Fundamental solutions for variable density two-dimensional elastodynamic problems

Fundamental solutions in the form of free-space Green's functions are developed for a class of two-dimensional, variable density elastodynamic problems. These fundamental solutions are evaluated by means of a coordinate transformation based on conformal mapping in conjunction with wave decomposition, which allows for both vertical and lateral heterogeneities, and can be used within the context of a boundary integral equation formulation analogous to that originally proposed by Cruse and Rizzo (J Math Anal Appl 22 (1968) 244). Finally, a numerical example serves to illustrate the methodology developed herein.     

The stress response of a semi-infinite micropolar granular material subject to a concentrated force normal to the boundary

We consider the problem of a two dimensional semi-infinite granular material subject to a concentrated or point force normal to the boundary. This boundary value problem was originally solved for a classical elastic material by Flamant in 1892 and, hence, is also known as the Flamant problem (Johnson [8]). In this paper, the granular material is considered as an elastic micropolar or Cosserat continuum and is represented by a particular form of the general constitutive law derived in Walsh and Tordesillas [29]. The stress distribution predicted by the model is in good agreement with experimental data for small strains. In particular, two important features that are captured by the proposed model are: (i) the presence of tensile stress response regions, and (ii) the dependence of the stresses on the microstructural properties, i.e. the particles normal, tangential and rotational stiffness constants. The proposed analysis utilizes two new stress functions, similar to Airys stress functions in classical elastic theory.The support of the US Army Research Office through a grant to AT (Grant No. DAAD19-02-1-0216) and the Melbourne Research and Development Grant scheme is gratefully acknowledged. We thank our reviewers for their useful suggestions and insightful comments.     

Effects of suction-injection-combination (SIC) on the onset of Rayleigh-Benard magnetoconvection in a micropolar fluid

The effects of suction-injection-combination (SIC) and magnetic field on the linear stability analysis of Rayleigh-Benard convection in a horizontal layer of an Boussinesq micropolar fluid is studied using a Rayleigh-Ritz techinque. The eigenvalues are obtained for free-free, rigid-free and rigid-rigid velocity boundary combinations with isothermal and adiabatic temperature conditions on the spin-vanishing boundaries. The eigenvalues are also obtained for lower rigid isothermal and upper free adiabatic boundaries with vanishing spin. The influence of various micropolar fluid parameters on the onset of convection has been analysed. It is found that the effect of Prandtl number on the stability of the system is dependent on the SIC being pro-gravity or anti-gravity. A similar Pe-sensitivity is found in respect of the critical wave number. It is observed that the micropolar fluid layer heated from below is more stable compared to the classical fluid layer.     

Linear micropolar elastic crack-tip fields under mixed mode loading conditions

Micropolar elasticity laws provide a possibility to describe constitutive properties of materials for which internal length scales may become important. They are characterized by the presence of couple stresses and nonsymmetric Cauchy stress tensor. Beyond the classical displacement field, the kinematical variables are augmented by a so-called microrotation field and its gradient, the latter introducing an internal length scale in the theory. For an isotropic, linear micropolar elastic material, the near-tip asymptotic field solutions for mode I and mode II cracks are derived. It is shown that these solutions behave similar to those according to the so-called couple stress theory, which has been investigated by Huang et al. (1997a), or similar to those derived for cellular materials by Chen et al. (1998). In particular, the singular fields have an order of singularity r ^–1/2 and are governed by some amplitude factors, having the meaning of stress intensity factors as in the classical linear elastic theory. The effect of material parameters on the stress intensity factors is studied by applying the finite element method to calculate the values of the stress intensity factors for an edge-cracked specimen of finite width.     

Fundamental solutions for thick sandwich plates

A method to evaluate the fundamental solutions of thick sandwich plates by the use of the Fourier transform technique is developed. The fundamental solution is presented in the Fourier integral form and the calculation of this solution is discussed in detail. A special care is taken for representation and calculation of the hypersingular integrals entering in this presentation.     

Impact of nonuniform heated plate on double-diffusive natural convection of micropolar fluid in a square cavity with Soret and Dufour effects

A numerical work has been carried out to study the effect of heated plate on double diffusive natural convection in a cavity with the presence of Soret and Dufour effects. The vertical left and right sidewalls of the cavity are maintained at constant cold temperatures while the lower and upper walls are considered insulated. The influence of pertinent parameters such as Rayleigh number, Schmidt number, vortex viscosity parameter, Soret and Dufour coefficients and plate non-uniformity parameter on the flow and heat transfer characteristics has been examined. Numerical results show that the heat and mass transfer rate increases with the rise of the Rayleigh number and Schmidt number. It is found that the heat and mass transfer rate are considerably suppressed by the vortex viscosity parameter. In addition, it is observed that the average Nusselt number increases and Sherwood number decreases with increasing Soret and Dufour effects.     

Finite element solution of mixed convection micropolar flow driven by a porous stretching sheet

This paper presents a finite element solution for the mixed convection micropolar flow driven by a porous stretching sheet with uniform suction. The governing partial differential equations are solved numerically by the using finite element method and the results have been compared with those obtained by using the quasi-linearization method. The effect of surface conditions on the velocity, microrotation as well as for temperature functions has been studied. It is noticed that the micropolar fluids help in the reduction of drag forces and also act as a cooling agent.     

Axisymmetric flow of two fluids in a pulsating pipe

The motion of a viscous thread surrounded by an annular viscous layer inside a pulsating cylindrical pipe whose radius is a periodic function of time is investigated. At zero Reynolds number, a stagnation-point-type solution may be written down in closed form. A Floquet linear stability analysis for Stokes flow reveals the pulsations either decrease or increase the growth rate of longwave disturbances depending on the initial radius of the thread. For a moderate-sized initial thread radius, increasing the amplitude of the pulsations decreases the critical wavenumber for instability to below the classical Rayleigh threshold. Increasing the viscosity contrast, so that the fluid in the annular layer becomes more viscous than the fluid in the thread, tends to decrease the growth rate of disturbances. In the second part of the paper, the basic stagnation-point-type flow at arbitrary Reynolds number is computed using a numerical method on the assumption that the interface is a circular cylinder at all times. During the motion, either the thread radius tends to increase and the thickness of the annular layer decreases, or else the thread tends to thin and the thickness of the annular layer increases, depending upon the initial conditions and the parameter values. For a judicious choice of initial condition, a time-periodic exact solution of the Navier–Stokes equations is identified.     

Overall elastoplastic property for micropolar composites with randomly oriented ellipsoidal inclusions

Overall linear and non-linear properties for micropolar composites containing 3D and in-plane randomly oriented inclusions are examined with an analytical micromechanical method. This method is based on Eshelby solution for a general ellipsoidal inclusion in a micropolar media and secant moduli method. The influence of inclusion’s shape, size and orientation on the classical effective moduli, yielding surface and non-linear stress and strain relation are examined. The results show that the effective moduli and non-linear stress–strain curves are always higher for micropolar composites than the corresponding classical composites. When the inclusion’s size is sufficiently large, the classical results can be recovered.     

Numerical study for micropolar flow over a stretching sheet

The paper presents a numerical study of the steady flow of a micropolar fluid flow from a stretching sheet. Approximate analytical solution of high nonlinear momentum, angular momentum and confluent hypergeometric similarity solution of the heat transfer equation are obtained for a particular case when the vortex viscosity is neglected. Accuracy of the analytical solution is verified by numerical solutions obtained by employing finite element and Chebyshev finite difference methods. The good agreement between the numerical results of both methods, together with an excellent agreement with the analytical solutions for the special case, ensures the reliability of the obtained results. The velocity, microrotation and temperature functions are shown graphically and the effect of the permeability parameter is studied.     

Identification of material parameters of micropolar theory for composites by homogenization method

The objective of the present work is to develop an analytical homogenization method to derive higher-order material parameters of micropolar theory. With help of Airy’s stress function, the constitutive equations for a homogenized micropolar medium are analytically established by considering a cylindrical representative volume element (RVE) subjected to quadratic and cubic boundary displacement conditions. Both porous and composite materials are considered, an analytical relation between the intrinsic length and the microstructural parameters is given.     

Fundamental solutions for dynamic BE analyses of incompressible problems

The dilatational wave velocity theoretically tends to infinity as Poisson's ratio approaches 0·5. This infinite wave velocity can cause serious numerical difficulty in boundary element analyses of dynamic incompressible problems. This paper shows that when Poisson's ratio equals 0·5 Stokes' solutions are independent of the dilatational wave velocity. Consequently, by using the modified fundamental solutions, the boundary element method can effectively analyse dynamic incompressible problems without special difficulty.     

Fundamental solution of a circular rigid punch problem for a half plane

A fundamental solution (a Green's function) of a circular rigid punch problem is derived. The punch on a half plane contacts smoothly at both ends of the contact region and a concentrated force or a point dislocation apply at an arbitrary point. Two cases where the punch is kept vertically and the punch is inclined are considered. Complex stress functions and a mapping function are used and a closed form solution is derived. Some results of stress distribution, contact length and resultant moment on the contact region are shown in figures.     

Finite element analysis of combined heat and mass transfer in hydromagnetic micropolar flow along a stretching sheet

This paper presents a finite element solution of the problem of heat and mass transfer in a hydromagnetic flow of a micropolar fluid past a stretching sheet. The transformed equations for the flow regime are solved numerically by using finite element method. The effect of important parameters namely magnetic field parameter, material parameter, Eckert number and Schmidt number over velocity, microrotation, temperature and concentration functions has been studied. It has been observed that the magnetic field parameter has the effect of reducing the velocity and increasing the microrotation, temperature and concentration while the micropolar parameter has the opposite effect on these functions except temperature function. Temperature increases with the increase in Eckert number and concentration decreases with the increase in Schmidt number.     

磁流体浸没物磁场力分析及磁浮特性

物体浸没于磁流体中表现出磁浮特性，对其受力状态分析是准确描述其悬浮状态的前提和基础。基于非线性磁流体应力张量模型和稳态Bernoulli方程，建立磁流体中浸没物受力模型。借助非磁性体受力模型的简化计算方法以及磁性体与磁流体之间的多场效应关系，分别对非磁性体、磁性体两类浸没物在磁流体中所受磁浮力进行分析，结果表明非磁性浸没物在磁流体中仅受到外加磁场贡献的一次磁浮力，而磁性浸没物除受到一次磁浮力外，还受到其自身激发磁场贡献的二次磁浮力．永磁体在磁流体中位置决定的磁浮力满足一定条件时，永磁体能够自悬浮于磁流体中。     

Exact solutions for two steady inviscid bubbles in the slow viscous flow generated by a four-roller mill

The nonlinear free-boundary problem of finding the equilibrium shapes of two equal-sized two-dimensional inviscid bubbles with surface tension situated in a polynomially-singular slow viscous flow is solved in terms of closed-form formulae. The singular flow is taken to be within the class of those realizable at the centre of a four-roller mill apparatus. The associated flow field is also found explicitly. These solutions allow investigation of the bubble shapes and associated streamline patterns as functions of the far-field asymptotic conditions. In certain regimes, the bubbles are found to exhibit both near-cusps and a characteristic dimpling as they draw closer together. The results provide the first instances of exact solutions involving two interacting bubbles in an unbounded Stokes flow.     

Analytical Model Development for Stokes-Type Settling in a Solidifying Fluid

Analytical modeling was used to track the mudline displacement of a simulated fluid dispersion that underwent a simulated linear increase in the fluid viscosity as a function of time following an induction period. Modeling results indicate two regimes of settling behavior from the driving force of density differences between the particles and the fluid. A linear mudline movement was first determined in the regime of constant viscosity, and a logarithmic retarded mudline movement was numerically determined following a linear rise in fluid viscosity. The results show that the combination of settling and solidification leads to a transitional mudline movement similar to batch settling curves described by early settling theories.     

Analytical Model Development for Stokes-Type Settling in a Solidifying Fluid

Analytical modeling was used to track the mudline displacement of a simulated fluid dispersion that underwent a simulated linear increase in the fluid viscosity as a function of time following an induction period. Modeling results indicate two regimes of settling behavior from the driving force of density differences between the particles and the fluid. A linear mudline movement was first determined in the regime of constant viscosity, and a logarithmic retarded mudline movement was numerically determined following a linear rise in fluid viscosity. The results show that the combination of settling and solidification leads to a transitional mudline movement similar to batch settling curves described by early settling theories.     

Revisiting Stokes first and second problems for fluids with pressure-dependent viscosities

We revisit the numerical study carried out by Srinivasan and Rajagopal [1] and we show that the long time solutions to the first and second Stokes problem for fluids with pressure dependent viscosities can be given by exact formulae. Moreover we show that the transient solutions – thus for example the solutions studied by numerical means by Srinivasan and Rajagopal [1] – must rapidly converge to these long time solutions.