Naghdi Theory Research Articles

Application of high-level Green–Naghdi theory to sill-controlled flows

Application of high-level Green–Naghdi theory to sill-controlled flows

Moored elastic sheets under the action of nonlinear waves and current

This study is concerned with the interaction between nonlinear water waves and uniform current with moored, floating elastic sheets, resembling floating solar panels, floating airports, tunnels and bridges, and floating energy systems. The Green–Naghdi theory is applied for the nonlinear wave–current motion, the thin plate theory is used to determine the deformations of the elastic sheet and Hooke’s law defines the effect of the mooring lines. The horizontal displacement of the floating sheet is determined by substituting the forces induced by the fluid flow and the tensions generated in the mooring lines into the equations of motion of the floating body. The resulting governing equations, boundary and matching conditions are solved in two dimensions with a finite-difference technique. The results are compared with the available numerical data. Overall, very good agreement is observed. In the model developed here, the sheet is allowed to drift due to the wave–current impact, and hence the mooring lines partially restrict both deformation and the horizontal motions of the sheet. The influence of the mooring lines on the dynamics of the floating sheet is assessed in terms of wave- and current-induced elastic deformations and surge movements of the sheet. It is demonstrated that the mooring lines attached to the leading and trailing edges of the sheet can be highly effective in mitigating the horizontal oscillations and vertical elastic deformations of the floating sheet subjected to the wave and current actions. Special attention is given to the horizontal periodic motions of the sheet, which are analysed by use of a Fourier transform technique. It is shown that the moored elastic sheet can oscillate at a frequency different from its exciting frequency as a result of restoring forces from the mooring lines, exciting resonance when both frequencies meet. Extensive study in a broad range of sheet parameters, mooring stiffnesses and wave–current conditions established the location of resonant regimes of different configurations of the moored systems. Analysis of wave reflection and transmission coefficient revealed that mooring lines of increasing stiffness intensify the wave reflection and, consequently, result in smaller energy transformation downwave.

Decay estimate of the viscoelastic plate with type II heat conduction in the whole space

This paper is concerning the problem determined by a viscoelastic plate when the heat is determined by the type II Green–Naghdi theory. Meanwhile the usual modeling is the combination of a conservative equation for the displacement and a dissipative equation for the temperature, we here change the role of the components and we propose a conservative component for the heat conduction, but a dissipative component for the displacement. In this situation, we consider this problem in the whole space and derive the decay estimates of solutions. Furthermore, the optimality of the decay estimate is discussed.

Electro-magnetic Field Effect on an Elastic Thermo-microstretch Porous Media Immersed in an Infinite Inviscid Liquid via Three-Phase Lag Model

PurposeThe aim of this work is to study porous thermo-elastic behavior in a half-space elastic micro-stretch media enveloped in an unbounded inviscid liquid with a magnetic field, utilizing three-phase lag model (3PHL) and Green–Naghdi theory of type III (G–N III).MethodsThe analytical technique used to obtain the ordinary differential equations was normal mode analysis. The Gauthier composite element is employed in an implementation to start comparing the observations—magnetic field on thermo-micro-stretch elastic porous medium completely covered in the endless fluid of G–N III theory with those for the model of 3PHL.ResultsThe magnetic field has been observed to make a significant impact over all physical quantities. Also, comparisons have been made for four various frequency values which explained that the frequency has an important role on the physical quantities.ConclusionOverall, the research on the magnetic field effect on thermo-micro-stretch porous media immersed in a liquid has several potential practical implications and engineering applications in various fields, such as biomedical, environmental, chemical, and energy engineering.

The effect of linear shear current on head-on collision of solitons

Head-on collision of two solitary waves in the presence of linear shear currents is studied by the use of the High-Level Green–Naghdi (HLGN) theory. The finite difference method is used to solve the HLGN model in the time-domain simulation. The initial values are obtained by the steady solution of solitary waves in the presence of linear shear currents. Shear currents with different velocities are considered to assess their effect on the solitary-wave collision. Three aspects of the head-on collision process in the presence of shear current are studied, namely, the wave elevation, velocity field, and particle trajectory. Results show that the background linear shear current significantly affects the wave elevation, velocity field, and particle trajectory during the head-on collision. It is observed that in the presence of the current, the wave elevation is narrower near the maximum surface displacement and is wider near the still-water level. It is also shown that near the seafloor, the horizontal velocity is opposite of the current direction, while it is following the current direction near the free surface. The opposite shear current results in the formation of a vortex in the fluid field. At the point of the collision, the vortex appears at a lower vertical position and shifts upstream of the current direction. Following the particle trajectories in the presence of the shear current, it is observed that the particles do not return to their initial positions after the head-on collisions, and the loop motions of the particles become smaller with larger current velocities.

Green-Naghdi Theory, Part B: Green-Naghdi Equations for Deep Water Waves

“Green — Naghdi Theory, Part A: Green — Naghdi (GN) Equations for Shallow Water Waves” have investigated the linear dispersion relations of high-level GN equations in shallow water. In this study, the GN equations for deep water waves are investigated. In the traditional GN equations for deep water waves, the velocity distribution assumption involves only one representative wave number. Herein, a new velocity distribution shape function with multiple representative wave numbers is employed. Further, we have derived the three-dimensional GN equations and analyzed the linear dispersion relations of the GN-3 and GN-5 equations. In this study, the finite difference method is used to simulate focus waves in the time domain. Additionally, the GN-5 equations are used to validate the wave profile and horizontal velocity distribution along water depth for different focused waves.

Damped waves under nonlocal Euler–Bernoulli and extended Green–Naghdi II theories in radiating thermoelastic nanobeams

We consider the problem of modelling nanobeams that dissipate thermal energy by radiation. We approach such a problem in a one-dimensional case by discussing the behavior of nonlocal nanobeams based on the Euler–Bernoulli assumptions. With these premises, we propose a thermoelastic model that takes into account the effects of thermal energy radiation to the external environment, employing an extension of the type II Green–Naghdi (GN-II) theory. We also deepen the formulated theoretical model making use of wave-form solutions, to highlight the presence of dissipative effects.

Thermal convection in a higher-gradient Navier–Stokes fluid

We discuss models for flow in a class of generalized Navier–Stokes equations. The work concentrates on producing models for thermal convection, analysing these in detail, and deriving critical Rayleigh and wave numbers for the onset of convective fluid motion. In addition to linear instability theory we present a careful analysis of fully nonlinear stability theory. The theories analysed all possess a bi-Laplacian term in addition to the normal spatial derivative term. The theories discussed are Stokes couple stress theory, dipolar fluid theory, Green–Naghdi theory, Fried–Gurtin–Musesti theory, and a second theory of Fried and Gurtin. We show that the Stokes couple stress theory and the Fried–Gurtin–Musesti theory involve the same partial differential equations while those of Green–Naghdi and dipolar theory are similar. However, we concentrate on boundary conditions which are crucial to understand all five theories and their differences.

Exponential stability for a thermoelastic Bresse system: Theoretical and numerical study

This study aims to investigate numerically and theoretically a thermoelastic Bresse system where the heat conduction given by the Green and Naghdi theories. Theoretically, we establish the existence and uniqueness of solutions as well as the exponential stability regardless of the system's parameters. Numerically, we present a finite element approximation, and we show the discrete energy decay. At the end, we provide some numerical results with an error estimate based on additional regularity of the solution.

Thomson effect and two‐temperature theory on a micropolar thermoelastic medium with voids under Green–Naghdi theory

AbstractThis paper aims to investigate the effect of two‐temperature and the Thomson effect in a micropolar thermoelastic medium with the void problem in the context of Green–Naghdi (G–N) theory. The normal mode analysis is used to obtain the analytical expressions of the considered variables. The non‐dimensional displacement, temperature, micro‐rotation, the change in the volume fraction field, and stress of the material are obtained and illustrated graphically. The variations of these quantities have been depicted graphically in the G–N theories of type‐II and III. From numerical calculations, the effects of the Thomson parameter and two‐temperature parameter on a homogeneous, isotropic, micropolar thermoelastic material with voids are revealed and discussed. Some particular cases of interest are deduced from the present investigation.

Magneto-thermoelasticity Green–Naghdi theory with memory-dependent derivative in the presence of a moving heat source

In the present work, a mathematical model of the Green-Naghdi thermoelasticity theory of type III (GN-III) with memory-dependent derivative (MDD) heat transfer for a perfectly conducting isotropic media has been constructed. The state-space and Laplace transform techniques are adopted for the solution of a half-space problem in the presence of a moving heat source with constant velocity. The inversion of the Laplace transforms is carried out using a numerical approach. Numerical results for all fields are given and illustrated graphically. Comparison is made with the results predicted by coupled thermoelasticity (DCT). The influences of MDD parameters and heat source speed on all fields are examined.

Drift of elastic floating ice sheets by waves and current: Multiple sheets

A nonlinear theoretical model for deformations, oscillations, and drift motions of multiple elastic ice sheets in shallow waters due to combined nonlinear waves and uniform current is presented. The model is based on the Green–Naghdi theory for the fluid motion and the thin plate theory for the deformation of the ice sheets. In principle, there are N number of the floating sheets with arbitrary lengths, drafts, and rigidities, which may be located at arbitrary distances from each other. Nonlinear waves of solitary and cnoidal types are considered, and there are no restrictions on the wave properties (wave height or wave period). The sheets, located at different positions, are shown to drift with different speeds, but surge in most of the wave conditions with equal amplitudes. It is shown systematically that wavelength and spacing between the sheets are the critical parameters determining the drift response of a set of freely floating ice sheets. When wavelength is equal to the distance between the centers of the sheets, they bend and drift in resonance, causing the largest wave reflection. The ambient current is found to affect the drift motion of the sheets nonlinearly. This work complements the part I paper of the same title, where drift motion of a single ice sheet was investigated.

Shallow Flows over Curved Beds: Application of the Serre–Green–Naghdi Theory to Weir Flow

AbstractThe Serre–Green–Naghdi (SGN) theory is used widely in maritime hydraulics for the computation of water waves from shallow to intermediate water depths. However, it has been largely ignored ...

Drift of elastic floating ice sheets by waves and current, part I: single sheet

The drift motion of a freely floating deformable ice sheet in shallow water subjected to incident nonlinear waves and uniform current is studied by use of the Green–Naghdi theory for the fluid motion and the thin plate theory for an elastic sheet. The nonlinear wave- and current-induced forces are obtained by integrating the hydrodynamic pressure around the body. The oscillations and translational motion of the sheet are then determined by substituting the flow-induced forces into the equation of motion of the body. The resulting governing equations, boundary and matching conditions are solved in two dimensions with a finite difference technique. The surge and drift motions of the sheet are analysed in a broad range of body parameters and various wave-current conditions. It is demonstrated that wavelength to sheet length ratio plays an important role in the drift response of the floating sheet, while the sheet mass and rigidity have comparatively less impact. It is also observed that while the presence of the ambient current changes the drift speed significantly (almost linearly), it has little to no effect on its oscillations. However, under the same ambient current, the drift speed changes remarkably by the wave period (or wavelength).

Finite Crack in a Thermoelastic Transversely Isotropic Medium Under Green-Naghdi Theory

In this paper, we have studied a model of finite linear Mode-I crack in a thermoelastic transversely isotropic medium under Green Naghdi theory. The crack is subjected to a prescribed temperature and a known tensile stress. The plane boundary surface is considered as isothermal and all the field variables are sufficiently smooth. The heat conduction equation is written under two temperature theory (2TT) for Green Naghdi model which contains absolute temperature as well as conductive temperature. The analytical expressions of displacement components, stress components and temperature variables are obtained by normal mode analysis and matrix inversion method. Comparisons have been made within Green Naghdi (G-N) theory of type I, type II and type III for displacement, stress and absolute temperature variables against the crack width for a transversely isotropic material (Cobalt) by virtues of graphs. Also, Comparison have been made among displacement, thermal stress and absolute temperature for different depths.

Strain gradient and Green–Naghdi-based thermoelastic wave propagation with energy dissipation in a Love–Bishop nanorod resonator under thermal shock loading

The thermoelastic wave propagation analysis is developed in a Love–Bishop nanorod resonator employing the strain gradient elasticity and Green–Naghdi theories with energy dissipation for the first time. The assumed nanorod resonator is subjected to thermal shock loading. The variational principle is applied to derive the coupled system of partial differential equations for temperature and displacement fields in a Love–Bishop nanorod. An analytical solution is proposed to solve the governing equations in Laplace domain. To obtain the temporal variation of fields’ variables, a proper Laplace inversion technique is employed in the problem. The size effects in the nano-sized Love–Bishop rods are taken into account with five small-scale parameters including three higher-order materials length parameters, the micro-length inertia and thermal parameters. Two types of thermal shock loadings such as laser pulse-induced suddenly increasing temperature are utilized in this work, and the laser shock-induced thermoelastic wave propagations in both temperature and displacement fields are illustrated. The effects of the higher-order materials length parameters and the micro-length inertia and thermal parameters on the propagation of thermal and elastic waves are studied in detail.

General decay for a viscoelastic-type Timoshenko system with thermoelasticity of type III

We consider a one-dimensional thermoelastic Timoshenko system, where the heat conduction is given by Green and Naghdi theories and acting on shear force. We prove that the unique damping given by the memory is sufficiently strong to stabilize the system exponentially, depending on a new relationship between the coefficients of the system and under some assumptions on the kernel of the memory term. In fact, we establish a general decay result, of which exponential and polynomial decay results are special cases.

Gaussian thermal shock-induced thermoelastic wave propagation in an FG multilayer hybrid nanocomposite cylinder reinforced by GPLs and CNTs

In this paper, a new modified micromechanical model is proposed to obtain the mechanical and thermal properties of a reinforced functionally graded (FG) multilayer hybrid nanocomposite cylinder for thermoelastic wave propagation analysis with energy dissipation using Green–Naghdi theory. The FG multilayer hybrid nanocomposite cylinder is assumed to be under Gaussian thermal shock loading and also it is reinforced by graphene platelets (GPLs) and carbon nanotubes (CNTs). The cylinder is assumed to be made of multi-layers (sub-cylinders), and each layer is reinforced by a uniform distribution of GPLs and CNTs. A modified micromechanical model is proposed to calculate the thermo-mechanical properties with nonlinear grading patterns of the GPLs and CNTs along the radial direction of the cylinder. The nonlinear grading patterns of the GPLs and CNTs distributions along the radial direction of the whole cylinder can be created using a suitable arrangement of the layers (sub-cylinders). An effective meshless collocation method based on the generalized finite difference (GFD) method and the Newmark method are employed to solve the coupled governing equations of the GN-based coupled thermoelasticity analysis. The effects of the key parameters such as the weight fraction of the GPLs and CNTs and volume fraction index on the thermoelastic wave propagations and dynamic behaviors of the field variables are studied in detail. Also, the effects of the reinforcement by GPLs and/or CNTs in a hybrid nanocomposite on the transient behaviors of cylinder in both temperature and displacement fields are compared. The accuracy and stability of the presented meshless method and also the proposed new modified micromechanical model is verified by the published reference data.

Magneto-thermoelastic wave propagation in a finitely conducting medium: A comparative study for three types of thermoelasticity I, II, and III

The present work is concerned with the propagation of magneto-thermoelastic plane waves of assigned frequency in a homogeneous isotropic finitely conducting elastic medium permeated by a uniform magnetic field. The problem has been formulated under the purview of Green and Naghdi theory of thermoelasticity of type-III (GN-III). Interactions among elastic, thermal and magnetic fields have been taken into account. A dispersion relation is derived to investigate the nature of waves propagating in the medium. Perturbation technique has been employed to obtain the solution of the dispersion relation, when the magneto-thermoelastic coupling parameter is small. This enables to identify three different types of waves. Phase velocity, specific loss and penetration depth have been discussed. Using derived analytical expressions, numerical results have been computed for the variation of several physical quantities of interest. The impact of the magnetic field has been studied. Computational results have been presented graphically for all the three cases GN-I, GN-II, GN-III. A comparative study has been performed for these three cases. The study reveals that in the case of GN-II theory, coupled magneto-thermoelastic waves are unattenuated and non-dispersive in contrast to the cases of GN-I and GN-III theories.

Thermomechanical Waves in an Axisymmetric Rotating Disk Using Refined Green–Naghdi Models

In this paper, four thermoelasticity models are applied to investigate the thermo-mechanical waves in an axisymmetric rotating disk. For this purpose, some modifications of Green and Naghdi’s generalized thermoelasticity theory have been made. These novel dual- and triple-phase-lag models are presented. The closed-form solution governing equations of thermomechanical waves in the rotating disk has been presented. Some validation results are tabulated to compare the simple and refined dual- and triple-phase lag models based upon the Green–Naghdi theory. The effects of dimensionless time and angular speed on all variables are investigated. The validity of results is acceptable and benchmark results are tabulated to aid other researchers in their upcoming comparisons.