Impermeability Condition Research Articles

Finite element model to investigate the dynamic instability of ring stiffened conical shells subjected to flowing fluid

In this study, the vibration stability (i.e., static divergence) and critical velocity of fluid-conveying, ring-stiffened, truncated conical shells are investigated under various boundary conditions. The shell is characterized using Sanders’ theory, while the fluid is modeled using a velocity potential approach with the impermeability condition at the fluid-shell interface. Using linear superposition, the natural frequencies corresponding to each flow velocity are determined by satisfying the dynamic characteristic equation and boundary conditions. Critical velocities are identified where the natural frequencies vanish, indicating static divergence. Parametric studies are conducted to investigate the effect of ring stiffeners on the critical velocities with respect to the semi-cone angle, number of rings, and boundary conditions. The proposed model is validated through comparison with published data. It is found that the rings significantly affect the stability of the cone under different boundary conditions. Instability in stiffened shells occurs at higher critical fluid velocities than in unstiffened shells across all boundary conditions. An increase in the vertex angle leads to a decrease in critical flow discharge.

Optimal distributed control for a Cahn-Hilliard-Darcy system with mass sources, unmatched viscosities and singular potential

We study a Cahn-Hilliard-Darcy system with mass sources, which can be considered as a basic, though simplified, diffuse interface model for the evolution of tumor growth. This system is equipped with an impermeability condition for the averaged velocity u as well as homogeneous Neumann boundary conditions for the phase function ϕ and the chemical potential µ. The source term in the convective Cahn-Hilliard equation contains a control R that can be thought as a drug or a nutrient in applications. Our goal is to study a distributed optimal control problem in the two-dimensional setting with a cost functional of tracking-type. In the physically relevant case with unmatched viscosities for the binary fluid mixtures and a singular potential, we first prove the existence and uniqueness of a global strong solution with ϕ being strictly separated from the pure phases ±1. This well-posedness result enables us to characterize the control-to-state mapping S: R→ϕ. Then we obtain the existence of an optimal control, the Frechet differentiability of S and first-order necessary optimality conditions expressed through a suitable variational inequality for the adjoint variables. Finally, we prove the differentiability of the control-to-costate operator and establish a second-order sufficient condition for the strict local optimality.

Numerical modeling and analysis of fluid-filled truncated conical shells with ring stiffeners

This study uses a hybrid finite element method to predict dynamic behavior of truncated conical shells with ring stiffeners under fluid loading. The proposed approach combines classical shell theory and the finite element method, making use of displacement functions derived from exact solutions of Sanders’ shell equilibrium equations for conical shells. The analysis of the shell-fluid interface involves leveraging the velocity potential, Bernoulli’s equation, and impermeability conditions to determine an explicit expression for fluid pressure. To the best of our knowledge, this paper is the first to compare the methods applied to ring-stiffened shells against other numerical and experimental findings. Our results on conical shells in various conditions, with and without ring stiffeners, are largely consistent with previous findings. This study also explores the influence of geometric parameters, stiffener quantity, cone angle, and applied boundary conditions on the natural frequency of fluid-loaded ring-stiffened conical shells. The paper concludes with a discussion of several useful implications for further research.

Shape optimization for the Stokes system with threshold leak boundary conditions

This paper discusses the process of optimizing the shape of systems that are controlled by the Stokes flow with threshold leak boundary conditions. In the theoretical part it focuses on studying the stability of solutions to the state problem in relation to a specific set of domains. In order to facilitate computation, the slip term and impermeability condition are regulated. In the computational part, the optimized portion of the boundary is defined using Bézier polynomials, in order to create a finite dimensional optimization problem. The paper also includes numerical examples to demonstrate the computational efficiency of this approach.

Nonlinear geometric fluid-structure interaction model of multilayered sandwich plates in contact with unbounded or bounded fluid flow

Based on the Kirchhoff-Love theory and von Kármán's geometric nonlinearity, the nonlinear vibration analysis of multilayered sandwich plates under ideal flow is presented. Rectangular plates are considered to be laminated of orthotropic composite layers as upper and lower face sheets as well as a central lightweight core made of functionally graded (FG), anisogrid lattice or metal foam materials. Three coupled equations of motion are reformulated into one decoupled governing equation in terms of transverse deflection (w) using the method of inverse differential operator. For the first time, an impermeability condition is developed that considers the moderately large deflection of the contact surface in order to correct the pressure distribution of fluid flow. Based on the potential flow theory, closed-form expressions of hydrodynamic pressure are obtained for different fluid boundary conditions including infinite depth of fluid, flow bounded by a rigid wall and flow between two identical elastic plates. Then, the Galerkin and multiple-scale perturbation methods are used to obtain closed-form solution of nonlinear natural frequency in terms of maximum vibration amplitude. Finally, case studies are presented to investigate the effects of geometric ratios, boundary condition, material distribution, layup, characteristics of flow and geometric nonlinearity on the response quantities. Findings indicate that the large deformation term in the developed impermeability condition has significant effect on the pressure distribution and its effect is more pronounced for thicker plates with larger mode number in the flow direction. Therefore, the novel nonlinear fluid-structure interaction model should be considered to study flow-induced dynamic behavior of sandwich plates with different boundary conditions. In addition, the snap-through behavior is seen in the nonlinear frequency-upstream speed curves and the incipient points of dynamic instability can vary remarkably with the fluid boundary conditions.

Численное исследование двухмерного симметричного течения несжимаемой жидкости с зависящей от координат вязкостью между частично неровными плоскостями

The structure of a symmetric two-dimensional unsteady flow between two planes having a periodic step-like irregularity is investigated by numerical methods using the finite element package Freefem++. The kinematic viscosity of the fluid is chosen as typical for turbulent flows, depending on the coordinates as 𝜇 = 𝜇0(𝜂2(𝑥) − 𝑧2), 𝜂(𝑥) = ℎ(𝑥) + 𝛿, where 𝜇0 is the characteristic viscosity, 𝛿 is the roughness, ℎ(𝑥) is the function defining the boundary 𝑧 = 𝜂(𝑥) (profiles of planes mirror-symmetric with respect to 𝑧 = 0), 𝑥, 𝑧 are longitudinal and transverse coordinates along the main flow. In contrast to the usual parabolic profile of the Poiseuille flow between the planes, with the specified choice of viscosity, the flow profile is logarithmic and has a singularity at the boundary of the region, to prevent which the roughness of the boundary is introduced. A distinctive feature of the problem under study is the setting of boundary conditions on an irregular part of the boundary. On plane sections of the boundary, we maintain the usual non-slip conditions for a viscous liquid, but on the irregular part of the boundary we set only the impermeability conditions of the boundary for the liquid (no sticking!). Computational experiment - numerical solution of the Navier-Stokes equations for a viscous incompressible fluid using a modified penalty method, showed that for a wide set of parameters, the structure of the flow described by the initial non-stationary complete problem and the quasi-stationary flow described by the asymptotic model constructed in the presented paper, for which there is an exact solution, are in good agreement - in the non-stationary flow, a stable regular system of vortices is established concentrated near those surface irregularities for which the boundary has a negative curvature (`deepening’of boundary).

Stokes problem with the Coulomb stick–slip boundary conditions in 3D: Formulations, approximation, algorithms, and experiments

The paper deals with the approximation and numerical realization of the Stokes system in 3D with Coulomb’s slip boundary conditions. The weak velocity–pressure formulation leads to an implicit inequality type problem which is discretized by the P1+bubble/P1 elements. To regularize the discrete non-smooth slip term and to release the discrete impermeability condition the duality approach is used. For numerical realization of the resulting saddle-point problem two strategies are proposed, namely (i) its fixed-point formulation solved by the method of successive approximations (ii) the direct numerical solution of the saddle-point problem. The semi-smooth Newton method is used to solve non-smooth equations appearing in both these approaches.

Conservation, impermeability and potential vorticity in relativistic magnetohydrodynamics

Abstract The conservation and impermeability conditions are reformulated utilising differential forms and generalised to spacetime. The thermodynamic and electromagnetic potential vorticity scalar fields are defined for relativistic magnetohydrodynamics and their evolution equations are derived.

Front-tracking simulation of the wetting behavior of an impinging droplet using a relaxed impermeability condition and a generalized Navier boundary condition

In this paper, we present a new method for simulating the wetting behavior of a droplet impinging on a solid wall. It works continuously from the high-speed regime (capillary number ∼ 1) to the low-speed regime (capillary number < 10−2). We combined front-tracking with a generalized Navier boundary condition (GNBC)—to represent low-speed wetting that takes into account slip at the molecular scale—with a rolling-belt model to represent high-speed wetting due to the rolling motions of fluid elements. Usually, a rolling-belt model requires both a threshold distance and compulsory switching from the high-speed regime. To overcome this defect, we utilized the modified relaxed impermeability condition originally proposed by Lukyanov and Pryer (Langmuir, Vol. 33, (2017), pp. 8582–8590.) for the rolling-belt model. By applying this condition, interface markers very near the wall can have a velocity normal to the wall as they approach it in reproducing the rolling motion. There is consequently no need of either an adjustable threshold distance or a compulsory switch-off of the rolling-belt model. We confirmed the results obtained using the proposed simulation method by comparing them with published experimental data. The validity and applicability of the present method are confirmed by the very good agreement of the simulations with the experimental data for a variety of conditions.

Improvement of the stability and accuracy of solid-wall immersed boundary schemes for the linearized Euler equations using boundary constraints

A Cartesian cut-cell method with explicit high-order finite-difference stencils for solid walls at internodal wall positions has been developed to solve the linearized Euler equations. For this, additional boundary constraints besides the physical impermeability condition have been incorporated into Hermite-based finite-difference stencils near the boundary. We demonstrate that our methodology and the ghost point formulations of Tam & Dong (1994) and Gloerfelt (2001) are equivalent when a wall intersects with a grid point. Other than Tam & Kurbatskii's (1997) extrapolation-based cut-cell treatment, our formulation can be regarded as a straightforward generalization of the above-mentioned ghost point formulations. A numerical reflection problem demonstrates improved accuracy in 1D for high-order schemes of sixth- and 12th-order. The scattering of a plane wave with 8 PPW for a 7-point-, and 5 PPW for an 11-point boundary scheme at a 2D-cylinder in fluid at rest demonstrates the extension of the proposed method to higher dimensions. For the 1D case, stability analysis has been carried out to show the significant impact of using boundary constraints to improve the numerical stability of a boundary scheme. In the multidimensional case, weak instabilities have to be suppressed by selective filtering. Our new formalism for boundary methods enables the assessment of the propagation characteristics of modified 1D-boundary stencils for first derivatives by Fourier analysis. It is shown that the incorporation of boundary constraints improves the modified wavenumber signature of the boundary stencils significantly.

Free vibrational behavior of functionally graded carbon nanotube composite fluid-filled cylindrical-conical shell: Analytical and experimental investigation

This investigation analyzes the effects of internal static fluid on the free vibrational behavior of joined shells made of carbon nanotube (CNT) fiber‐reinforced composite. Various types of functionally graded CNTs composite with temperature dependence are considered. Classic shell theory and Love’s assumptions combined with Hamilton’s principle are used to obtain motion equations. Bernoulli equation, velocity potential, and impermeability condition are employed to the coupling effects of fluid pressure with motion equations of shells. Continuity conditions in the joint point of conical and cylindrical sections are supposed to couple equations of two parts. The generalized differential quadrature (GDQ) method is employed to solve equations for various boundary conditions. Experimental modal tests on a sample made by the 3-D printer and prior others’ research are used to validate the accuracy of results. Consequently, parametric studies are expressed, such as longitudinal and circumferential mode numbers, semi-vertex angle, distributions of CNTs, CNTs fraction volumes, fluid density, shell thickness, and operating temperature. According to the results, although fluid decreases the frequency generally, the process of changing frequency in terms of other parameters is similar.

On nonlinear free vibration of externally compressible fluid-loaded sandwich cylindrical shells: Curvature nonlinearity in bending and impermeability condition

A nonlinear fluid–structure interaction (FSI) model is presented for nonlinear vibration analysis of sandwich cylindrical shells subjected to an external compressible flow by considering the curvature nonlinearity in impermeability condition and bending. The sandwich shells are made of two face sheets and a central core of advanced materials including functionally graded (FG), metal foam, and anisogrid lattice composite. Based on the Kirchhoff–love hypotheses with the geometric nonlinearities in the normal strain and curvature of mid-surface, one decoupled nonlinear integral–differential equation is obtained for axisymmetric bending vibration of sandwich cylindrical shells. For the first time, the nonlinear impermeability condition is developed in order to take into account the large deformation of thin structures in FSI modeling. In both subsonic and supersonic regimes of steady irrotational compressible flow, a closed-form expression for nonlinear pressure distribution is presented according to the conservation of mass and generalized Bernoulli equation of extended potential theory. Using the Galerkin’s method in conjunction with the multiple-scale perturbation technique, the decoupled equation is solved to obtain nonlinear frequency and time–history response. Finally, parametric studies are carried out to investigate the effects of geometric ratios, boundary conditions, fluid specifications, and material variation on the responses of sandwich shells with different layups such as [Al; FG core; ZrO2], [Al; foam core; ZrO2], [Al; lattice core; Al]. Significant differences in pressure variation due to curvature nonlinearity are observed for short shells with larger vibration amplitude and larger number of half-waves in the axial direction. Therefore, the linear vibration analysis is inadequate for analyzing short sandwich cylindrical shells surrounded by an external fluid flow especially in the case of movable ends.

Effect of solid waste landfill leachate contaminants on hydraulic conductivity of landfill liners.

Landfilling is one of the most widely used methods to reduce the impact on the environment and human health by ensuring the management of solid wastes. For the process in question to be called landfill, the landfill leachate must be controlled and liner impermeability conditions must be provided. For this reason, compacted clay liners (CCL) and geosynthetic clay liners (GCL) with very low hydraulic conductivity are often used as hydraulic barriers in landfills to prevent the risk of leachate mixing with groundwater. However, as a result of various interactions between leachate-clay liners, changes occur in the hydraulic conductivity of the liners. In this review, the change (increase/decrease) in the hydraulic conductivity of the landfill liners caused by the contaminants in the leachate composition and the mechanisms responsible for this change were examined. In addition, deficiencies in the literature on this subject were identified and directions for future studies were presented.

Характеристическая задача Коши стандартного вида для описания истечения политропного газа в вакуум с косой стенки

Начально-краевая задача для системы уравнений газовой динамики, решение которой описывает разлет политропного газа в вакуум с косой стенки в пространстве автомодельных переменных $x/t$, $y/t$ в общем несогласованном случае, приведена к характеристической задаче Коши стандартного вида в пространстве новых независимых переменных $\vartheta$, $\zeta$. Уравнение $\vartheta=0$ задает характеристическую поверхность, через которую двойная волна примыкает к известному решению - центрированной волне Римана. Уравнение $\zeta=0$ означает, что за новую координатную ось выбирается косая стенка, на которой выполняется условие непротекания. Для этой новой начально-краевой задачи в отличие от известного решения аналогичной задачи, полученного С. П. Баутиным и С. Л. Дерябиным в пространстве специальных переменных, доказана теорема существования и единственности решения системы уравнений газовой динамики в пространстве физических автомодельных переменных в виде сходящегося бесконечного ряда. Описан алгоритм построения коэффициентов ряда.

Nonlinear vibrations and stability of rotating cylindrical shells conveying annular fluid medium

This study aims to investigate the nonlinear dynamic characteristics of annular cylinders coupled with a fluid medium to identify the effective parameters on stability margins at different rotation speeds. To achieve this, stability and nonlinear vibrations of rotating cylindrical shells containing incompressible annular fluid are considered. The fluid medium is bounded by a rigid external cylinder. Sanders–Koiter kinematic assumptions are utilized to determine the geometrically nonlinear structural equations of motion. These equations are then used to investigate finite amplitude vibrations of various fluid-loaded shells at different states. A penalty approach is introduced by using boundary springs with arbitrary stiffness to simulate any desired set of end conditions. Both driven and companion modes of vibration are included in the solution algorithm to capture the traveling wave response. Fluid equations are determined by using the linearized Navier–Stokes equations. The two sets of governing equations are coupled through the flow impermeability condition on the internal shell’s interface and the zero fluid particle velocity condition on the external boundary. Nonlinear numerical solutions to rotating shells with extremal fluid loading are conducted by the Runge–Kutta direct integration technique. The minimum number of driven/companion modes is determined and the results are verified with the available literature. Results show a significant exchange of energy between the driven and companion modes which consequently initiates a traveling wave response in rotating shell configurations. Moreover, the centrifugal loading and external excitation are shown to be effective on the oscillation amplitude as rotation speed changes leading to alteration of the response quality from periodic to chaotic.

Control Problem Related to 2D Stokes Equations with Variable Density and Viscosity

We study an optimal control problem for the stationary Stokes equations with variable density and viscosity in a 2D bounded domain under mixed boundary conditions. On in-flow and out-flow parts of the boundary, nonhomogeneous Dirichlet boundary conditions are used, while on the solid walls of the flow domain, the impermeability condition and the Navier slip condition are provided. We control the system by the external forces (distributed control) as well as the velocity boundary control acting on a fixed part of the boundary. We prove the existence of weak solutions of the state equations, by firstly expressing the fluid density in terms of the stream function (Frolov formulation). Then, we analyze the control problem and prove the existence of global optimal solutions. Using a Lagrange multipliers theorem in Banach spaces, we derive an optimality system. We also establish a second-order sufficient optimality condition and show that the marginal function of this control system is lower semi-continuous.

Brenier–Schrödinger problem on compact manifolds with boundary

ABSTRACT We consider the Brenier–Schrödinger problem on compact manifolds with boundary. In the spirit of a work by Arnaudon, Cruzeiro, Léonard, and Zambrini, we study the kinetic property of regular solutions and obtain a link to the Navier–Stokes equations with an impermeability condition. Moreover, by a method of taking quotients by reflection groups, we enhance the class of models for which the problem admits a unique solution.

A benchmark problem to evaluate implementational issues for three-dimensional flows of incompressible fluids subject to slip boundary conditions

We consider flows of an incompressible Navier–Stokes fluid in a tubular domain with Navier’s slip boundary condition imposed on the impermeable wall. We focus on several implementational issues associated with this type of boundary conditions within the framework of the standard Taylor-Hood mixed finite element method and present the computational results for flows in a tubular domain of finite length with one inlet and one outlet. In particular, we present the details regarding variants of the Nitsche method concerning the incorporation of the impermeability condition on the wall. We also find that the manner in which the normal to the boundary is numerically implemented influences the nature of the computational results. As a benchmark, we set up steady flows in a tube of finite length and compare the computational results with the analytical solutions. Finally, we identify various quantities of interest, such as the dissipation, wall shear stress, vorticity, pressure drop, and provide their precise mathematical definitions. We document how well these quantities are computationally approximated in the case of the benchmark.Although the geometry of the benchmark is simple, the correct computational results require careful selection of numerical methods and surprisingly non-trivial computational resources. Our goal is to test, using the setting with a known analytical solution, a robust computational tool that would be suitable for computations on real complex geometries that have relevance to problems in engineering and medicine. The model parameters in our computations are chosen based on flows in large arteries.

Numerical investigation of double-diffusive convection at vibrations

The problem of convection in a rectangular vertical cell, located in the gravity field and under the action of high-frequency vertical vibrations, is solved using the direct numerical simulation. At the initial moment of time, the system is formed by two horizontal layers of miscible liquids: the lower (heavy) layer is an aqueous solution of sodium chloride, while the upper one is a sugar solution. A two-dimensional formulation is considered; ANSYS Fluent software package is used as a solver. The impermeability and no-slip conditions are satisfied at the cell boundaries. As a result of the difference in the diffusion coefficients of dissolved species, layers with unstable stratification are formed over time near the contact zone. This leads to the onset of convection in the form of thin ascending and descending fingers. The vibrations, on average, lead to a decrease in the growth rate of fingers and the velocity of convection. In the case of relatively high vibrational overloads, a more regular convective structure forms in the diffusion layer leading to a more pronounced mixing of fluids.

Dual strategies for solving the Stokes problem with stick–slip boundary conditions in 3D

The paper deals with the numerical realization of the 3D Stokes flow subject to threshold slip boundary conditions. The weak velocity–pressure formulation leads to an inequality type problem that is approximated by a mixed finite element method. The resulting algebraic system is non-smooth. Besides the pressure, three additional Lagrange multipliers are introduced: the discrete normal stress releasing the impermeability condition and two discrete shear stresses regularizing the non-smooth slip term. Eliminating the discrete velocity component we obtain the minimization problem for the smooth functional, expressed in terms of the pressure, the normal, and the shear stresses. This problem is solved either by a path following variant of the interior point method or by the semi-smooth Newton method. Numerical scalability is illustrated by computational experiments.