A NEW APPROXIMATE ANALYTICAL SOLUTION OF THE IDEAL POTENTIAL FLOW AROUND A CIRCULAR CYLINDER BETWEEN TWO PARALLEL FLAT PLATES

In ref. [1], Lin obtained an approximate analytical solution of the ideal potential flow around a circular cylinder between two parallel flat flates. In this paper, the author shows that one may obtain the result coinciding with that obtained in ref. [1] by making use of the Shvez's method. Morever, we can obtain a more accurate result than that obtained in ref. [1], if we make use of the improved Shvez's method. Some calculating examples are presented.     

THE APPROXIMATE SOLUTION OF THE SELF-SIMILAR PROBLEM FOR RADIAL FLOW OF NON-NEWTONIAN FLUIDS THROUGH POROUS MEDIA

In this paper, we study the approximate solution of the self-simikar problem for radial flow of non-Newtonian fluids through porous media. Assuming that the fluids obey the exponential function law, we obtain an exact solution for the exponent n=0 and compare it with the approximate solution in ref.[1]. For n>1 and n<1, we obtain respectively approximate solutions. Some exampls are presented.     

LAMINAR BOUNDARY LAYER BETWEEN TWO PLANES PERPENDICULAR TO EACH OTHER

In this paper, we obtain a third-order approximate solution for the laminar boundarylayer between two plans perpendicular to each other.In boundary layer equations, the viscous and the inertial terms have the same quantitystep.In this paper, at first, supposing that the inertial terms are bigger than the viscousterms, we solve the boundary layer equations, and then we suppose that the viscous termsare bigger than the inertial terms. At last. we take the mean value as the valid solution of theboundary layer equations.The first-and the second-order approximate solutions obtained in this paper coincdewith the results in ref[1],while the third-order solution obtained in this paper is better thanthat in ref [1].     

Flow on oscillating rectangular duct for Maxwell fluid

This paper presents an analysis for the unsteady flow of an incompressible Maxwell fluid in an oscillating rectangular cross section.By using the Fourier and Laplace transforms as mathematical tools,the solutions are presented as a sum of the steady-state and transient solutions.For large time,when the transients disappear,the solution is represented by the steady-state solution.The solutions for the Newtonian fluids appear as limiting cases of the solutions obtained here.In the absence of the frequency of oscillations,we obtain the problem for the flow of the Maxwell fluid in a duct of a rectangular cross-section moving parallel to its length.Finally,the required time to reach the steady-state for sine oscillations of the rectangular duct is obtained by graphical illustrations for different parameters.Moreover,the graphs are sketched for the velocity.     

THE THEORY OF SIMILARITY AND THE PROFILE OF VERTICAL DISTRIBUTION OF CONCENTRATION OF SAND PARTICLES

In this article,we treat the problem of two-dimensionaluniform steady channel flow of turbid water with theory ofsimilarity.Under the condition of similarity of turbulentfluctuation velocity and fluctuation of concentration ofsand particles,we obtain the profile of the vertical dis-tribution of concentration of sand particles.This profileof vertical distribution of concentration of sand particlesis slightly different from that obtained by diffusion theory,and departs from that obtained by gravitational theory.     

EXISTENCE AND COMPARISON RESULTS OF SOLUTIONS FOR NONLINEAR RANDOM INTEGRAL AND DIFFERENTIAL EQUATIONS RELATIVE TO WEAK TOPOLOGY

In this paper,we prove several existence theorems of random solutions to nonlinearrandom Volterra integral equations under the weak topology of Banach spaces.Then,asapplications,we obtain the existence theorems of weak random solutions to randomdiffrential equations. Existence of extremal random solutions and a random comparisontheorem for these random equations are also obtained.Our theorems imaprove and extendthe corresponding resulls in [4,5,10,11,12].     

THE SCHRDINGER EQUATION OF THIN SHELL THEORIES

This work is the continuation of the discussions of[50]and[51].In this paper:(A)The Love-Kirchhoff equation of small deflection problem for elastic thin shellwith constant curvature are classified as the same several solutions of Schr(?)dingerequation,and we show clearly that its form in axisymmetric problem;(B)For example for the small deflection problem,we extract the general solution ofthe vibration problem of thin spherical shell with equal thickness by the force in centralsurface and axisymmetric external field,that this is distinct from ref.[50]in variable.Today the variable is a space-place,and is not time;(C)The von Kármán-Vlasov equation of large deflection problem for shallow shellare classified as the solutions of AKNS equations and in it the one-dimensional problem isclassified as the solution of simple Schr(?)dinger equation for eigenvalues problem,and wetransform the large deflection of shallow shell from nonlinear problem into soluble linearproblem.     

ON PERFECTLY STRESS FIELD AT A MIXED-MODE CRACK TIP UNDER PLANE AND ANTI-PLANE STRAIN

In [1],under the condition that all the perfectly plastic stress components at a crack tipare functions of θ only,making use of equilibrium equations.stress-strain rate relations,compatibility equations and yield condition,Lin derived the general analytical expressionsof the perfectly plastic stress field at a mixed-mode crack tip under plane and anti-planestrain.But in [1] there were several restrictions on the proportionality factor λ in thestress-strain rate relations,such as supposing that λ is independent of θ andsupposing that λ=c or cr~(-1).In this paper,we abolish these restrictions.The cases in [1],λ=cr(?)(n=0 or-1)are the special cases of this paper.     

The Schrödinger equation in theory of plates and shells with orthorhombic anisotropy

This work is the continuation of the discussion of Refs. [1-5]. In this paper:[A] The Love-Kirchhoff equations of vibration problem with small deflection for orthorhombic misotropic thin shells or orthorhombic anisotropic thin plates on Winkler’s base are classified as several of the same solutions of Schrodmger equation, and we can obtain the general solutions for the two above-mentioned problems by the method in Refs. [1] and [3-5].[B] The. von Karman-Vlasov equations of large deflection problem for shallow shells with orthorhombic anisotropy (their special cases are the von Harmon equations of large deflection problem for thin plates with orthorhombic anisotropy) are classified as the solutions of AKNS equation or Dirac equation, and we can obtain the exact solutions for the two abovementioned problems by the inverse scattering method in Refs. [4-5].The general solution of small deflection problem or the exact solution of large deflection problem for the corrugated or rib-reinforced plates and shells as special cases is included in this paper.     

Decay rates of higher-order norms of solutions to the Navier-Stokes-Landau-Lifshitz system

In this paper, we investigate a system of the incompressible Navier-Stokes equations coupled with Landau-Lifshitz equations in three spatial dimensions. Under the assumption of small initial data, we establish the global solutions with the help of an energy method. Furthermore, we obtain the time decay rates of the higher-order spatial derivatives of the solutions by applying a Fourier splitting method introduced by Schonbek(SCHONBEK, M. E. L~2 decay for weak solutions of the Navier-Stokes equations. Archive for Rational Mechanics and Analysis, 88, 209–222(1985)) under an additional assumption that the initial perturbation is bounded in L~1(R~3).     

Approximate solutions of the Alekseevskii–Tate model of long-rod penetration

The Alekseevskii–Tate model is the most successful semi-hydrodynamic model applied to long-rod penetration into semi-infinite targets. However, due to the nonlinear nature of the equations, the rod(tail) velocity, penetration velocity, rod length, and penetration depth were obtained implicitly as a function of time and solved numerically By employing a linear approximation to the logarithmic relative rod length, we obtain two sets of explicit approximate algebraic solutions based on the implicit theoretica solution deduced from primitive equations. It is very convenient in the theoretical prediction of the Alekseevskii–Tate model to apply these simple algebraic solutions. In particular, approximate solution 1 shows good agreement with the theoretical(exact) solution, and the first-order perturbation solution obtained by Walters et al.(Int. J. Impac Eng. 33:837–846, 2006) can be deemed as a special form of approximate solution 1 in high-speed penetration. Meanwhile, with constant tail velocity and penetration velocity approximate solution 2 has very simple expressions, which is applicable for the qualitative analysis of long-rod penetration. Differences among these two approximate solutions and the theoretical(exact) solution and their respective scopes of application have been discussed, and the inferences with clear physical basis have been drawn. In addition, these two solutions and the first-order perturbation solution are applied to two cases with different initial impact velocity and different penetrator/target combinations to compare with the theoretical(exact) solution. Approximate solution 1 is much closer to the theoretical solution of the Alekseevskii–Tate model than the first-order perturbation solution in both cases, whilst approximate solution 2 brings us a more intuitive understanding of quasi-steady-state penetration.     

Electrokinetic energy conversion of electro-magneto-hydro-dynamic nanofluids through a microannulus under the time-periodic excitation

In this work, the effects of externally applied axial pressure gradients and transverse magnetic fields on the electrokinetic energy conversion(EKEC) efficiency and the streaming potential of nanofluids through a microannulus are studied. The analytical solution for electro-magneto-hydro-dynamic(EMHD) flow is obtained under the condition of the Debye-H¨uckel linearization. Especially, Green's function method is used to obtain the analytical solutions of the velocity field. The result shows that the velocity distribution is characterized by the dimensionless frequency ?, the Hartmann number Ha, the volume fraction of the nanoparticles φ, the geometric radius ratio a, and the wall ζ potential ratio b. Moreover, the effects of three kinds of periodic excitations are compared and discussed. The results also show that the periodic excitation of the square waveform is more effective in increasing the streaming potential and the EKEC efficiency. It is worth noting that adjusting the wall ζ potential ratio and the geometric radius ratio can affect the streaming potential and the EKEC efficiency.     

Nonstationary random vibration of one-degree and multidegree of freedom systems

In ref.[1]for nonstationary random processes such aswe defined the spectral densityS_ξ(t,ω)=|f(t,ω)|~2S(ω)where Z(A)is an orthogonal random measure,S(ω)is the spectral density of thestationary random processf(t,ω)is a complex function with two real variables,it is called modulation functionand it is satisfied byIn ref[1]we obtained the main results:for the linear dynamic systems which isdescribed by such equationswe obtained under certain conditions thatwherehere W(t,τ)is the response function of the systems to the unit pulse.This paper is continued on ref.[1].We obtain some results for the one-degree andmultidegree of freedom systems.     

Circular Shallow Spherical Shells with Central Circular Hole Under Simultaneous Actions of Arbitrary Unsteady Temperature Field and Arbitrary Dynamic Normal Load

In this paper,under assumption that tempeature is linearly distributed along the thickness of theshell,we deal with problems as indicated in the title and obtain general solutions of them which areexpressed in analytic form.In the first part,we investigate free vibration of circular shallow spherical shells with circularholes at the center under usual arbitrary boundary conditions.As an example,we calculate fundamen-tal natural frequency of a circular shallow spherical shell whose edge is fixed(m=0).Results we get areexpressed in analytic form and check well with E.Reissner’s[1].Method for calculating frequencyequation is recently suggested by Chien Wei-zang and is to be introduced in appendix3.In the second part,we investigate forced vibration of shells as indicated in the title under arbitr-ary harmonic temperature field and arbitrary harmonic dynamic normal load.In the third part,we investigate forced vibration of the above mentioned shells with initialconditions under arbitrary unsteady temperature     

Circular shallow spherical shells with central circular hole under simultaneous actions of arbitrary unsteady temperature field and arbitrary dynamic normal load

In this paper, under assumption that tempeature is linearly distributed along the thickness of the shell, we deal with problems as indicated in the title and obtain general solutions of them which are expressed in analytic form.In the first part, we investigate free vibration of circular shallow spherical shells with circular holes at the center under usual arbitrary boundary conditions. As an example, we calculate fundamental natural frequency of a circular shallow spherical shell whose edge is fixed (m=0). Results we get are expressed in analytic form and check well with E. Reissner’s [1]. Method for calculating frequency equation is recently suggested by Chien Wei-zang and is to be introduced in appendix 3.In the second part, we investigate forced vibration of shells as indicated in the title under arbitrary harmonic temperature field and arbitrary harmonic dynamic normal load.In the third part, we investigate forced vibration of the above mentioned shells with initial conditions under arbitrary unsteady temperature field and arbitrary normal load.In appendix 1 and 2, we discuss how to express displacement boundary conditions with stress function and boundary conditions in the case m=1.     

THE CREEPING MOTION IN THE ENTRY REGION OF A SEMI-INFINITE CIRCULAR CYLINDRICAL TUBE

In 1969,Lew and Fung[1]considered the inlet flow into a se-mi-infinite circular cylinder at low Reynolds number.Dagan etal.[2]in1982 obtained a series solution for the creeping motionthrough a pore of finite length directly.The numerical resultsobtained in[1]also describe the entrance flow in a tube of afinite length as the Fourier integrals in the general solutions arereplaced by Fourier series.In the present paper,the Fourier in-tegralss are evaluated numerically and the velocity,pressure dis-tribution and the stream function in the entry region of a semi-infinite circular cylindrical tube is close to the factor1.3 sug-gested by Lew and Fung[1].The collocation technique applied inthe present paper is shown to converge rapidly and it should beuseful in other similar problems.     

ON THE STOKES ENTRY FLOW INTO A SEMI-INFINITE CIRCULAR CYLINDRICAL TUBE

The problem of Stokes entry flow into a semi-infinite circular cylindrical tube wasstudied in this paper.A new kind of series solutions was derived.Their evident differencefrom the solutions in References[1,2]is that the present solutions don’t involve infiniteintegral.So they are favourable for calculation.We calculated an example by allocatedmethod and obtained satisfied results.     

ON THE GENERAL EQUATION AND THE GENERAL SOLUTION IN PROBLEMS FOR PLASTODYNAMICS WITH RIGID-PLASTIC MATERIAL

This work is the continuation of the discussion of refs.[1-2].We discuss thedynamics problems of ideal rigid—plastic material in the flow theory of plasticity in thispaper.From introduction of the theory of functions of complex variable under Dirac-Paulirepresentation we can obtain a group of the so-called“general equations”(i.e.have twoscalar equations)expressed by the stream function and the theoretical ratio.In this paperwe also testify that the equation of evolution for time in plastodynamics problema is neitherdissipative nor disperive,and the eigen-equation in plastodynamics problems is a stationarySchr(?)dinger equation,in which we take partial tensor of stress-increment as eigenfunctionsand take theoretical ratio as eigenvalues.Thus,we turn nonlinear plastodynamics problemsinto the solution of linear stationary Schr(?)dinger equation,and from this we can obtain thegeneral solution of plastodynamics problems with rigid-plastic material.     

ONE-DIMENSIONAL NONUNIFORM TEMPERATURE FLOW WITH HEAT TRANSFER BY CONVECTION

In ref.[1],V.E.Najenov studied the conditions that when the viscosity of the liquid isan exponential function of temperature,the pipe flow,having steady heat transfer,is one-dimensional and with nonuniform temperature.For plane canal circular pipe he stillstudied the velocity and the temperture fields.In this paper,the author presents two new methods for solving the same problem.Themethod as in ref.[1]may be regarded as the natural branch of the methods of this paper.One of our new methods only can solve the same problem as in ref.[1]and the complexdegree of its computing process is nearly the same as that in ref.[1].But the other can gobeyond the studying scope of ref.[1],namely,for the case that the curvatures ofcircumference of the cross section of the pipe are not equivalent everywhere,the problemmay also be solved.     

THE SCHR(?)DINGER EQUATION IN THEORY OF PLATES AND SHELLS WITH ORTHORHOMBIC ANISOTROPY

This work is the continuation of the discussion of Refs.[1-5].In this paper:[A] The Love-Kirchhoff equations of vibration problem with small deflection fororthorhombic anisotropic thin shells or orthorhombic anisotropic thin plates on Winkler’sbase are classified as several of the same solutions of Schr?dinger equation.and we canobtain the general solutions for the two above-mentioned problems by the method in Refs.[1]and[3-5].[B]The von Kármán-Vlasov equations of large deflection problem for shallow shellswith orthorhombic anisotropy(their special cases are the von Kármán equations of largedeflection problem for thin plates with orthorhombic anisotropy)are classified as thesolutions of AKNS equation or Dirac equation,and we can obtain the exact solutions forthe two abovementioned problems by the inverse scattering method in Refs.[4-5].The general solution of small deflection problem or the exact solution of largedeflection problem for the corrugated or rib-reinforced plates and shells as special c