RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅵ)—CONSERVATION LAWS OF MASS AND INERTIA

IntroductionThisworkisadirectcontinuationandasupplementofRefs .[1～8] .InRefs.[1～8]thecoupledbalancelawsandequationsofmomentum ,angularmomentumandenergyaswellasthenewHamiltonprinciple,principleofvirtualpowerandNoethertheoremhavebeenpresented .However,thecoupledconservationlawsofmassandinertiahavenotbeenreestablishedyet.Thepurposeofthispaperistoreestablishtheconservationlawsandequationsofmassandinertiaandtocombinethemwiththecoupledbalancelawsandequationsofmomentum ,angularmomentum ,energyand…     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅵ)—CONSERVATION LAWS OF MASS AND INERTIA

The purpose is to reestablish the coupled conservation laws, the local conservation equations and the jump conditions of mass and inertia for polar continuum theories. In this connection the new material derivatives of the deformation gradient, the line element, the surface element and the volume element were derived and the generalized Reynolds transport theorem was presented. Combining, these conservation laws of mass and inertia with the balance laws of momentum, angular momentum and energy derived in our previous papers of this series, a rather complete system of coupled basic laws and principles for polar continuum theories is constituted on the whole. From this system the coupled nonlocal balance equations of mass, inertia, momentum, angular momentum and energy may be obtained by the usual localization. Contributed by DAI Tian-min, Original Member of Editorial Committee, AMM Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931≈)     

Peristaltic flow of Johnson-Segalman fluid in asymmetric channel with convective boundary conditions

This work is concerned with the peristaltic transport of the Johnson-Segalman fluid in an asymmetric channel with convective boundary conditions. The mathematical modeling is based upon the conservation laws of mass, linear momentum, and energy. The resulting equations are solved after long wavelength and low Reynolds number are used. The results for the axial pressure gradient, velocity, and temperature profiles are obtained for small Weissenberg number. The expressions of the pressure gra-dient, velocity, and temperature are analyzed for various embedded parameters. Pumping and trapping phenomena are also explored.     

Turbulent flow field and heat transfer in a heated circular channel under a reciprocating motion

This study is to simulate the turbulent flow field and heat transfer when cold fluid flows through a finite-length circular channel, which meantime undergoes a reciprocating motion using a numerical method. The mass, momentum and energy conservation equations and turbulent k- equations are derived for a turbulent flow field in a reciprocating coordinate by a coordinate transformation from the stationary coordinate system. The SIMPLE-C scheme is used to investigate heat transfer in a cooling channel undergoing a reciprocating motion by changing parameters and cooling mediums. The parameters are Reynolds number (7170 to 20000) and Pulsating number (1.55 to 4.65). The results show that the averaged Nusselt number increase with increasing both Reynolds number and Pulsating number, and the averaged Nusselt number of cooling oil is lower than that of water at the same incoming flow rate for both the stationary and reciprocating channel.     

一类多体系统非完整运动最优规划的拟牛顿算法

研究带有非完整约束的一类多体系统运动规划问题。多体系统中的非完整约束通常是由不可积的速度约束或不可积的守恒律引起。在系统动量和动量矩守恒情况下,动力学方程降阶为非完整形式约束方程,系统的控制问题可转化为无漂移系统的非完整运动规划问题。文中首先导出具有多体开链系统的非完整运动模型。利用最优控制理论和最优化技术,采用输入参数化的方法将连续的最优控制问题转化为离散的最优控制问题,提出一种非完整多体系统运动规划的拟牛顿算法。最后将该方法用于自由漂浮的空间三连杆机构,仿真结果验证了该方法的有效性。     

The Motion of a Fluid–Rigid Disc System at the Zero Limit of the Rigid Disc Radius

We consider the two-dimensional motion of the coupled system of a viscous incompressible fluid and a rigid disc moving with the fluid, in the whole plane. The fluid motion is described by the Navier–Stokes equations and the motion of the rigid body by conservation laws of linear and angular momentum. We show that, assuming that the rigid disc is not allowed to rotate, as the radius of the disc goes to zero, the solution of this system converges, in an appropriate sense, to the solution of the Navier–Stokes equations describing the motion of only fluid in the whole plane. We also prove that the trajectory of the centre of the disc, at the zero limit of its radius, coincides with a fluid particle trajectory.     

A numerical continuous model for the hydrodynamics of fluid particle systems

In order to understand the hydrodynamic interactions that can appear in a fluid particle motion, an original method based on the equations governing the motion of two immiscible fluids has been developed. These momentum equations are solved for both the fluid and solid phases. The solid phase is assumed to be a fluid phase with physical properties, such as its behaviour can be assimilated to that of pseudo‐rigid particles. The only unknowns are the velocity and the pressure defined in both phases. The unsteady two‐dimensional momentum equations are solved by using a staggered finite volume formulation and a projection method. The transport of each particle is solved by using a second‐order explicit scheme. The physical model and the numerical method are presented, and the method is validated through experimental measurements and numerical results concerning the flow around a circular cylinder. Good agreement is observed in most cases. The method is then applied to study the trajectory of one settling particle initially off‐centred between two parallel walls and the corresponding wake effects. Different particle trajectories related to particulate Reynolds numbers are presented and commented. A two‐body interaction problem is investigated too. This method allows the simulation of the transport of particles in a dilute suspension in reasonable time. One of the important features of this method is the computational cost that scales linearly with the number of particles. Copyright © 1999 John Wiley & Sons, Ltd.     

全弹性平面运动机器人的高精度运动控制和振动抑制算法研究

为实现对基座、关节和臂均存在弹性的空间机器人运动高精度控制及多重振动抑制,建立了基座、关节和臂全弹性空间机器人动力学模型,并采用运动有限维PD重复学习控制及振动同步抑制方案进行研究.首先,利用线性弹簧、扭转弹簧和欧拉-伯努力梁理论,假设模态法和动量守恒定律,采用拉格朗日方程建立了弹性基座、柔性关节和柔性臂空间机器人动力...     

Optimal control of nonholonomic motion planning for a flee-falling cat

The nonholonomic motion phnning of a free-falling cat is investigated.Nonholonomicity arises in a free-falling cat subject to nonintegrable angle velocity constraints or nonintegrable conservation laws.When the total angular momentum is zero,the motion equation of a free-falling cat is established based on the model of two symmetric rigid bodies and conservation of angular momentum.The control of system can be converted to the problem of nonholonomic motion planning for a free-falling cat.Based on Ritz approximation theory,the Gauss-Newton method for motion planning by a falling cat is proposed.The effectiveness of the numerical algorithm is demonstrated through simulation on model of a free-falling cat.     

Optimal control of nonholonomic motion planning for a free-falling cat

The nonholonomic motion planning of a free-falling cat is investigated. Non-holonomicity arises in a free-falling cat subject to nonintegrable angle velocity constraints or nonintegrable conservation laws. When the total angular momentum is zero, the motion equation of a free-falling cat is established based on the model of two symmetric rigid bodies and conservation of angular momentum. The control of system can be converted to the problem of nonholonomic motion planning for a free-falling cat. Based on Ritz approximation theory, the Gauss-Newton method for motion planning by a falling cat is proposed. The effectiveness of the numerical algorithm is demonstrated through simulation on model of a free-falling cat.     

Inertial steady flow of a bed of granular material down a surface with microrelief with allowance for finite time of intergranular contacting

An inertial flow of a granular material can be described by the laws of conservation of mass, momentum, and energy of random motion of solid particles by invoking some closing relations. In this work, these closing relations are inferred from the dimensional theory. The system of equations obtained is used to determine characteristics of a steady flow of a bed of a granular material down an inclined surface with a microrelief for various Richardson numbers and finite contact times of the particles during their collisions. Novosibirsk Military Institute, Novosibirsk 630103. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 6, pp. 128–132, November–December, 1999.     

Transient flow of magnetized Maxwell nanofluid: Buongiorno model perspective of Cattaneo-Christov theory

The present research article is devoted to studying the characteristics of Cattaneo-Christov heat and mass fluxes in the Maxwell nanofluid flow caused by a stretching sheet with the magnetic field properties. The Maxwell nanofluid is investigated with the impact of the Lorentz force to examine the consequence of a magnetic field on the flow characteristics and the transport of energy. The heat and mass transport mechanisms in the current physical model are analyzed with the modified versions of Fourier’s and Fick’s laws, respectively. Additionally, the well-known Buongiorno model for the nanofluids is first introduced together with the Cattaneo-Christov heat and mass fluxes during the transient motion of the Maxwell fluid. The governing partial differential equations (PDEs) for the flow and energy transport phenomena are obtained by using the Maxwell model and the Cattaneo-Christov theory in addition to the laws of conservation. Appropriate transformations are used to convert the PDEs into a system of nonlinear ordinary differential equations (ODEs). The homotopic solution methodology is applied to the nonlinear differential system for an analytic solution. The results for the time relaxation parameter in the flow, thermal energy, and mass transport equations are discussed graphically. It is noted that higher values of the thermal and solutal relaxation time parameters in the Cattaneo-Christov heat and mass fluxes decline the thermal and concentration fields of the nanofluid. Further, larger values of the thermophoretic force enhance the heat and mass transport in the nanoliquid. Moreover, the Brownian motion of the nanoparticles declines the concentration field and increases the temperature field. The validation of the results is assured with the help of numerical tabular data for the surface velocity gradient.     

Stokes’ first problem for some non-Newtonian fluids: Results and mistakes

The well-known problem of unidirectional plane flow of a fluid in a half-space due to the impulsive motion of the plate it rests upon is discussed in the context of the second-grade and the Oldroyd-B non-Newtonian fluids. The governing equations are derived from the conservation laws of mass and momentum and three correct known representations of their exact solutions given. Common mistakes made in the literature are identified. Simple numerical schemes that corroborate the analytical solutions are constructed.     

Global Strong Solutions for the Two-Dimensional Motion of an Infinite Cylinder in a Viscous Fluid

In this paper, we consider a two-dimensional fluid-rigid body problem. The motion of the fluid is modelled by the Navier-Stokes equations, whereas the dynamics of the rigid body is governed by the conservation laws of linear and angular momentum. The rigid body is supposed to be an infinite cylinder of circular cross-section. Our main result is the existence and uniqueness of global strong solutions.     

One-dimensional model for microscale shock tube flow

A one-dimensional model for the numerical simulation of transport effects in small-scale, i.e., low Reynolds number, shock tubes is presented. The conservation equations have been integrated in the lateral directions and three-dimensional effects have been introduced as carefully controlled sources of mass, momentum and energy, into the axial conservation equations. The unsteady flow of gas behind the shock wave is reduced to a quasi-steady flow by choosing a coordinate system attached to the shock. The boundary layer problem is thereby reduced to a laminar solution, similar to the Blasius solution, with the exception that the wall velocity can be nonzero. The resulting one-dimensional equations are then solved numerically using a two-step Lax-Wendroff/ MacCormack scheme with flux correction transport. For validation purposes, comparisons are performed against previously published shock structure and low Reynolds number shock tube experiments; good agreement is observed. The model has been used to predict the performance of a 10μm shock tube and the result of this simulation shows the possibility of shock wave disappearance at lower pressure ratios for a micro-scale shock tube.      

Conservation laws and group properties of equations of isentropic gas motion

A system of equations of isentropic gas motion with n ⩾ 2 is classified in terms of zero-order conservation laws with the use of the method of A-operators. New conservation laws are found to be valid only for potential isentropic motion of the Chaplygin gas. In this case, the greatest number of nontrivial conservation laws is obtained, with n scalar conservation laws being nonlocal. Additional properties of symmetry of the considered equations associated with these conservation laws are indicated.     

Boundary layer flow of Maxwell fluid due to torsional motion of cylinder: modeling and simulation

This paper investigates the boundary layer flow of the Maxwell fluid around a stretchable horizontal rotating cylinder under the influence of a transverse magnetic field. The constitutive flow equations for the current physical problem are modeled and analyzed for the first time in the literature. The torsional motion of the cylinder is considered with the constant azimuthal velocity E. The partial differential equations (PDEs) governing the torsional motion of the Maxwell fluid together with energy transport are simplified with the boundary layer concept. The current analysis is valid only for a certain range of the positive Reynolds numbers. However, for very large Reynolds numbers, the flow becomes turbulent. Thus, the governing similarity equations are simplified through suitable transformations for the analysis of the large Reynolds numbers. The numerical simulations for the flow, heat, and mass transport phenomena are carried out in view of the bvp4c scheme in MATLAB. The outcomes reveal that the velocity decays exponentially faster and reduces for higher values of the Reynolds numbers and the flow penetrates shallower into the free stream fluid. It is also noted that the phenomenon of stress relaxation, described by the Deborah number, causes to decline the flow fields and enhance the thermal and solutal energy transport during the fluid motion. The penetration depth decreases for the transport of heat and mass in the fluid with the higher Reynolds numbers. An excellent validation of the numerical results is assured through tabular data with the existing literature.     

漂浮基柔性空间机器人的鲁棒控制及振动抑制

讨论了漂浮基柔性空间机器人系统的动力学建模、运动控制算法设计以及关节、臂双重柔性振动的分级主动抑制问题. 利用系统动量、动量矩守恒关系和拉格朗日-假设模态法对系统进行动力学分析,建立系统动力学方程. 基于奇异摄动法,将系统分解为表示系统刚性运动部分的慢变子系统, 表示由柔性臂引起的系统柔性运动部分的快变子系统1和表示由柔性关节引起的系统柔性运动部分的快变子系统2. 针对慢变子系统提出一种鲁棒控制方法来补偿系统参数的不确定性和柔性关节引起的转动误差,实现系统期望运动轨迹的渐近跟踪;针对快变子系统1采用线性二次型最优控制器来抑制由柔性臂引起的系统柔性振动;针对快变子系统2设计了基于机械臂和电机转子的转角速度差值的反馈控制器来抑制由柔性关节引起的系统柔性振动. 因此,系统的总控制律为以上3个子系统控制律的综合. 最后通过仿真实验证明了所提出的混合控制方法的有效性.      

Impact of Non-uniform Properties on Governing Equations for Fluid Flows in Porous Media

The macroscopic governing equations of a compressible multicomponents flow with non-uniform viscosity and with mass withdrawal (due to heterogeneous reactions) in a porous medium are developed. The method of volume averaging was used to transform local (or microscopic) governing equations into averaged (or macroscopic) governing equations. The impacts of compressibility, non-uniform viscosity, and mass withdrawal on the form of the averaged equations and on the value of the macroscopic transport coefficients were investigated. The results showed that the averaged mass conservation equation is significantly affected by mass withdrawal when a specific criterion on the size of the domain is respected. The results also showed that the form of the averaged momentum equations is not affected by mass withdrawal, by compressibility effects or by non-uniform viscosity, provided that the Reynolds number at the pore level is small. Nonetheless, the velocity field is affected by the heterogeneous reaction via the averaged mass conservation equation, and also by viscosity variations due to the presence of the volume-averaged viscosity (which value changes with position) in the averaged momentum equations. A new closure variable definition was proposed to formulate the closure problem, which avoided the need to solve an integro-differential equation in the closure problem. This formulation was used to show that the permeability tensor only depends on the geometry of the porous medium. In other words, that tensor is independent on whether the fluid is compressible/incompressible, has uniform/non-uniform viscosities, and whether mass withdrawal due to heterogeneous reactions is present/absent.     

Finite element simulations of incompressible flow past a heated/cooled sphere

This paper reports numerical simulation of the flow past a heated/cooled sphere. A Galerkin finite element method is used to solve the 3D incompressible Boussinesq equations in primitive variable form. Numerical simulations of flow around the sphere for a range of Grashof numbers and moderate Reynolds numbers, were conducted. The drag coefficient for adiabatic flow shows good agreement with standard correlations over the range of the Reynolds numbers investigated. It is shown that the drag can vary considerably with heating of the sphere and that computational fluid dynamics methods can be used to derive constitutive laws for macroscopic momentum and heat exchange in multiphase flow. © 1998 John Wiley & Sons, Ltd.