized using a local induction approximation (LIA). The regularized self-induced velocity field is then shown to be proportional to the mean curvature vector field of the membrane but rotated by 90 degrees in the plane of normals. Next, the Hamiltonian membrane model is presented. The symplectic structure for this model is derived from a general formula for vorticity distributions due to Marsden and Weinstein (1983). Finally, the dynamics of the four-form $\omega \wedge \omega$ is examined. It is shown that Ertel's vorticity theorem in $R^3$, for the constant density case, can be viewed as a special case of the dynamics of this four-form.Comment: Submitted to Journal of Mathematical Physic

Poisson brackets for the dynamically coupled system of a free boundary and a neutrally buoyant rigid body in a body-fixed frame

The fully coupled dynamic interaction problem of the free surface of an incompressible fluid and a rigid body beneath it, in an inviscid, irrotational framework and in the absence of surface tension, is considered. Evolution equations of the global momenta of the body+fluid system are derived. It is then shown that, under fairly general assumptions, these evolution equations combined with the evolution equation of the free-surface, referred to a body-fixed frame, is a Hamiltonian system. The Poisson brackets of the system are the sum of the canonical Zakharov bracket and the non-canonical Lie-Poisson bracket. Variations are performed consistent with the mixed Dirichlet-Neumann problem governing the system

Geometric phases for corotating elliptical vortex patches

We describe a geometric phase that arises when two elliptical vortex patches corotate. Using the Hamiltonian moment model of Melander, Zabusky, and Styczek [J. Fluid Mech. 167, 95–115 (1986)] we consider two corotating uniform elliptical patches evolving according to the second order truncated equations of the model. The phase is computed in the adiabatic setting of a slowly varying Hamiltonian as in the work of Hannay [J. Phys. A 18, 221–230 (1985)] and Berry [Proc. R. Soc. London, Ser. A 392, 45–57 (1984)]. We also discuss the geometry of the symplectic phase space of the model in the context of nonadiabatic phases. The adiabatic phase appears in the orientation angle of each patch—it is similiar in form and is calculated using a multiscale perturbation procedure as in the point vortex configuration of Newton [Physica D 79, 416–423 (1994)] and Shashikanth and Newton [J. Nonlinear Sci. 8, 183–214 (1998)], however, an extra factor due to the internal stucture of the patch is present. The final result depends on the initial orientation of the patches unlike the phases in the works of Hannay and Berry [J. Phys. A 18, 221–230 (1985)]; [Proc. R. Soc. London, Ser. A 392, 45–57 (1984)]. We then show that the adiabatic phase can be interpreted as the holonomy of a connection on the trivial principal fiber bundle pi:T2×S1-->S1, where T2 is identified with the product of the momentum level sets of two Kirchhoff vortex patches and S1 is diffeomorphic to the momentum level set of two point vortex motion. This two point vortex motion is the motion that the patch centroids approach in the adiabatic limit

The Hamiltonian structure of a two-dimensional rigid circular cylinder interacting dynamically with N point vortices

This paper studies the dynamical fluid plus rigid-body system consisting of a two-dimensional rigid cylinder of general cross-sectional shape interacting with N point vortices. We derive the equations of motion for this system and show that, in particular, if the vortex strengths sum to zero and the rigid-body has a circular shape, the equations are Hamiltonian with respect to a Poisson bracket structure that is the sum of the rigid body Lie–Poisson bracket on Se(2)*, the dual of the Lie algebra of the Euclidean group on the plane, and the canonical Poisson bracket for the dynamics of N point vortices in an unbounded plane. We then use this Hamiltonian structure to study the linear and nonlinear stability of the moving Föppl equilibrium solutions using the energy-Casimir method

Sintering characteristics of red mud compact

It is beyond doubt that activity of primary industries often yields substantial amounts of byproducts. The disposal in the original industrial site is favoured by economic reasons though traditional storage in nearby dumps can be impractical owing to the considerable masses involved and environmental restrictions. The local exploitation of these by-products is therefore a growing technological aspect of basic industries and one tenable option is their re-use as starting materials for other productions.This huge amount of industrial yproducts/wastes which is becoming a client for increasing environmental pollution & generation of a huge amount of unutilized resources. With a view to the above, this research is aimed at finding out utilization of such things/material/industrial byproducts for value added applications & also helps to solve the environmental problems. The present piece of my research work aims at, to provide a valued input/utilization to industrial byproduct/waste. An emblematic case is the ‘red mud’ discharged by industry producing alumina from bauxite: alkaline digestion of 2.5 t of bauxite affords alumina and ≈1.5 t of red mud ,so that an average Al2O3 productivity of 5×105 t year−1 involves a mass of by-products of ≈7.5×105 t year−1 discharged as slurry retaining variable water contents. This amount is composed of Fe and Ti oxides, behaving as chemically inert matter, with variable percentages of nominal SiO2,Al2O3 and Na2O. The material is available as a watery mixture which settles slowly and may easily be conveyed from station to station by continuous fluid-carrying machinery.The recycling of the mud, after curing or high temperature annealing — up to 1200°C — for large-rate daily mass consumption industries such as bricks and tile kilns has been put forward in a number of papers[2,3,4,5,6,7,8,9]Most of the above reports appear fragmentary and, to some extent surprising, characterization work is limited to the elemental analysis of the raw material and the identification of the crystalline phases in dried samples. However, the definition of thermal behaviour in a wide working range of temperatures appears mandatory for a feasible exploitation of the mud in high temperature applications. Indeed, the reactivity of red mud components on heating may promote ceramization and shrinkage and, apart from other qualities, may affect the mechanical features of clay-based items fabricated with bauxite-waste addition.Accordingly, we focus here on the thermal behaviour of the mud, the solid-state transformations and solid–liquid phase transitions. The use of thermal analysis coupled with of X-ray diffraction methods seemed well suitable for the problem at hand. The present study is a part of a long-term project on the exploitation of red mud as a clay additive for the ceramic industry or as a compound for self-binding mortars in the fabrication of stoneware

Non-invasive determination of external forces in vortex-pair-cylinder interactions

Expressions for the conserved linear and angular momenta of a dynamically coupled fluid + solid system are derived. Based on the knowledge of the flow velocity field, these expressions allow the determination of the external forces exerted on a body moving in the fluid such as, e.g., swimming fish. The verification of the derived conserved quantities is done numerically. The interaction of a vortex pair with a circular cylinder in various configurations of motions representing a generic test case for a dynamically coupled fluid + solid system is investigated in a weakly compressible Navier-Stokes setting using a Cartesian cut-cell method, i.e., the moving circular cylinder is represented by cut cells on a moving mesh. The objectives of this study are twofold. The first objective is to show the robustness of the derived expressions for the conserved linear and angular momenta with respect to bounded and discrete data sets. The second objective is to study the coupled dynamics of the vortex pair and a neutrally buoyant cylinder free to move in response to the fluid stresses exerted on its surface. A comparison of the vortex-body interaction with the case of a fixed circular cylinder evidences significant differences in the vortex dynamics. When the cylinder is fixed strong secondary vorticity is generated resulting in a repeating process between the primary vortex pair and the cylinder. In the neutrally buoyant cylinder case, a stable structure consisting of the primary vortex pair and secondary vorticity shear layers stays attached to the moving cylinder. In addition to these fundamental cases, the vortex-pair-cylinder interaction is studied for locomotion at constant speed and locomotion at constant thrust. It is shown that a similar vortex structure like in the neutrally buoyant cylinder case is obtained when the cylinder moves away from the approaching vortex pair at a constant speed smaller than the vortex pair translational velocity. Finally, the idealized symmetric settings are complemented by an asymmetric interaction of a vortex pair and a cylinder. This case is discussed for a fixed and a neutrally buoyant cylinder to show the validity of the derived relations for multi-dimensional body dynamics

Sintering Behavior of Red Mud Compact

It is beyond doubt that activity of primary industries often yields substantial amounts of byproducts. The disposal in the original industrial site is favoured by economic reasons though traditional storage in nearby dumps can be impractical owing to the considerable masses involved and environmental restrictions. The local exploitation of these by-products is therefore a growing technological aspect of basic industries and one tenable option is their re-use as starting materials for other productions. This huge amount of industrial byproducts/wastes which is becoming a client for increasing environmental pollution & generation of a huge amount of unutilized resources. With a view to the above, this research is aimed at finding out utilization of such things/materials/industrial byproducts for value added applications & also helps to solve the environmental problems. The present piece of my research work aims at, to provide a valued input/utilization to industrial byproduct/waste

The motion of the 2D hydrodynamic Chaplygin sleigh in the presence of circulation

We consider the motion of a planar rigid body in a potential flow with circulation and subject to a certain nonholonomic constraint. This model is related to the design of underwater vehicles. The equations of motion admit a reduction to a 2-dimensional nonlinear system, which is integrated explicitly. We show that the reduced system comprises both asymptotic and periodic dynamics separated by a critical value of the energy, and give a complete classification of types of the motion. Then we describe the whole variety of the trajectories of the body on the plane.Comment: 25 pages, 7 figures. This article uses some introductory material from arXiv:1109.321

The Dynamics of a Rigid Body in Potential Flow with Circulation

We consider the motion of a two-dimensional body of arbitrary shape in a planar irrotational, incompressible fluid with a given amount of circulation around the body. We derive the equations of motion for this system by performing symplectic reduction with respect to the group of volume-preserving diffeomorphisms and obtain the relevant Poisson structures after a further Poisson reduction with respect to the group of translations and rotations. In this way, we recover the equations of motion given for this system by Chaplygin and Lamb, and we give a geometric interpretation for the Kutta-Zhukowski force as a curvature-related effect. In addition, we show that the motion of a rigid body with circulation can be understood as a geodesic flow on a central extension of the special Euclidian group SE(2), and we relate the cocycle in the description of this central extension to a certain curvature tensor.Comment: 28 pages, 2 figures; v2: typos correcte