Equilibrium Configurations of a 3D Fluid-Beam Interaction Problem

Instabilities in a rotor system partially filled with a fluid can have an exponentially increasing amplitude, and this can cause catastrophic damage. Numerous theoretical models have been proposed, and numerous experiments have been conducted to investigate the mechanisms of this phenomenon. However, the explanation of the existence of the first unstable region induced by a viscous incompressible fluid is unclear, and only one solving method, a standard finite difference procedure, was proposed in 1991 for solving the instabilities in a system containing a symmetric rotor partially filled with a viscous incompressible fluid. To better understand the mechanisms of the instability induced by the viscous fluid, based on the linearized two-dimensional Navier–Stokes equations, this system's differential equations are transferred to solve the characteristic equations with boundary conditions. A Matlab boundary value problem (BVP) solver bvp5c proposed in 2008 is an efficient tool to solve this problem by uncoupling the boundary conditions with unknown initial guess. Applying this approach to a rotor system allows the instability regions to be obtained. In this study, first, the radial and tangential velocities and pressure fluctuations along the radial direction of a disk filled with fluid were examined. Then, parametric analysis of the effect of the Reynolds number R e c r, filling ratio H, damping ratio C, and mass ratio m on the system's stability was conducted. Using this calculation method allowed the first exploration of some new laws regarding the instabilities. These results will benefit the further understanding of the existence of the first unstable region of a rotor partially filled with a viscous incompressible fluid.

High-order fluid solver based on a combined compact integrated RBF approximation and its fluid structure interaction applications

Instability of a rotor partially filled with viscous incompressible fluid will cause the amplitudes of perturbations to increase exponentially. Many models of an isotropic rotor partially filled with fluid have been proposed to investigate its stability. However, the bifurcation of an anisotropic rotor partially filled with viscous incompressible fluid is complicated, which has rarely been studied. To investigate this problem, a continuous model is first established for the isotropic case and the hydrodynamic forces are calculated. The D-decomposition method is then used to determine the stable and unstable regions of the isotropic rotor. An analytical prediction method is then proposed in this paper, and the results for stable and unstable regions are the same as those obtained with the D-decomposition method. Then, this novel analytical prediction model is applied to an anisotropic rotor partially filled with viscous incompressible fluid, and the stable and unstable regions are analyzed. One isotropic and two anisotropic conditions are compared to verify the correctness of the proposed analytical method. The results show that the dimensionless damping and stiffness have significant effects on the stability of an anisotropic rotor partially filled with viscous incompressible fluid; in particular, it is found that there exists a single stable region for low values of the dimensionless damping coefficient and stiffness. Furthermore, the bifurcation law of different anisotropic parameters is first explored, which can provide theoretical guidance for the chosen external stiffness and damping coefficients.

In this chapter, we study the hydrodynamics of a viscous incompressible fluid. The Lagrangian hydrodynamical systems (LHSs) of a viscous incompressible fluid were introduced in [39] as generalizations of those of an ideal incompressible fluid. Namely, these LHSs were defined as systems on the group of volume-preserving diffeomorphisms with an additional right-invariant force field which depends on the velocity of the fluid. In the tangent space at id, the force field is v△ where △ is the Laplace-de Rham operator and v is the viscosity coefficient. Note, however, that the operator △ does not preserve the space of H8-smooth vector fields. In fact, △sends H8-smooth vector fields to fields which belong to a broader Sobolev class. As a consequence, the method relies heavily on the theory of partial differential equations, leading to the loss of many natural geometric properties of the LHSs of an ideal incompressible fluid (Chap. 8) in the passage to a viscous incompressible fluid.

In this paper, we study the existence and long-time behavior of global strong solutions to a system describing the mixture of two viscous incompressible Newtonian fluids of the same density. The system consists of a coupling of Navier-Stokes and Cahn-Hilliard equations. We first show the global existence of strong solutions in several cases. Then we prove that the global strong solution of our system will converge to a steady state as time goes to infinity. We also provide an estimate on the convergence rate.

The solvability of a certain two-dimensional boundary-value problem for the system of the Navier-Stokes equations, describing the steady (partially common) motion of two heavy viscous incompressible capillary fluids with free noncompact boundaries, is proved.

Numerical investigation is performed of the problem on stratified flow of two viscous incompressible fluids in a horizontal (and slightly inclined to the horizontal) plane-parallel channel. An algorithm is developed for the determination of the interface between fluids. The control volume approach and the SIMPLER algorithm are used to obtain the results. The following characteristic quantities are determined and analyzed in all versions of calculations for the water-oil system: flow field, pressure, fluid interface, friction on the channel walls, velocity profiles in the channel cross sections.

ABSTRACTThis paper is devoted to a consideration of the following problem: Given a static state in which an incompressible viscous fluid is arranged in horizontal strata and the density is a function of the vertical coordinate only, to determine the initial manner of development of an infinitesimal disturbance. The mathematical problem is reduced to one in characteristic values in a fourth-order differential equation and a variational principle characterizing the solution is enunciated. The particular case of two uniform fluids of different densities (but the same kinematic viscosity) separated by a horizontal boundary is considered in some detail. The mode of maximum instability in case the upper fluid is more dense and the manner of decay in case the lower fluid is more dense are determined; and the results of the calculations are illustrated graphically. Gravity waves (obtained in the limit when the density of the upper fluid is zero) are also treated.

Pressure correction DRBEM solution for heat transfer and fluid flow in incompressible viscous fluids

Обсуждается разрешимость переопределенной системы уравнений тепловой конвекции в приближении Буссинеска. Система уравнений Обербека-Буссинеска, дополненная уравнением несжимаемости, является переопределенной. Количество уравнений превосходит количество неизвестных функций, поскольку изучаются неоднородные слоистые потоки вязкой несжимаемой жидкости (одна из компонент вектора скорости тождественно равна нулю). Проведено исследование разрешимости нелинейной системы уравнений Обербека-Буссинеска. Исследование разрешимости переопределенной системы нелинейных уравнений в частных производных Обербека-Буссинеска осуществлялось при помощи построения нескольких частных точных решений. Приведен новый класс точных решений для описания трехмерных нелинейных слоистых течений вертикальной завихренной вязкой несжимаемой жидкости. Вертикальная компонента завихренности в невращающейся жидкости генерируется неоднородным полем скоростей на нижней границе бесконечного горизонтального слоя жидкости. Конвекция в вязкой несжимаемой жидкости индуцируется линейными источниками тепла. Основное внимание уделено исследованию свойств поля скоростей течения. Исследована зависимость структуры этого поля от величины вертикальной закрутки. Показано, что одна из компонент вектора скорости при ненулевой вертикальной закрутке допускает расслоение на пять зон по толщине рассматриваемого слоя (четыре застойные точки). Анализ поля скоростей показал, что кинетическая энергия жидкости может дважды принимать нулевой значение по толщине слоя.

In this paper, the convergence of incompressible monopolar viscous non-Newtonian fluids is investigated in 3D periodic domain. We obtain the conclusion that the solutions of non-Newtonian fluids converge to the solutions of Navier-Stokes equation in the sense of L2-norm, as the viscosity goes to zero and the initial data belong to V.

The author proved the existence of a unique solution of the Navier-Stokes equations for a viscous incompressible and compressible fluids. A method is also proposed for the search for this solution, which consists in moving along the gradient to the unique saddle point of some functional. It should be noted the complexity software implementation for its. For a viscous incompressible fluid, a simple method for finding a solution is proposed, consisting in moving along a gradient to a single point of a convex functional. An explanation is proposed for the mechanism of the appearance of turbulent flows. A method for solving the Navier-Stokes equations with turbulence is proposed. For the stationary case, experimental programs have been developed in the MATLAB system

We consider heat-conducting viscous incompressible (not necessarily Newtonian) fluids under the general Stokesian constitutive hypotheses. Given a natural and mild condition on the stress tensor at vanishing velocity, which is satisfied for Newtonian fluids, we discuss the stability behavior of stationary states at which the fluid is at rest and at constant temperature. In particular, we prove the existence of global small strong solutions for rather general isothermal non-Newtonian fluids. We also study bifurcation problems and show that subcritical bifurcations can occur. This effect can be seen only if the full energy equation is taken into consideration, that is, if the energy dissipation term is not dropped, as is done in the usual Boussinesq approximation. Bibliography: 29 titles.

A new type of wave behaviour is found for third order waves in a compressible inviscid dipolar fluid. Several stability-like results are presented for the theories of a viscous incompressible dipolar fluid and a mixture of two viscous incompressible fluids.

It is known that convection contributes to the natural mixing of melts due to the uneven distribution of their density. This mixing can be enhanced by various devices and by changing the conditions surrounding the flow region of the medium, e.g. by changing the pressure at the boundaries of the flow region of such fluid media. A pressure difference inside the region filled with a viscous fluid can induce its flow. A classic example of such a flow is the Poiseuille flow. Moreover, if the pressure distribution depends also on horizontal coordinates (i.e., there is a pressure difference not only in the vertical direction), then longitudinal pressure gradients also generate additional flows, which are superimposed on the classical flow. The paper studies the effect of pressure on the shear convective flow of a viscous incompressible binary fluid in a horizontal layer. To describe such flows, we use a system of equations of concentration convection, which includes the equation of motion of a viscous binary fluid, the equation of a concentration change, and the incompressibility equation. The solution of the system of constitutive equations is sought with the use of the class of generalized solutions, in which the velocities depend only on the vertical coordinate, the pressure and concentration being also linearly dependent on the longitudinal (horizontal) coordinates. As the boundary conditions, it is assumed that the no-slip condition is met at the lower impermeable boundary, that the upper boundary of the layer is motionless, and that the distribution of salinity and pressure is specified on it. The solution of the formulated boundary value problem is a set of polynomial functions. The highest-degree polynomial describes background pressure. The study of the properties of background pressure and the longitudinal pressure gradients is in the focus of attention. It is shown that background pressure decreases strictly monotonically with moving away from the lower boundary of the layer, regardless of the control parameters of the boundary value problem. In this case, the longitudinal pressure gradients are also described by strictly monotonic functions, but the nature of the monotonicity is determined by the values of the longitudinal concentration gradients specified at the upper boundary of the fluid layer. The relevant findings are illustrated.