Nonlinear instability of two dielectric viscoelastic fluids

2004 ◽  
Vol 82 (12) ◽  
pp. 1109-1133 ◽  
Author(s):  
Galal M Moatimid
Keyword(s):  
Boundary Conditions ◽  
Electric Fields ◽  
Multiple Scales ◽  
Landau Equation ◽  
Equations Of Motion ◽  
Nonlinear Boundary Conditions ◽  
Interfacial Wave ◽  
Surface Evolution ◽  
Stability Diagrams ◽  
Viscoelastic Effects

A weakly nonlinear interfacial wave propagating between two dielectric fluids and influenced by an oblique electric field is studied. The analysis considers the surface tension and viscoelastic effects. Due to the presence of streaming and viscoelasticity, a mathematical simplification is considered. The viscoelastic contribution is demonstrated through the boundary conditions. Therefore, the equations of motion are solved in the absence of the viscoelastic effects. The solutions of the linearized equations of motion under the nonlinear boundary conditions lead to a nonlinear characteristic equation governing the surface evolution. This equation is accomplished by utilizing cubic nonlinearity. Taylor theory is adopted to expand the characteristic nonlinear equation in the light of the multiple-scales technique. The perturbation analysis produces two levels of the solvability conditions, which are used to construct the Ginzburg–Landau equation. Stability criteria are discussed both theoretically and computationally in which stability diagrams are obtained. Under appropriate data choices, we can recover some reported works as limiting cases. The effects of the orientation of the electric fields on the stability configuration in linear as well as nonlinear approaches are discussed. PACS Nos.: 47.65.+a, 47.20.–k, 47.50.+d

The Influence of Nonlinear Boundary Conditions on the Nonplanar Autoparametric Responses of an Inextensible Beam

2001 ◽  
Author(s):  
Haider N. Arafat ◽  
Ali H. Nayfeh
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Nonlinear Behavior ◽  
Nonlinear Boundary Conditions ◽  
Nonlinear Springs ◽  
Out Of Plane

Abstract The nonplanar responses of a beam clamped at one end and restrained by nonlinear springs at the other end is investigated under a primary resonance base excitation. The beam’s geometry and the springs’ linear stiffnesses are such that the system possesses a one-to-one autoparametric resonance between the nth in-plane and out-of-plane modes. The beam is modeled using Euler-Bernoulli theory and includes cubic geometric and inertia nonlinearities. The objective is to assess the influence of the nonlinear boundary conditions on the beam’s oscillations. To this end, the method of multiple scales is directly applied to the integral-partial-differential equations of motion and associated boundary conditions. The result is a set of four nonlinear ordinary-differential equations that govern the slow dynamics of the system. Solutions of these modulation equations are then used to characterize the system’s nonlinear behavior.

Response variations of a cantilever beam–tip mass system with nonlinear and linearized boundary conditions

2018 ◽  
Vol 25 (3) ◽  
pp. 485-496 ◽  
Author(s):  
Vamsi C. Meesala ◽  
Muhammad R. Hajj
Keyword(s):  
Boundary Conditions ◽  
Cantilever Beam ◽  
Nonlinear Boundary ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Boundary Conditions ◽  
Distributed Parameter ◽  
Governing Equations ◽  
Mass System ◽  
Tip Mass

The distributed parameter governing equations of a cantilever beam with a tip mass subjected to principal parametric excitation are developed using a generalized Hamilton's principle. Using a Galerkin's discretization scheme, the discretized equation for the first mode is developed for simpler representation assuming linear and nonlinear boundary conditions. The discretized governing equation considering the nonlinear boundary conditions assumes a simpler form. We solve the distributed parameter and discretized equations separately using the method of multiple scales. Through comparison with the direct approach, we show that accounting for the nonlinear boundary conditions boundary conditions is important for accurate prediction in terms of type of bifurcation and response amplitude.

Nonlinear electrohydrodynamic Rayleigh–Taylor instability with mass and heat transfer: effect of a normal field

1994 ◽  
Vol 72 (9-10) ◽  
pp. 537-549 ◽  
Author(s):  
Abou El Magd A. Mohamed ◽  
Abdel Raouf F. Elhefnawy ◽  
Y. D. Mahmoud
Keyword(s):  
Heat Transfer ◽  
Multiple Scales ◽  
Landau Equation ◽  
Finite Thickness ◽  
Surface Charges ◽  
Stability Diagrams ◽  
Dielectric Fluids ◽  
Mass And Heat Transfer ◽  
The Stability ◽  
Cubic Nonlinear Schrödinger Equation

The nonlinear electrohydrodynamic stability of two superposed dielectric fluids with interfacial transfer of mass and heat is presented for layers of finite thickness. The fluids are subjected to a normal electric field in the absence of surface charges. Using a technique based on the method of multiple scales it is shown that the evolution of the amplitude is governed by a Ginzburg–Landau equation. When the mass and heat transfer are neglected, the cubic nonlinear Schrödinger equation is obtained. Further, it is shown that, near the marginal state, a nonlinear diffusion equation is obtained in the presence of mass and heat transfer. The various stability criteria are discussed both analytically and numerically and the stability diagrams are obtained.

Nonlinear Electrohydrodynamic Stability of Two Superposed Walters B′ Viscoelastic Fluids in Relative Motion Through Porous Medium

2013 ◽  
Vol 29 (4) ◽  
pp. 569-582 ◽  
Author(s):  
M. F. El-Sayed ◽  
N. T. Eldabe ◽  
M. H. Haroun ◽  
D. M. Mostafa
Keyword(s):  
Surface Tension ◽  
Porous Medium ◽  
Electric Fields ◽  
Multiple Scales ◽  
Landau Equation ◽  
Three Dimensions ◽  
Surface Charges ◽  
Fluid Velocities ◽  
Medium Permeability ◽  
The Stability

ABSTRACTA nonlinear stability of two superposed semi-infinite Walters B′ viscoelastic dielectric fluids streaming through porous media in the presence of vertical electric fields in absence of surface charges at their interface is investigated in three dimensions. The method of multiple scales is used to obtain a Ginzburg-Landau equation with complex coefficients describing the behavior of the system. The stability of the system is discussed both analytically and numerically in linear and nonlinear cases, and the corresponding stability conditions are obtained. It is found, in the linear case, that the surface tension and medium permeability have stabilizing effects, and the fluid velocities, electric fields and kinematic viscoelastici-ties have destabilizing effects, while the porosity of porous medium and kinematic viscosities have dual role on the stability. In the nonlinear case, it is found that the fluid velocities, kinematic viscosities, kinematic viscoelasticities, surface tension and porosity of porous medium have stabilizing effects; while the electric fields and medium permeability have destabilizing effects.

Vibration Response and Parametric Instability in Beams With Combined Quadratic and Cubic Material Nonlinearities

1995 ◽  
Author(s):  
David Chelidze ◽  
Kambiz Farhang ◽  
Tyler J. Selstad
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Multiple Scales ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Response Amplitude ◽  
Nonlinear Material ◽  
Mode Shapes ◽  
Cross Sectional

Abstract Parametric stability in beams with combined quadratic and cubic material nonlinearities is examined. A general mathematical model is developed for parametrically excited beams accounting for their nonlinear material characteristic. Second- and forth-order nonlinear differential equations are found to govern the axial and transverse motions, respectively. Expansions for displacements are assumed in terms of the linear undamped free-oscillation modes. Boundary conditions are applied to the expansions for displacements to determine the mode shapes. Multiplying the equations of motion by the corresponding shape functions, accounting for their orthogonal properties, and integrating over the beam length, a set of coupled nonlinear differential equations in the time-dependent modal coefficients is obtained. Utilizing the method of multiple scales, frequency response as well as response versus excitation amplitude are obtained for two beams of different cross sectional areas. Results are presented for three boundary conditions. It is found that, qualitatively, the response is similar for all the boundary conditions. Quantitative comparison of the cases considered indicate that the highest response amplitude occurs for the cantilever beam with the end mass. The bifurcation points for simply supported beam occur at lower excitation parameter value. It is apparent that more slender columns have larger response amplitude.

Model of vortex flow of charge in a vessel with turbine impeller

1987 ◽  
Vol 52 (8) ◽  
pp. 1888-1904
Author(s):  
Miloslav Hošťálek ◽  
Ivan Fořt
Keyword(s):  
Boundary Conditions ◽  
Equations Of Motion ◽  
Theoretical Data ◽  
Eddy Diffusion ◽  
Quantitative Agreement ◽  
Creeping Flow ◽  
Model Parameters ◽  
Mean Flow ◽  
Two Dimensional Flow ◽  
Vorticity Transport

A theoretical model is described of the mean two-dimensional flow of homogeneous charge in a flat-bottomed cylindrical tank with radial baffles and six-blade turbine disc impeller. The model starts from the concept of vorticity transport in the bulk of vortex liquid flow through the mechanism of eddy diffusion characterized by a constant value of turbulent (eddy) viscosity. The result of solution of the equation which is analogous to the Stokes simplification of equations of motion for creeping flow is the description of field of the stream function and of the axial and radial velocity components of mean flow in the whole charge. The results of modelling are compared with the experimental and theoretical data published by different authors, a good qualitative and quantitative agreement being stated. Advantage of the model proposed is a very simple schematization of the system volume necessary to introduce the boundary conditions (only the parts above the impeller plane of symmetry and below it are distinguished), the explicit character of the model with respect to the model parameters (model lucidity, low demands on the capacity of computer), and, in the end, the possibility to modify the given model by changing boundary conditions even for another agitating set-up with radially-axial character of flow.

Ordinary Differential Equations with Nonlinear Boundary Conditions

2002 ◽  
Vol 9 (2) ◽  
pp. 287-294
Author(s):  
Tadeusz Jankowski
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Existence Results ◽  
Monotone Iterative Technique ◽  
Lower And Upper Solutions ◽  
Iterative Technique ◽  
Monotone Iterative

Abstract The method of lower and upper solutions combined with the monotone iterative technique is used for ordinary differential equations with nonlinear boundary conditions. Some existence results are formulated for such problems.

Nonlinear Vibrations of Two-Span Composite Laminated Plates with Equal and Unequal Subspan Lengths

2017 ◽  
Vol 9 (6) ◽  
pp. 1485-1505
Author(s):  
Lingchang Meng ◽  
Fengming Li
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Laminated Plates ◽  
Laminated Plate ◽  
Composite Laminated Plate ◽  
Vibration Localization ◽  
Composite Laminated Plates ◽  
Localization Phenomenon ◽  
Harmonic Resonance

AbstractThe nonlinear transverse vibrations of ordered and disordered two-dimensional (2D) two-span composite laminated plates are studied. Based on the von Karman's large deformation theory, the equations of motion of each-span composite laminated plate are formulated using Hamilton's principle, and the partial differential equations are discretized into nonlinear ordinary ones through the Galerkin's method. The primary resonance and 1/3 sub-harmonic resonance are investigated by using the method of multiple scales. The amplitude-frequency relations of the steady-state responses and their stability analyses in each kind of resonance are carried out. The effects of the disorder ratio and ply angle on the two different resonances are analyzed. From the numerical results, it can be concluded that disorder in the length of the two-span 2D composite laminated plate will cause the nonlinear vibration localization phenomenon, and with the increase of the disorder ratio, the vibration localization phenomenon will become more obvious. Moreover, the amplitude-frequency curves for both primary resonance and 1/3 sub-harmonic resonance obtained by the present analytical method are compared with those by the numerical integration, and satisfactory precision can be obtained for engineering applications and the results certify the correctness of the present approximately analytical solutions.

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