On the Swimming of a "Pod "

A "pod" is a medical device consisting of a small hollow magnetattached to a plastic "tail". If it is placed in a blood vesselof a patient, and an alternating magnetic field applied, themagnet oscillates angularly and the plastic tail causes it toswim. The purpose of the device is to deliver medicaments, whichmay have been placed in the hollow magnet, or perform othertasks at any desired location in the main circulatory system.This paper seeks to analyse the swimming process and suggestdesign characteristics for efficient swimming. The analysis can also be regarded as a contribution to the generaltheory of swimming of elastic bodies excited by an externalforce.     

Stabilization of the non-trivial relative equilibria of a gyrostat with an elastic element in a circular orbit

The motion of a gyrostat, regarded as a rigid body, in a circular Kepler orbit in a central Newtonian force field is investigated in a limited formulation. A uniformly rotating statically and dynamically balanced flywheel is situated in the rigid body. A uniform elastic element, which, during the motion of the system, is subjected to small deformations, is rigidly connected to the rigid body-gyrostat body. The problem is discretized without truncating the corresponding infinite series, based on a modal analysis or using a certain specified system of functions, for example, of the assumed forms of the oscillations, which depend on the spatial coordinates and which satisfy appropriate boundary-value problems of the linear theory of elasticity. The elastic element is specified in more detail (a rod, plate, etc.), as well as its mass and stiffness characteristics and the form of the fastening, and the choice of the system of functions is determined. Non-trivial relative equilibria of the system (the state of rest with respect to an orbital system of coordinates when the elastic element is deformed) is sought approximately on the basis of a converging iteration method, described previously. It is shown, using Routh's theorem, that by an appropriate choice of the gyrostatic moment and when certain conditions, imposed on the system parameters are satisfied, one can stabilize these equilibria (ensure that they are stable).     

An analysis of the swimming problem of a singly flagellated micro-organism in an MHD fluid flowing through a porous medium

The focus of the present work is concerned with the study of the swimming of microscopic organisms that use a single flagellum for propulsion in a magnetohydrodynamic (MHD) fluid flowing through a porous medium. The flow is modelled by appropriate equations and the organism is modelled by an infinite flexible but inextensible transversely waving sheet, which represents approximately the flagellum. The governing equations subject to appropriate boundary conditions are solved analytically. Expressions for the velocity of propulsion of the microscopic organism are obtained. We show that as the MHD character of the fluid is removed the results match those of an earlier analysed problem of propulsion through a fluid in a porous medium. In addition, in the final case of a simple viscous fluid (absence of magnetic field), we show that as the permeability becomes large the results reduce to the swimming of such organisms in a viscous fluid (discounting the pores and the MHD character).     

The hysteretic event in the computation of magnetization

Summary Simulations of magnetic and magnetostrictive behavior based on micromagnetic theory exhibit hysteresis. These systems have a highly nonlinear character involving both short range anisotropy and elastic fields and dispersive demagnetization fields. Hysteresis occurs even in the absence of an imposed dynamical mechanism, for example, a Landau-Lifshitz-Gilbert dissipative equation for the magnetic moment, and is symptomatic of the way the system navigates a path through local minima of its energy space. It is not sensitive to the particular method: We implement continuation based on the conjugate gradient method, although the same results were obtained by, for example, a Newton’s method. The phenomenon is robust: Computational experiments confirm that the shape of the loop is invariant over several decades of mesh refinement. Our experience has led us to hold that optimization procedures have the propensity to become marooned at local extrema when applied to nonconvex situations and that this presents a fundamental challenge to analysis. Understanding and controlling such phenomena present the opportunity to develop predictive tools and diagnostics. For example, since the energy picture is mesh-independent, computing on a fairly coarse grid suffices to establish its character.     

Hydromagnetic flow near an oscillating porous flat plate

A solution in the closed form has been obtained for the flow of a viscous incompressible fluid of small electrical conductivity near an infinite insulated porous flat plate oscillating harmonically in its own plane in the presence of a transverse magnetic field of uniform strength fixed relative to the fluid. Small uniform suction has been imposed along the surface of the plate. This is a generalisation of the result of Ong and Nicholls for the hydromagnetic flow near an oscillating solid flat plate.     

A mixed formulation for 3D magnetostatic problems: theoretical analysis and face-edge finite element approximation

Summary. A mixed field-based variational formulation for the solution of threedimensional magnetostatic problems is presented and analyzed. This method is based upon the minimization of a functional related to the error in the constitutive magnetic relationship, while constraints represented by Maxwell's equations are imposed by means of Lagrange multipliers. In this way, both the magnetic field and the magnetic induction field can be approximated by using the most appropriate family of vector finite elements, and boundary conditions can be imposed in a natural way. Moreover, this method is more suitable than classical approaches for the approximation of problems featuring strong discontinuities of the magnetic permeability, as is usually the case. A finite element discretization involving face and edge elements is also proposed, performing stability analysis and giving error estimates. Received January 23, 1998 / Revised version received July 23, 1998 / Published online September 24, 1999     

A lattice Boltzmann model for Maxwell’s equations

In this paper, a lattice Boltzmann model for the Maxwell’s equations is proposed by taking separate sets of distribution functions for the electric and magnetic fields, and a lattice Boltzmann model for the Maxwell vorticity equations with third order truncation error is proposed by using the higher-order moment method. At the same time, the expressions of the equilibrium distribution function and the stability conditions for this model are given. As numerical examples, some classical electromagnetic phenomena, such as the electric and magnetic fields around a line current source, the electric field and equipotential lines around an electrostatic dipole, the electric and magnetic fields around oscillating dipoles are given. These numerical results agree well with classical ones.     

Chaotic motion of magnetotactic bacteria

The motion of a magnetotactic bacterium submitted to an oscillating magnetic field was studied. The nature of the U-turn process, which occurred when the magnetic field was reversed, was investigated. It is analytically shown that this process presents a chaotic behavior. When the magnetic field is reversed the bacterium may decide if the turning will be to the right side or to the left side. Such choice is highly sensitive to the initial conditions, making it impossible to predict which side will be taken in the U-turn.     

On Love-type waves in a finitely deformed magnetoelastic layered half-space

In this paper, the propagation of Love-type waves in a homogeneously and finitely deformed layered half-space of an incompressible non-conducting magnetoelastic material in the presence of an initial uniform magnetic field is analyzed. The equations and boundary conditions governing linearized incremental motions superimposed on an underlying deformation and magnetic field for a magnetoelastic material are summarized and then specialized to a form appropriate for the study of Love-type waves in a layered half-space. The wave propagation problem is then analyzed for different directions of the initial magnetic field for two different magnetoelastic energy functions, which are generalizations of the standard neo-Hookean and Mooney?CRivlin elasticity models. The resulting wave speed characteristics in general depend significantly on the initial magnetic field as well as on the initial finite deformation, and the results are illustrated graphically for different combinations of these parameters. In the absence of a layer, shear horizontal surface waves do not exist in a purely elastic material, but the presence of a magnetic field normal to the sagittal plane makes such waves possible, these being analogous to Bleustein?CGulyaev waves in piezoelectric materials. Such waves are discussed briefly at the end of the paper.     

Exact solution for the pulsating finite gap Dean flow

Analytical solutions of the Navier–Stokes equations for the fully developed laminar flow in a cylindrical annulus, when an oscillating circumferential pressure gradient is imposed (finite gap oscillating Dean flow), are presented. The solution for the case of steady flow, which has been given by Goldstein, is obtained as a limit case of the oscillating flow when the frequency of the oscillating pressure gradient tends to zero. The pulsating flow solution is obtained by the superposition of the constant and oscillating pressure gradient solutions.     

弹性动脉血管中血液流动特性的模拟和分析

研究了两个不同的非牛顿血液流动模型:低粘性剪切简单幂律模型和低粘性剪切及粘弹性振荡流的广义Maxwell模型．同时利用这两个非牛顿模型和牛顿模型,研究了磁场中刚性和弹性直血管中血液的正弦型脉动．在生理学条件下,大动脉中血液的弹性对其流动性态似乎并不产生影响,单纯低粘性剪切模型可以逼真地模拟这种血液流动．利用高剪切幂律模型模拟弹性血管中的正弦型脉动流,发现在同一压力梯度下,与牛顿流体相比较,幂律流体的平均流率和流率变化幅度都更小．控制方程用Crank-Niclson方法求解．弹性动脉中血液受磁场作用是产生此结果的直观原因．在主动脉生物流的模拟中,与牛顿流体模型比较,发现在匹配流率曲线上,幂律模型的平均壁面剪切应力增大,峰值壁面剪切应力减小．讨论了弹性血管横切磁场时的血液流动,评估了血管形状和表面不规则等因素的影响．     

DNS of MHD turbulence in liquid metals based on the least dissipative modes

We investigate numerically the incompressible Navier-Stokes equations under an externally imposed magnetic field. The results obtained for the simplified geometry of a three dimensional periodic box and show a lower computational complexity and stronger relationship between the physical reality and its numerical representation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

磁弹性耦合效应引起的铁磁直杆磁场中振动频率的改变

基于磁弹性广义变分原理和Hamilton原理,对处于外加磁场中的软铁磁体,建立了磁弹性动力学理论模型．分别通过关于铁磁杆磁标势和弹性位移的变分运算,获得了包含磁场和弹性变形的所有基本方程,并给出描述磁弹性耦合作用的磁体力和磁面力．采用摄动技术和Galerkin方法,将所建立的磁弹性理论模型用于外加磁场中铁磁直杆的振动分析．结果表明,由于磁弹性耦合效应,外加磁场将对铁磁杆的振动频率产生影响:当铁磁杆的振动位移沿着磁场方向时,其频率减小并出现磁弹性屈曲失稳;当铁磁杆的振动位移垂直于磁场方向时,其频率将会增大．理论模型能够很好地解释已有实验观测的振动频率改变现象．     

Well-posedness of a 2D swimming model governed in the nonstationary Stokes fluid by multiplicative controls

We introduce and discuss the well-posedness of a schematic simplified mathematical model of an abstract object ‘swimming’ in the 2D nonstationary Stokes fluid. The object consists of finitely many subsequently connected small sets, each of which can act upon a pair of the adjacent sets in a rotation fashion with the purpose to generate its fish-like or rowing motion. The structural integrity of the object is maintained by respective elastic Hooke's forces. Such models are of interest in biological and engineering applications dealing with propulsion systems in fluids. Mathematics-wise, the model equations are described by a complex highly nonlinear hybrid system of partial differential equations and ordinary differential equations, which include a fluid equation and an equation describing the motion of the object in it. Our study is linked to viewing the swimming process as the one governed by multiplicative (bilinear) controls.     

Modeling a swimming fish with an initial boundary value problem: Unsteady maneuvers of an elastic plate with internal force generation

In order to model unsteady maneuvers in swimming fish, we develop an initial-boundary value problem for a fourth-order hyperbolic partial differential equation in which the fish's body is treated as an inhomogeneous elastic plate. The model is derived from the three-dimensional equations of elastic dynamics, and is essentially a simple variant of the classical Kirchhoff model for a dynamic plate. The model incorporates body forces generating moment to simulate muscle force generation in fish. The initial-boundary value problem is reduced to a beam model in one spatial dimension and formulated computationally using finite differences. Interaction with the surrounding water is represented by nonlinear viscous damping. Two example applications using simple but physically reasonable physiological parameters are presented and interpreted. One models the acceleration from rest to steady swimming, the other a rapid turn from rest.     

Stokes drag on axially symmetric bodies: a new approach

In this paper a new approach to evaluate the drag force in a simple way on a restricted axially symmetric body placed in a uniform stream (i) parallel to its axis, (ii) transverse to its axis, is advanced when the flow is governed by the Stokes equations. The method exploits the well-known integral for evaluating the drag on a sphere. The method not only provides the value of the drag on prolate and oblate spheroids and a deformed sphere in axial flow which already exists in literature but also new results for a cycloidal body, an egg shaped body and a deformed sphere in transverse flow. The salient results are exhibited graphically. The limitations imposed on the analysis because of the lack of fore and aft symmetry in the case of an eggshaped body is also indicated. It is also seen that the analysis can be extended to calculate the couple on a body rotating about its axis of symmetry.     

A BOUNDARY INTEGRAL METHOD FOR COMPUTING ELASTIC MOMENT TENSORS FOR ELLIPSES AND ELLIPSOIDS

The concept of elastic moment tensor occurs in several interesting contexts, in particular in imaging small elastic inclusions and in asymptotic models of dilute elastic composites. In this paper, we compute the elastic moment tensors for ellipses and ellipsoids by using a systematic method based on layer potentials. Our computations reveal an underlying elegant relation between the elastic moment tensors and the single layer potential.     

Exploration of the encased nanocomposites functionally graded porous arches: Nonlinear analysis and stability behavior

This paper explores the nonlinear stability mechanism of the functionally graded porous (FGP) arch reinforced by graphene nanocomposites. Both the pores and the nanocomposites are distributed symmetrically to the mid-surface of the arch but not uniformly in the cross-section so that the bending stiffness can be best improved. The arch is confined in an elastic medium with a radially-pointed concentrated load at the crown position. The confinement of the medium results in a symmetrical deformed shape of the arch, which can be described by an admissible displacement function. Associated with the thin-walled arch theory and the principle of minimum potential energy, analytical predictions are obtained to express the critical buckling load, as well as the hoop force and bending moment. Subsequently, a numerical model is developed to simulate the medium and the arch in ABAQUS software. By introducing the modified arc-length method, the equilibrium paths of the encased arch are traced. After comparison in terms of the critical buckling load and the equilibrium paths, it is found the numerical results are in good accordance with the analytical solutions. Finally, particular attention is paid to the parameters that may impact the buckling load, such as the porosity coefficient, the weight fraction, the central angle, the geometry of the Graphene platelets (GPLs) et al.     

Oscillations of the inertia moment of a finite Fermi system in the cranking model framework

In the framework of the cranking model with the potential of an anisotropic harmonic oscillator, we rigorously calculate how the moment of inertia of a finite Fermi system depends on the chemical potential at finite temperatures in the adiabatic limit analytically. We show that this dependence involves smooth and oscillating components. We find analytic expressions for these components at arbitrary temperatures and axial deformation frequencies. We show that oscillations of the moment of inertia increase as the spherical limit is approached and decrease exponentially as the temperature increases.     

Propagating waves in elastic material with voids subjected to electro-magnetic interactions

Constitutive relations and field equations are developed for an elastic solid with voids subjected to electro-magnetic field. The linearized form of the relations and equations are presented separately when medium is subjected to a large magnetic field and when it is subjected to a large electric field. The possibility of propagation of time harmonic plane waves in an infinite elastic solid with voids has been explored. It is found that when the medium is subjected to large magnetic field, there exist two coupled longitudinal waves propagating with distinct speeds and a transverse wave mode. However, when the medium is subjected to a large electric field, there may propagate five basic waves comprising of four coupled longitudinal waves propagating with distinct speeds and a lone transverse wave. The effects of magnetic and electric fields are observed on the propagation characteristics of the existing waves. Under the limiting cases of frequency and for different electric conductive materials, the speeds of various waves are investigated. The phase speeds of different waves and their corresponding attenuations have been computed against the frequency parameter and depicted graphically for a specific material.