Non-linear wave modulation in a prestressed viscoelastic thin tube filled with an inviscid fluid

In the present work, the propagation of weakly non-linear waves in a prestressed thin viscoelastic tube filled with an incompressible inviscid fluid is studied. Considering that the arteries are initially subjected to a large static transmural pressure P₀ and an axial stretch λ_z and, in the course of blood flow, a finite time-dependent displacement is added to this initial field, the governing non-linear equation of motion in the radial direction is obtained. Using the reductive perturbation technique, the propagation of weakly non-linear, dispersive and dissipative waves is examined and the evolution equations are obtained. Utilizing the same set of governing equations the amplitude modulation of weakly non-linear and dissipative but strongly dispersive waves is examined. The localized travelling wave solution to these field equations are also given.     

MHD flow and heat transfer of a UCM fluid over a stretching surface with variable thermophysical properties

In this paper we investigate the effects of temperature-dependent viscosity, thermal conductivity and internal heat generation/absorption on the MHD flow and heat transfer of a non-Newtonian UCM fluid over a stretching sheet. The governing partial differential equations are first transformed into coupled non-linear ordinary differential equation using a similarity transformation. The resulting intricate coupled non-linear boundary value problem is solved numerically by a second order finite difference scheme known as Keller-Box method for various values of the pertinent parameters. Numerical computations are performed for two different cases namely, zero and non-zero values of the fluid viscosity parameter. That is, 1/?? _r ??0 and 1/?? _r ??0 to get the effects of the magnetic field and the Maxwell parameter on the velocity and temperature fields, for several physical situations. Comparisons with previously published works are presented as special cases. Numerical results for the skin-friction co-efficient and the Nusselt number with changes in the Maxwell parameter and the fluid viscosity parameter are tabulated for different values of the pertinent parameters. The results obtained for the flow characteristics reveal many interesting behaviors that warrant further study on the non-Newtonian fluid phenomena, especially the UCM fluid phenomena. Maxwell fluid reduces the wall-shear stress.     

Saint-Venant end effects for plane deformations of linear piezoelectric solids

The recent developments in smart structures technology have stimulated renewed interest in the fundamental theory and applications of linear piezoelectricity. In this paper, we investigate the decay of Saint-Venant end effects for plane deformations of a piezoelectric semi-infinite strip. First of all, we develop the theory of plane deformations for a general anisotropic linear piezoelectric solid. Just as in the mechanical case, not all linear homogeneous anisotropic piezoelectric cylindrical solids will sustain a non-trivial state of plane deformation. The governing system of four second-order partial differential equations for the two in-plane displacements and electric potential are overdetermined in general. Sufficient conditions on the elastic and piezoelectric constants are established that do allow for a state of plane deformation. The resulting traction boundary-value problem with prescribed surface charge is an oblique derivative boundary-value problem for a coupled elliptic system of three second-order partial differential equations. The special case of a piezoelectric material transversely isotropic about the poling axis is then considered. Thus the results are valid for the hexagonal crystal class 6mm. The geometry is then specialized to be a two-dimensional semi-infinite strip and the poling axis is the axis transverse to the longitudinal direction. We consider such a strip with sides traction-free, subject to zero surface charge and self-equilibrated conditions at the end and with tractions and surface charge assumed to decay to zero as the axial variable tends to infinity. A formulation of the problem in terms of an Airy-type stress function and an induction function is adopted. The governing partial differential equations are a coupled system of a fourth and third-order equation for these two functions. On seeking solutions that exponentially decay in the axial direction one obtains an eigenvalue problem for a coupled system of fourth and second-order ordinary differential equations. This problem is the piezoelectric analog of the well-known eigenvalue problem arising in the case of an anisotropic elastic strip. It is shown that the problem can be uncoupled to an eigenvalue problem for a single sixth-order ordinary differential equation with complex eigenvalues characterized as roots of transcendental equations governing symmetric and anti-symmetric deformations and electric fields. Assuming completeness of the eigenfunctions, the rate of decay of end effects is then given by the real part of the eigenvalue with smallest positive real part. Numerical results are given for PZT-5H, PZT-5, PZT-4 and Ceramic-B. It is shown that end effects for plane deformations of these piezoceramics penetrate further into the strip than their counterparts for purely elastic isotropic materials.     

On non-linear vibration analyses of continuous systems with quadratic and cubic non-linearities

This paper examines the validity of non-linear vibration analyses of continuous systems with quadratic and cubic non-linearities. As an example, we treat a hinged-hinged Euler-Bernoulli beam resting on a non-linear elastic foundation with distributed quadratic and cubic non-linearities, and investigate the primary (Ω≈ω_n) and subharmonic (Ω≈2ω_n) resonances, in which Ω and ω_n are the driving and natural frequencies, respectively. The steady-state responses are found by using two different approaches. In the first approach, the method of multiple scales is applied directly to the governing equation that is a non-linear partial differential equation. In the second approach, we discretize the governing equation by using Galerkin's procedure, and then apply the shooting method to the obtained ordinary differential equations. In order to check the validity of the solutions obtained by the two approaches, they are compared with the solutions obtained numerically by the finite difference method.     

Non-linear dynamics of wind turbine wings

The paper deals with the formulation of non-linear vibrations of a wind turbine wing described in a wing fixed moving coordinate system. The considered structural model is a Bernoulli-Euler beam with due consideration to axial twist. The theory includes geometrical non-linearities induced by the rotation of the aerodynamic load and the curvature, as well as inertial induced non-linearities caused by the support point motion. The non-linear partial differential equations of motion in the moving frame of reference have been discretized, using the fixed base eigenmodes as a functional basis. Important non-linear couplings between the fundamental blade mode and edgewise modes have been identified based on a resonance excitation of the wing, caused by a harmonically varying support point motion with the circular frequency ω. Assuming that the fundamental blade and edgewise eigenfrequencies have the ratio of ω₂/ω₁?2, internal resonances between these modes have been studied. It is demonstrated that for ω/ω₁?0.66,1.33,1.66 and 2.33 coupled periodic motions exist brought forward by parametric excitation from the support point in addition to the resonances at ω/ω₁?1.0 and ω/ω₂?1.0 partly caused by the additive load term.     

On the Parametric Resonance of Columns Carrying Concentrated Masses

Abstract

The effect of various parameters upon the region of dynamic instability of a uniform simply supported column carrying n concentrated masses and subjected to an axial periodic force at one end is presented. The effect of axial inertia of the mass per unit length of the column is also included. This problem is reduced to a coupled system of two homogeneous partial differential equations of second order with periodic coefficients which can be written in the form of a matric differential equation of the Mathieu-Hill type. A solution methodology is developed and successfully demonstrated through numerical examples.     

Free Convection Boundary Layer Flow Past a Vertical Surface in a Porous Medium with Temperature-Dependent Properties

Numerical solutions for the free convection heat transfer in a viscous fluid at a permeable surface embedded in a saturated porous medium, in the presence of viscous dissipation with temperature-dependent variable fluid properties, are obtained. The governing equations for the problem are derived using the Darcy model and the Boussinesq approximation (with nonlinear density temperature variation in the buoyancy force term). The coupled non-linearities arising from the temperature-dependent density, viscosity, thermal conductivity, and viscous dissipation are included. The partial differential equations of the model are reduced to ordinary differential equations by a similarity transformation and the resulting coupled, nonlinear ordinary differential equations are solved numerically by a second order finite difference scheme for several sets of values of the parameters. Also, asymptotic results are obtained for large values of | f _w|. Moreover, the numerical results for the velocity, the temperature, and the wall-temperature gradient are presented through graphs and tables, and are discussed. It is observed that by increasing the fluid variable viscosity parameter, one could reduce the velocity and thermal boundary layer thickness. However, quite the opposite is true with the non-linear density temperature variation parameter.     

Numerical studies of slow viscous rotating flow past a sphere. III

The flow of steady incompressible viscous fluid rotating about the z-axis with angular velocity ω and moving with velocity u past a sphere of radius a which is kept fixed at the origin is investigated by means of a numerical method for small values of the Reynolds number Re_ω. The Navier–Stokes equations governing the axisymmetric flow can be written as three coupled non-linear partial differential equations for the streamfunction, vorticity and rotational velocity component. Central differences are applied to the partial differential equations for solution by the Peaceman–Rachford ADI method, and the resulting algebraic equations are solved by the ‘method of sweeps’. The results obtained by solving the non-linear partial differential equations are compared with the results obtained by linearizing the equations for very small values of Re_ω. Streamlines are plotted for Ψ = 0·05, 0·2, 0·5 for both linear and non-linear cases. The magnitude of the vorticity vector near the body, i.e. at z = 0·2, is plotted for Re_ω = 0·05, 0·24, 0·5. The correction to the Stokes drag as a result of rotation of the fluid is calculated.     

A stream function finite element solution for two-dimensional natural convection with accurate representation of nusselt number variations near a corner

A finite element stream function formulation is presented for the solution to the two-dimensional double-glazing problem. Laminar flow with constant properties is considered and the Boussinesq approximation used. A restricted variational principle is used, in conjunction with a triangular finite element of C¹ continuity, to discretize the two coupled governing partial differential equations (4th order in stream function and second order in temperature). The resulting non-linear system of equations is solved in a segregated (decoupled) manner by the Newton-Raphson linearizing technique. Results are produced for the standard test case of an upright square cavity. These are for Rayleigh numbers in the range 10³?10⁵, with a Prandtl number of 0.71. Comparisons are made with benchmark results presented at the 1981 International Comparison study in Venice. In the discussion of results, emphasis is placed on the variation of local Nusselt number along the isothermal walls, particularly near the corner. This reveals a noticeable source of error in the evaluation of the maximum Nusselt number by lower order discretization methods.     

On non-linear dynamics of a coupled electro-mechanical system

Electro-mechanical devices are an example of coupled multi-disciplinary weakly non-linear systems. Dynamics of such systems is described in this paper by means of two mutually coupled differential equations. The first one, describing an electrical system, is of the first order and the second one, for mechanical system, is of the second order. The governing equations are coupled via linear and weakly non-linear terms. A classical perturbation method, a method of multiple scales, is used to find a steady-state response of the electro-mechanical system exposed to a harmonic close-resonance mechanical excitation. The results are verified using a numerical model created in MATLAB Simulink environment. Effect of non-linear terms on dynamical response of the coupled system is investigated; the backbone and envelope curves are analyzed. The two phenomena, which exist in the electro-mechanical system: (a)?detuning (i.e. a natural frequency variation) and (b)?damping (i.e. a decay in the amplitude of vibration), are analyzed further. An applicability range of the mathematical model is assessed.     

Analytical and numerical solutions of a coupled non-linear system arising in a three-dimensional rotating flow

Hydromagnetic flow between two horizontal plates in a rotating system, where the lower is a stretching sheet and the upper is a porous solid plate (in the presence of a magnetic field), is analyzed. The equations of conservation of mass and momentum are reduced to a system of coupled non-linear ordinary differential equations. These basic non-linear differential equations, for the velocity field (f′,f,g), are solved numerically by using a fourth-order Runge-Kutta integration scheme. The numerical results thus obtained are validated by the analytical results (for small R) obtained by the perturbation technique and presented through graphs. Also, the effects of the non-dimensional parameters R, λ, M² and K² on the velocity field are discussed, and it is shown that for large K², the coriolis force and the magnetic field that act against the pressure gradient cause reverse flow.     

Soret and Dufour effects on Double-Diffusive Free Convection From a Corrugated Vertical Surface in a Non-Darcy Porous Medium

Combined heat and mass transfer process by natural convection from a wavy vertical surface immersed in a fluid-saturated semi-infinite porous medium due to Soret and Dufour effects for Forchheimer extended non-Darcy model has been analyzed. A similarity transformation followed by a wavy to flat surface transformation is applied to the governing coupled non-linear partial differential equations, and they are reduced to boundary layer equations. The obtained boundary layer equations are solved by finite difference scheme based on the Keller-Box approach in conjunction with block-tridiagonal solver. Detailed simulations are carried out for a wide range of parameters like Groshof number (Gr*), Lewis number (Le), Buoyancy ratio (B), Wavy wall amplitude (a), Soret number (S _r ), and Dufour number (D _f ). Comparison tables local and average Nusselt (Nu) number, local and average Sherwood (Sh) number plots are presented.     

Non-linear,non-steady burning of a solid propellent in three coordinate systems

The non-linear partial differential equation controlling the temperature distribution in a burning solid propellent in a rectangular, cylindrical or spherical coordinate system is transformed from a fixed coordinate system to a moving Lagrangian coordinate system, and then, using a similarity variable is further transformed to a non-linear, second order, ordinary differential equation. The burning wall temperature is time dependent, and the burning rate is an explicit function of the wall surface temperature.The boundary conditions for the resulting non-linear differential equation are given, and the necessary form of the burning rate is determined. Additionally, the linear partial differential equation for this burning solid propellent problem is also treated in all three coordinate systems. A non-linear example is given and solved, in all three coordinate systems, to illustrate the method.     

First integrals for equations with non-linearities of the Emden-Fowler type

An ad hoc procedure is given for obtaining first integrals of second order differential equations in which the non-linear term is a power of the dependent variable, as in the Emden-Fowler equation. The main theorem is a considerable extension of previous results along these lines. A corollary implies several known examples.     

Effect of temperature dependent viscosity on revolving axi-symmetric ferrofluid flow with heat transfer

The prime objective of the present study is to examine the effect of temperature dependent viscosity μ(T) on the revolving axi-symmetric laminar boundary layer flow of an incompressible, electrically non-conducting ferrofluid in the presence of a stationary plate subjected to a magnetic field and maintained at a uniform temperature. To serve this purpose, the non-linear coupled partial differential equations are firstly converted into the ordinary differential equations using well-known similarity transformations. The popular finite difference method is employed to discretize the non-linear coupled differential equations. These discretized equations are then solved using the Newton method in MATLAB, for which an initial guess is made with the help of the Flex PDE Solver. Along with the velocity profiles, the effects of temperature dependent viscosity are also examined on the skin friction, the heat transfer, and the boundary layer displacement thickness. The obtained results are presented numerically as well as graphically.     

Three-dimensional channel flow of second grade fluid in rotating frame

An analysis is performed for the hydromagnetic second grade fluid flow between two horizontal plates in a rotating system in the presence of a magnetic field.The lower sheet is considered to be a stret...     

A field method and its application to the theory of vibrations

The problem of integration of the differential equations of motion of a nonconservative dynamical system is replaced by an equivalent problem of finding a complete integral of a quasi-linear partial differential equation of the first order. In the second part, these complete integrals are combined with the two time scales perturbation method in the study of non-linear oscillatory motions.     

The prediction of time-dependent non-linear stresses in viscoelastic materials: I. Selection of constitutive equation and strain measure

The theories for the prediction of time-dependent, non-linear stresses in viscoelastic materials such as polymers are reviewed, and it is noted that the commonly observed stress non-linearity may be ascribed either, as is usually done, to memory-function non-linearity or, alternatively, to strain-measure non-linearity. To investigate the latter alternative whilst retaining a general memory-function non-linearity, a single-integral constitutive equation of the Bird—Carreau type is employed but with an arbitrary strain measure I in place of the normally employed Finger tensor F. This model includes as special cases a large proportion of the constitutive equations previously employed for predictive purposes and in particular with a linear memory function it is shown to be indistinguishable, with the normally conducted shear experiments, from the successful BKZ model.In the new model the shear component I₁₂ of the strain measure can be found from experimental results obtained in the startup of steady shear flow, without specification or restriction of memory-function non-linearity. The form of I₁₂ found from experiment is quite non-linear in shear a for ¦a¦> 2, and hence differs from the F tensor for which F₁₂ = a. The same form for I₁₂ found for a variety of polymer solutions and a polymer melt and consequently a simple function describing I₁₂ is proposed as a new, material-independent, strain measure.     

Non-linear flexural–torsional dynamic analysis of beams of arbitrary cross section by BEM

In this paper, a boundary element method is developed for the non-linear flexural–torsional dynamic analysis of beams of arbitrary, simply or multiply connected, constant cross section, undergoing moderately large deflections and twisting rotations under general boundary conditions, taking into account the effects of rotary and warping inertia. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse loading in both directions as well as to twisting and/or axial loading. Four boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to the angle of twist and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique leads to a system of non-linear coupled Differential–Algebraic Equations (DAE) of motion, which is solved iteratively using the Petzold–Gear Backward Differentiation Formula (BDF), a linear multistep method for differential equations coupled to algebraic equations. The geometric, inertia, torsion and warping constants are evaluated employing the Boundary Element Method. The proposed model takes into account, both the Wagner's coefficients and the shortening effect. Numerical examples are worked out to illustrate the efficiency, wherever possible the accuracy, the range of applications of the developed method as well as the influence of the non-linear effects to the response of the beam.