Mathematical modeling of rigid bodies motion through some non-Newtonian flows

The main novelty of this article is to reveal a weak formulation of rigid bodies motion arising from free boundaries evolution in visco-plastic flows governed by the incompressible Norton–Hoff operator with non-cylindrical coefficients. We supply the existence result of an interface between the two non-miscible fluids by the use of the non-smooth evolution theory. We prove that the fluid’s flow is converted to a rigid body when its viscosity is large enough. The established results are variants or extensions of the existing formulations.     

On the boundaries of the parametric resonance domain

The problem of stability for a system of linear differential equations with coefficients which are periodic in time and depend on the parameters is considered. The singularities of the general position arising at the boundaries of the stability and instability (parametric resonance) domains in the case of two and three parameters are listed. A constructive approach is proposed which enables one, in the first approximation, to determine the stability domain in the neighbourhood of a point of the boundary (regular or singular) from the information at this point. This approach enables one to eliminate a tedious numerical analysis of the stability region in the neighbourhood of the boundary point and can be employed to construct the boundaries of parametric resonance domains. As an example, the problem of the stability of the oscillations of an articulated pipe conveying fluid with a pulsating velocity is considered. In the space of three parameters (the average fluid velocity and the amplitude and frequency of pulsations) a singularity of the boundary of the stability domain of the “dihedral angle” type is obtained and the tangential cone to the stability domain is calculated.     

Effect of vertical vibrations on a two-layer system with a deformable interface

The effect of single- and two-frequency vibrations on the behavior of a system consisting of two homogeneous viscous fluids bounded by rigid walls is analyzed. It is assumed that the system as a whole is under vertical vibrations obeying a certain law. An eigenvalue problem is obtained in order to analyze the stability of the relative equilibrium. The case of finite frequencies and arbitrary modulation amplitudes is treated along with the case of high frequencies and small modulation amplitudes. In the former case, the parametric resonance domains are examined depending on the parameters of the system. In the latter case, the high-frequency vibration is shown to create effective surface tension, thus flattening the interface, and can suppress instability when the heavy fluid is over the light one.     

On some unsteady flows of a non-Newtonian fluid

Some properties of unsteady unidirectional flows of a fluid of second grade are considered for flows impulsively started from rest by the motion of a boundary or two boundaries or by sudden application of a pressure gradient. Flows considered are: unsteady flow over a plane wall, unsteady Couette flow, flow between two parallel plates suddenly set in motion with the same speed, flow due to one rigid boundary moved suddenly and one being free, unsteady Poiseuille flow and unsteady generalized Couette flow. The results obtained are compared with those of the exact solutions of the Navier–Stokes equations. It is found that the stress at time zero on the stationary boundary for the flows generated by impulsive motion of a boundary or two boundaries is finite for a fluid of second grade and infinite for a Newtonian fluid. Furthermore, it is shown that for unsteady Poiseuille flow the stress at time zero on the boundary is zero for a Newtonian fluid, but it is not zero for a fluid of second grade.     

Hydrodynamic Models and Confinement Effects by Horizontal Boundaries

Confinement effects by rigid boundaries in the dynamics of ideal fluids are considered from the perspective of long-wave models and their parent Euler systems, with the focus on the consequences of establishing contacts of material surfaces with the confining boundaries. When contact happens, we show that the model evolution can lead to the dependent variables developing singularities in finite time. The conditions and the nature of these singularities are illustrated in several cases, progressing from a single-layer homogeneous fluid with a constant-pressure free surface and flat bottom, to the case of a two-fluid system contained between two horizontal rigid plates, and finally, through numerical simulations, to the full Euler stratified system. These demonstrate the qualitative and quantitative predictions of the models within a set of examples chosen to illustrate the theoretical results.

     

Computation of flow-induced motion of floating bodies

A computational procedure for the prediction of motion of rigid bodies floating in viscous fluids and subjected to currents and waves is presented. The procedure is based on a coupled iterative solution of equations of motion of a rigid body with up to six degrees of freedom and the Reynolds-averaged Navier–Stokes equations describing the two- or three-dimensional fluid flow. The fluid flow is predicted using a commercial CFD package which can use moving grids made of arbitrary polyhedral cells and allows sliding interfaces between fixed and moving grid blocks. The computation of body motion is coupled to the CFD code via user-coding interfaces. The method is used to compute the 2D motion of floating bodies subjected to large waves and the results are compared to available experimental data, showing favorable agreement.     

Integrable billiards model important integrable cases of rigid body dynamics

Abstract—A generalized billiard is considered, in which a point moves on a locally flat surface obtained by isometrically gluing together several plane domains along boundaries being arcs of confocal quadrics. Under this motion, a point moves from one domain to another, passing through the glued boundaries. Many integrable cases of rigid body dynamics with appropriate parameter values at certain levels of integrals are modeled by classical or generalized billiards; in the paper, Liouville equivalence is proved by comparing Fomenko–Zieschang invariants.     

The problem of the stability of quasilinear flows with respect to perturbations of the hardening function

A formulation of the linearized boundary-value problem of the stability of a deformation process with respect to small perturbations of the hardening function (of the scalar constitutive relation of the material) is presented. The characteristic vector relations of the medium are assumed to be linear. The occurrence of rigid zones in the domain of the solid and the change in their boundaries in the perturbed motion are taken into account. A perfect rigid plastic deformation and the flow of a Newtonian fluid are considered explicitly as the basic flow. In the latter case, the equation of the asymptotic boundary of the rigid zone, which appears when there is a small variation in the yield stress and a transition to a viscoplastic material, is derived.     

On the fall of a heavy rigid body in an ideal fluid

We consider a problem about the motion of a heavy rigid body in an unbounded volume of an ideal irrotational incompressible fluid. This problem generalizes a classical Kirchhoff problem describing the inertial motion of a rigid body in a fluid. We study different special statements of the problem: the plane motion and the motion of an axially symmetric body. In the general case of motion of a rigid body, we study the stability of partial solutions and point out limiting behaviors of the motion when the time increases infinitely. Using numerical computations on the plane of initial conditions, we construct domains corresponding to different types of the asymptotic behavior. We establish the fractal nature of the boundary separating these domains.     

Spectral stability criterion and the Cauchy problem for the Hill equation at parametric resonance

An analytical solution to the Cauchy problem for the Hill equation is constructed by the second-order averaging method for three instability domains, stability domains near the boundaries with the instability domains, and on the boundaries themselves. An unstable exponentially decaying solution is found in the instability domains. A simple (convenient for applications) stability criterion for the trivial solution is formulated in the form of an inequality expressed in terms of the constant component, the amplitudes, and the frequencies of harmonics in the spectrum of the periodic coefficient of the Hill equation.     

Anisotropic Second-Order Fluid

A classification of fluids is presented in accordance with the terminology of the school of rational mechanics. Rheological equations of state are formulated for an anisotropic second-order fluid. A qualitative assumption is made in regard to the rheological behavior of a dilute suspension of rigid ellipsoids of revolution whose dispersion medium is modeled by a second-order fluid.     

Convective Instabilities in Anisotropic Porous Media

This paper addresses the problem of the onset of Rayleigh-Bénard convection in a porous layer using Brinkman's equation and anisotropic permeability. The critical Rayleigh number and wave number at marginal stabilities are calculated for both free and rigid boundaries. In both cases, it is noted that there exist ranges for which the stability criteria is intermediate to the low porosity Darcy approximation and to high porosity single viscous fluid. The permeability anisotropy is found to select the mode of instability.     

On stability and self-excited vibrations in fluid-structure-interaction

In flexible channels conveying fluid the steady state may loose stability by divergence or flutter. The aim of this contribution is to investigate the basic excitation mechanisms of flow-induced vibrations and to evaluate the influence of various parameters on the stability behaviour of the coupled problem. Therefore, a simple, yet general model is proposed. The fluid is assumed to be inviscid and irrotational and both incompressible and compressible flow is considered. It is guided by a planar, rectangular channel with a rigid wall and a thin, flexible wall. The latter is modelled as a one-parametric continuum on an elastic foundation, which exhibits bending and extensional stiffness. By examining the energy balance over one oscillation circle it is possible to reveal the mechanisms of energy transfer between the coupled components of the system. Based on this analysis a physical explanation of the arising instabilities is possible. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Viscous potential flow analysis of interfacial stability with mass transfer through porous media

The interfacial stability with mass transfer, surface tension, and porous media between two rigid planes will be investigated in the view of viscous potential flow analysis. A general dispersion relation is obtained. For Kelvin-Helmholtz instability, it is found that the stability criterion is given by a critical value of the relative velocity. On the other hand, in the absence of gravity the problem reduces to Brinkman model of the stability of two fluid layers between two rigid planes. Vanishing of the critical value of the relative velocity gives rise to a new dispersion relation for Rayleigh-Taylor instability. Formulas for the growth rates and neutral stability curve are also given and applied to air-water flows. The effects of viscosity, porous media, surface tension, and heat transfer are also discussed in relation to whether the system is potentially stable or unstable. The Darcian term, permeability’s and porosity effects are also concluded for Kelvin-Helmholtz and Rayleigh-Taylor instabilities. The relation between porosity and dimensionless relative velocity is also investigated.     

The stability domains of hamiltonian systems

Linear Hamiltonian systems with an arbitrary number of degrees of freedom, which depend smoothly on a vector of real parameters, are investigated. All possible singularities of the boundary of the stability domain of Hamiltonian systems of general position are determined and described for the case of two and three parameters. In the first approximation, the geometry of these singularities (the orientation in the parameter space, angles, etc.) is determined on the basis of the first derivative of the matrix of the system with respect to the parameters, as are the eigenvectors and generalized eigenvectors evaluated at the singular point. A detailed investigation is made of gyroscopic systems as a special case of Hamiltonian systems. As mechanical examples, an account is given of the problem of the stability of the oscillations of a tube through which a fluid is flowing, and of the stability of the motion of a two-body system. The tangent cones to the stability domains of these systems at singular points of the “cusp” and “dihedral angle” type, which arise on the boundaries of these domains, are found.     

Transient flow in a rotating porous annulus due to suction at the walls

A highly porous material occupies the annular region between two coaxial infinitely long cylinders. A viscous incompressible fluid fills this porous medium and is initially in a state of rigid rotation together with the medium. The flow has been disturbed by imposing suction/injection at the outer/inner cylindrical boundaries respectively. The Brinkman's law has been used to represent the fluid motion. The exact solution for the resulting unsteady flow is obtained by Laplace transformation technique. The transient evolution of the boundary layers and the response of steady boundary layers to the resistance of the medium are discussed.     

On the motion of rigid bodies in a viscous fluid

We consider the problem of motion of several rigid bodies in a viscous fluid. Both compressible and incompressible fluids are studied. In both cases, the existence of globally defined weak solutions is established regardless possible collisions of two or more rigid objects.     

Bénard instabilities in polar fluids

This paper employs continuum theory of polar fluids to investigate the onset of roll-type instabilities in a fluid confined between horizontal rigid boundaries and subject to a vertical temperature gradient. A Fourier series method is used to obtain an exact expression for the determination of the critical temperature gradient for the particular type of instability. Results are presented for a wide range of the various parameters in the theory.

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Résumé Cet article utilise la théorie du continuum des fluides polaires en vue d'étudier le début d'instabilités de type cellulaire dans un fluide compris entre des limites horizontales rigides et soumis à un gradient de température vertical. Afin de déterminer exactement le gradient de température critique pour le type d'instabilité en question, on utilise une méthode basée sur les séries de Fourier. On présente des résultats pour une large gamme des différents paramètres utilisés dans la théorie.

     

MHD Couette flow of a viscous stratified fluid of large conductivity

The MHD Couette flow of a viscous stratified fluid of large electrical conductivity with suction and injection at the plane boundaries is studied when the plane boundaries are maintained at different temperatures. The Oseen type governing equations are formulated using the method suggested by Greenspan for stratified fluids. Introducing the similarity variables, the linearised equations are solved to obtain the velocity and temperature distributions. The results show that the behaviour of velocity and temperature in fluids of large conductivity is different from the behaviour of velocity and temperature for fluids of finite conductivity. The effect of the magnetic field on the load capacity is investigated for the case when the width of the channel is small.     

The Panel Method for Rigid Section Group Added Mass Coefficients and Its Application to Fuel Assemblies北大核心CSCD

A numerical method based on the panel method was developed to calculate the added fluid mass coefficients of the rigid section group with arbitrarily complex shapes, and was successfully applied to the PWR fuel assemblies. The variation law of added mass coefficients with position deviations was analyzed in the seismic test of 1×5 fuel assemblies. The results show that, this method is suitable for the calculation of the added mass coefficients of rigid section groups with complex continuous boundaries. Compared with the gap between assemblies, the gap between baffles and assemblies has a dominant influence on the added mass coefficient. Regardless of the position deviation, the sum of the added mass coefficients of all assemblies and baffles in the assumed motion direction is approximately equal to –1, and that in the perpendicular direction is approximately equal to 0. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.