Stability for basic system of equations of atmospheric motion

The topological characteristics for the basic system of equations of atmospheric motion were analyzed with the help of method provided by stratification theory. It was proved that in the local rectangular coordinate system the basic system of equations of atmospheric motion is stable in infinitely differentiable function class. In the sense of local solution, the necessary and sufficient conditions by which the typical problem for determining solution is well posed were also given. Such problems as something about ＂speculating future from past＂ in atmospheric dynamics and how to amend the conditions for determining solution as well as the choice of underlying surface when involving the practical application were further discussed. It is also pointed out that under the usual conditions, three motion equations and continuity equation in the basic system of equations determine entirely the property of this system of equations.     

Stability for basic system of equations of atmospheric rhotion

The topological characteristics for the basic system of equations of atmo- spheric motion were analyzed with the help of method provided by stratification theory.It was proved that in the local rectangular coordinate system the basic system of equations of atmospheric motion is stable in infinitely differentiable function class.In the sense of local solution,the necessary and sufficient conditions by which the typical problem for determining solution is well posed were also given.Such problems as something about“speculating future from past”in atmospheric dynamics and how to amend the condi- tions for determining solution as well as the choice of underlying surface when involving the practical application were further discussed.It is also pointed out that under the usual conditions,three motion equations and continuity equation in the basic system of equations determine entirely the property of this system of equations.     

Impact of singularity of Navier-Stokes equation upon atmospheric motion equations

Some conclusions about the smooth function classes stability for the basic system of equations of atmospheric motion and instability for Navier-Stokes equation are summarized. On the basis of this, by taking the basic system of equations of atmospheric motion via Boussinesq approximation as example to explain in detail that the instability about some simplified models of the basic system of equations for atmospheric motion is caused by the instability of Navier-Stokes equation, thereby, a principle to guarantee the stability of simplified equation is drawn in simplifying the basic system of equations.     

Analytical solution of attitude motion for spacecraft with a slewing appendage

This paper investigates pitch motion and in orbital plane elastic vibration of a spacecraft with a flexible beam type appendage undergoing prescribed slew maneuver. The governing equations are transformed into a standard quasi-linear form, and then solved by Butenin's variation of parameters approach. Validity of the analytical solutions is assessed over a range of system parameters and initial conditions by comparing them with the results of numerical integration. The results show that they are very good approximations and provide extensive insight into the dynamical response of the system.     

Analytical solutions for some nonlinear evolution equations

IntroductionItiswell_knownthatmanyimportantdynamicsprocessescanbedescribedbyspecificnonlinearpartialdifferentialequations .Whenanonlinearpartialdifferentialequationisusedtodescribeaphysicalparameterthatshowssomekindsofpropagationoraggregationproperties,oneofthemostimportantphysicalmotivationsistosolvethepartialdifferentialequationwithacertaintypeoftravellingwavesolution .Inthepastseveraldecades,therehavebeenmanyattemptsinthisfieldbothbymathematiciansandphysicists[1]- [16 ],however,duetothecomp…     

One type of integrals for the equations of motion of higher-order nonholonomic systems

This paper presents one type of integrals and its condition of existence for theequations Of motion of higher-order nonholonomic systems,including I-order integral(generalized energy integral),2-order integral and p-order integral(P>2).All of theseintegrals can be constructed by the Lagrangianfunction of the system.Two examples aregiven to illustrate the application of the suggested method.     

Analytical solution of magnetohydrodynamic sink flow

An analytical solution to the famous Falkner-Skan equation for the magnetohydrodynamic (MHD) flow is obtained for a special case, namely, the sink flow with a velocity power index of −1. The solution is given in a closed form. Multiple solution branches are obtained. The effects of the magnetic parameter and the wall stretching parameter are analyzed. Interesting velocity profiles are observed with reversal flow regions even for a stationary wall. These solutions provide a rare case of the Falkner-Skan MHD flow with an analytical closed form formula. They greatly enrich the analytical solution for the celebrated Falkner-Skan equation and provide better understanding of this equation.     

Analytical solution of magnetohydrodynamic sink flow

An analytical solution to the famous Falkner-Skan equation for the magnetohydrodynamic (MHD) flow is obtained for a special case, namely, the sink flow with a velocity power index of -1. The solution is given in a closed form. Multiple solution branches are obtained. The effects of the magnetic parameter and the wall stretching parameter are analyzed. Interesting velocity profiles are observed with reversal flow regions even for a stationary wall. These solutions provide a rare case of the Falkner-Skan MHD ...     

Analytical formulas of solutions of geometrically nonlinear equations of axisymmetric plates and shallow shells

Starting from the step-by-step iterative method, the analytical formulas of solutions of the geometrically nonlinear equations of the axisymmetric plates and shallow shells, have been obtained. The uniform convergence of the iterative method, is used to prove the convergence of the analytical formulas of the exact solutions of the equations.     

Analytical solutions of singularities moving with an arbitrary path when two fluids are present

The derivations are carried out for the velocity potentials of singularities moving withan arbitrary path either in the upper fluid or in the lower fluid with or without a horizontalbottom when two fluids are present.In such a case,the pressure distribution is no longerequal to a constant or zero at the free interface.Taking the influence of an upper fluid uponthe lower fluid into consideration,a series of fundamental solutions in closed forms arepresented in this paper.     

Analytical solution of bending-compression column using different tension-compression modulus

Based on elastic theory of different tension-compression modulus, the analyttcal solution was deduced for bending-compression column subject to combined loadings by the flowing coordinate systemaxis, stress, strain andand phased integration methoddisplacement were developed,The formulations for the neutralthe finite element program was compiled for calculation, and the comparison between the result of finite element and analytical solution were given too. Finally, compare and analyze the result of different modulus and the same modulus, obtain the difference of two theories in result, and propose the reasonable suggestion for the calculation of this structure.     

A note on general solution of Stokes equations

Li et al. (2015) claim that it is sufficient to use two harmonic functions to express the general solution of Stokes equations. In this paper, we demonstrate that this is not true in a general case and that we in fact need three scalar harmonic functions to represent the general solution of Stokes equations (Venkatalaxmi et al., 2004).     

Analytical solution for laminar flow through leaky tube

The laminar flow through a leaky tube is investigated, and the momentum and conservation of energy equations are solved analytically. By using the Hagen-Poiseuille velocity profile and defining unknown functions for the axial and radial velocity components, the pressure and mass transfer equations are obtained, and their profiles are plotted according to different parameters. The results indicate that the axial velocity, the radial velocity, the mass transfer parameter, and the pressure in the tube decrease as the fluid moves along the tube.     

New scalar-type equilibrium equations of the general space force system

ＩｎｔｒｏｄｕｃｔｉｏｎＬｉｋｅｔｈｅｃｏｐｌａｎａｒｆｏｒｃｅｓｙｓｔｅｍ ,ａｎｙｇｅｎｅｒａｌｓｐａｃｅｆｏｒｃｅｓｙｓｔｅｍＦｉ(ｉ =1 ,2 ,… ,ｎ)ｃａｎｂｅｒｅｄｕｃｅｄｔｏａｐｏｉｎｔＯａｎｄｒｅｐｌａｃｅｄｂｙａｐｒｉｎｃｉｐａｌｖｅｃｔｏｒＲａｎｄａｐｒｉｎｃｉｐａｌｍｏｍｅｎｔＬ0 .Ｓｏｔｈａｔｔｈｅｖｅｃｔｏｒ＿ｔｙｐｅｎｅｃｅｓｓａｒｙａｎｄｓｕｆｆｉｃｉｅｎｔｃｏｎｄｉｔｉｏｎｓｆｏｒｅｑｕｉｌｉｂｒｉｕｍｏｆｔｈｅｇｉｖｅｎｆｏｒｃｅｓｙｓｔｅｍａｒｅＲ =0 ,　Ｌ0 =0 . (1 )…     

Analytical approximate 3D solution for the longitudinally vibrating cylinder

Summary The paper presents a three-dimensional approximate solution of a longitudinally vibrating cylinder. It is based on the frequency equation for free waves travelling along an infinitely long cylinder which follows from the radial boundary condition on the lateral surface of the cylinder. An equivalent longitudinal traction is calculated by integration of the normal stresses over the end cross section. The full solution of the longitudinally vibrating rod adapted to given axial boundary conditions is derived. The solution is compared to numerical results, and good agreement is obtained.     

Two types of new algorithms for finding explicit analytical solutions of nonlinear differential equations

ＩｎｔｒｏｄｕｃｔｉｏｎＦｏｒｓｅｖｅｒａｌｈｕｎｄｒｅｄｙｅａｒｓ,ｉｔｈａｓｂｅｅｎｂｏｔｈｉｍｐｏｒｔａｎｔａｎｄｄｉｆｆｉｃｕｌｔｐｒｏｊｅｃｔｔｏｃｏｎｓｔｒｕｃｔｔｈｅａｎａｌｙｔｉｃａｌｓｏｌｕｔｉｏｎｓｆｏｒｎｏｎｌｉｎｅａｒｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ.Ｍａｎｙｍａｔｈｅｍａｔｉｃｉａｎｓａｎｄｔｈｅｏｒｅｔｉｃａｌｐｈｙｓｉｃｉｓｔｓｈａｖｅｄｏｎｅａｌｏｔｏｆｗｏｒｋ[1～ 5 ].Ｈｏｗｅｖｅｒ,ｔｈｅｒｅｅｘｉｓｔｌｏｔｓｏｆｉｍｐｏｒｔａｎｔａｎｄｂｅｉｎｇｏｆｐ…     

Equations of motion for mechanical systems: A unified approach

This paper presents a unified framework from which emerge the Lagrange equations, the Gibbs-Appell Equations and the Generalized Inverse Equations for describing the motion of constrained mechanical systems. The unified approach extends the applicability of the first two approaches to systems where the constraints are non-linear functions of the generalized velocities and are not necessarily independent. Furthermore, the approach leads to the Explicit Gibbs-Appell Equations.     

Canonical forms of Nielsen's and Cenov's dynamical equations

In this paper, with Poincaré's formalism, and an indirect method, the canonical forms of the generalized equations of motion due to Nielsen and Cenov of a holonomic dynamical system in the velocity-phase space and the acceleration-phase space are obtained in terms of the Poincaré parameters This paper was presented at the International Congress of Mathematicians (ICM), 21–29 August, 1990, Kyoto University, Japan.     

Three-dimensional motion of a viscous incompressible fluid in a narrow tube

An approximate solution of the problem of unsteady motion of a viscous incompressible fluid in a long narrow deformable tube at low Reynolds numbers is obtained. Pressure oscillations and tube deformation are shown to be related by an integrodifferential equation. The solution obtained extends the Poiseuille solution in elliptic tubes to the case of comparatively arbitrary small deformations in terms of the tube length and angle. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 28–32, July–August, 2009.