Shock–Free Breakup of Droplets. Temporal Characteristics

In the present paper, we consider the shock–free breakup of droplets in their encounter with a layer (sheet) of a moving gas in the absence of pressure perturbations when the droplets are affected by a short U–shaped pulse of aerodynamic forces. Under a high pressure of the ambient gas medium p₀ = 20—80 bar, the droplets (ethanol or liquid oxygen) have a chance to break up after stay in a thing (2—5 mm thick) gas layer (jet) moving with a velocity of 1—10 m/sec. A distinctive feature of the process is that the characteristic time of droplet deformation and the period of natural oscillations coincide with the residence time for the droplets in the region of their interaction with the gas stream. Empirical formulas are proposed for determination of the total breakup time and the duration of the droplet disintegration stage in shock–free breakup.     

Free Boundary Problems for a Lotka–Volterra Competition System

In this paper we investigate two free boundary problems for a Lotka–Volterra type competition model in one space dimension. The main objective is to understand the asymptotic behavior of the two competing species spreading via a free boundary. We prove a spreading-vanishing dichotomy, namely the two species either successfully spread to the right-half-space as time \(t\) goes to infinity and survive in the new environment, or they fail to establish and die out in the long run. The long time behavior of the solutions and criteria for spreading and vanishing are also obtained. This paper is an improvement and extension of Guo and Wu (J Dyn Differ Equ 24:873–895, 2012).     

Paradox of “Self-Outflow” of a Free Liquid Jet

The effect of accumulation of reversible strains on the pressure head is analyzed theoretically for elastic and viscoelastic liquid flow from convergent channels. It is shown that, depending on the rheological features of the liquids, the pressure head can both increase and decrease as compared with the pressure determined by the Bernoulli formula. In particular, a situation in which a liquid flows out without any pressure head applied (paradox of self-outflow) is possible within the framework of the model. Transition to different viscoelastic liquid flow regimes as a function of the constitutive parameters is considered with reference to a channel with a sharp bottleneck (abrupt decrease in the cross-section).     

Secondary Instability of Large Scale Structure in Free Turbulent Shear Layer

ＳＥＣＯＮＤＡＲＹＩＮＳＴＡＢＩＬＩＴＹＯＦＬＡＲＧＥＳＣＡＬＥＳＴＲＵＣＴＵＲＥＩＮＦＲＥＥＴＵＲＢＵＬＥＮＴＳＨＥＡＲＬＡＹＥＲＺｈａｏＧｅｎｇ－ｆｕ（赵耕夫）（ＴｉａｎｊｉｎＵｎｉｖｅｒｓｉｔｙ，Ｔｉａｎｊｉｎ．３０００７２．Ｐ．Ｒ．Ｃｈｉｎａ...     

Discussion on“Rectangular Plates with Free Edges on Elastic Foundations”

This paper points out the former work does not fulfill the boundaryconditions that the concentrated force at the four corner points should notexist.Therefore,Ritz method adopted by the author concerned in the illus-trative example will not be convergent in the best way.Moreover,Garlerkinmethod which is illustrated in this paper may carry out incorrect results ifwe apply its formulae.We have proved that the boundary conditions whichgovern the concentrated force equal to zero at the four corner points areindispensable if the problem is properly set.     

Controlling the Free‐Surface Profile of Film Flow over Complex Topography

A twodimensional model describing nonisothermal viscous thin film flow over complex topography is considered. The model is based on the Navier–Stokes equations in the Oberbeck–Boussinesq approximation. A numerical analysis of the effect of thermal loading on the location of the film free surface is performed. It is shown that changing the substrate temperature function, it is possible to control the freesurface profile on separate topographical features. The results of solution of model problems are presented).     

Discussion on “Rectangular Plates with Free Edges on Elastic Foundations”

This paper points out the former work does not fulfill the boundary conditions that the concentrated force at the four corner points should not exist. Therefore, Ritz method adopted by the author concerned in the illustrative example will not be convergent in the best way. Moreover, Garlerkin method which is illustrated in this paper may carry out incorrect results if we apply its formulae. We have proved that the boundary conditions which govern the concentrated force equal to zero at the four corner points are indispensable if the problem is properly set.     

A Reaction–Diffusion–Advection Equation with Mixed and Free Boundary Conditions

We investigate a reaction–diffusion–advection equation of the form \(u_t-u_{xx}+\beta u_x=f(u)\) \((t>0,\,0<x<h(t))\) with mixed boundary condition at \(x=0\) and Stefan free boundary condition at \(x=h(t)\). Such a model may be applied to describe the dynamical process of a new or invasive species adopting a combination of random movement and advection upward or downward along the resource gradient, with the free boundary representing the expanding front. The goal of this paper is to understand the effect of advection environment and no flux across the left boundary on the dynamics of this species. For the case \(|\beta |<c_0\), we first derive the spreading–vanishing dichotomy and sharp threshold for spreading and vanishing, and then provide a much sharper estimate for the spreading speed of h(t) and the uniform convergence of u(t, x) when spreading happens. For the case \(|\beta |\ge c_0\), some results concerning virtual spreading, vanishing and virtual vanishing are obtained. Here \(c_0\) is the minimal speed of traveling waves of the differential equation.     

Free convection of nanofluid filled enclosure using lattice Boltzmann method （LBM）

The lattice Boltzmann method (LBM) is used to examine free convection of nanofluids. The space between the cold outer square and heated inner circular cylinders is filled with water including various kinds of nanoparticles: TiO₂, Ag, Cu, and Al₂O₃. The Brinkman and Maxwell-Garnetts models are used to simulate the viscosity and the effective thermal conductivity of nanofluids, respectively. Results from the performed numerical analysis show good agreement with those obtained from other numerical methods. A variety of the Rayleigh number, the nanoparticle volume fraction, and the aspect ratio are examined. According to the results, choosing copper as the nanoparticle leads to obtaining the highest enhancement for this problem. The results also indicate that the maximum value of enhancement occurs at λ = 2.5 when Ra = 10⁶ while at λ = 1.5 for other Rayleigh numbers.     

Longitudinal Free Vibration of Inhomogeneous Elastic Straight Strut with a Variable Cross Section

The scope of the present article,motivated by the case of the composite wooden propeller of anairplane,is to deal tentatively with the longitudinal free vibrationproblem of an elastic straight bar witha more general mathematical treatment.In this analysis,we have assigned to the modulus of elasticity,the bar cross section as well as themass per unit length of the bar an exponential function variation,and then found a general solution,wherein three parameters were considered as the main factors to affect the longitudinal free vibrationof the inhomogeneous elastic straight bar with a variable cross section.     

Vibration of Shallow Shells with Two Adjacent Edges Clamped and the Others Free∗

ABSTRACT

Considerable information is available in the published literature on the free vibration frequencies and mode shapes of rectangular flat plates having two adjacent edges clamped and the other two free. However, no results appear to have been published previously for shallow shells having such edge conditions. The present work uses the Ritz method with displacement components in the form of algebraic polynomials to obtain accurate frequencies. Frequencies are determined for the first eight modes of shallow shells having spherical, cylindrical, and hyperbolic paraboloi-dal curvatures and square planforms. Beginning with the plate, the curvatures are incrementally increased in each case to the limits of shallow     

Fokker–Planck Equations for a Free Energy Functional or Markov Process on a Graph

The classical Fokker–Planck equation is a linear parabolic equation which describes the time evolution of the probability distribution of a stochastic process defined on a Euclidean space. Corresponding to a stochastic process, there often exists a free energy functional which is defined on the space of probability distributions and is a linear combination of a potential and an entropy. In recent years, it has been shown that the Fokker–Planck equation is the gradient flow of the free energy functional defined on the Riemannian manifold of probability distributions whose inner product is generated by a 2-Wasserstein distance. In this paper, we consider analogous matters for a free energy functional or Markov process defined on a graph with a finite number of vertices and edges. If N ≧ 2 is the number of vertices of the graph, we show that the corresponding Fokker–Planck equation is a system of N nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. However, in contrast to stochastic processes defined on Euclidean spaces, the situation is more subtle for discrete spaces. We have different choices for inner products on the space of probability distributions resulting in different Fokker–Planck equations for the same process. It is shown that there is a strong connection but there are also substantial discrepancies between the systems of ordinary differential equations and the classical Fokker–Planck equation on Euclidean spaces. Furthermore, both systems of ordinary differential equations are gradient flows for the same free energy functional defined on the Riemannian manifolds of probability distributions with different metrics. Some examples are also discussed.     

On a Degenerate Free Boundary Problem and Continuous Subsonic–Sonic Flows in a Convergent Nozzle

This paper concerns the well-posedness of a boundary value problem for a quasilinear second order elliptic equation which is degenerate on a free boundary. Such problems arise when studying continuous subsonic–sonic flows in a convergent nozzle with straight solid walls. It is shown that for a given inlet being a perturbation of an arc centered at the vertex of the nozzle and a given incoming mass flux belonging to an open interval depending only on the adiabatic exponent and the length of the arc, there is a unique continuous subsonic–sonic flow from the given inlet with the angle of the velocity orthogonal to the inlet and the given incoming mass flux. Furthermore, the sonic curve of this continuous subsonic–sonic flow is a free boundary, where the flow is singular in the sense that while the speed is C ^1/2 Hölder continuous at the sonic state, the acceleration blows up at the sonic state.     

Estimate of the Latent Free Energy and Damage at the Tip of an Opening–Mode Crack

A method of estimating the latent elastic energy associated with the microinhomogeneity of the stress and plastic–strain fields inside the plastic zone localized near the tip of an opening–mode crack (Dugdale zone) under conditions of plane stresses is proposed. The microinhomogeneity of plastic flow upon small strain hardening is taken into account only in the form of considerable distortion of the geometry of the free surfaces of the plastic zone. The damage that developes because of release of the latent free energy is estimated depending on the magnitude of the crack opening.     

On the Propagation of Long‐Wave Perturbations in a Two‐Layer Free‐Boundary Rotational Fluid

A mathematical model for the propagation of longwave perturbations in a freeboundary shear flow of an ideal stratified twolayer fluid is considered. The characteristic equation defining the velocity of perturbation propagation in the fluid is obtained and studied. The necessary hyperbolicity conditions for the equations of motion are formulated for flows with a monotonic velocity profile over depth, and the characteristic form of the system is calculated. It is shown that the problem of deriving the sufficient hyperbolicity conditions is equivalent to solving a system of singular integral equations. The limiting cases of weak and strong stratification are studied. For these models, the necessary and sufficient hyperbolicity conditions are formulated, and the equations of motion are reduced to the Riemann integral invariants conserved along the characteristics.     

The ‘Missing’ Self-Similar Free Convection Boundary-Layer Flow Over a Vertical Permeable Surface in a Porous Medium

The free convection boundary-layer flow of a Darcy–Boussinesq fluid from a vertical permeable plate with an inverse-linear temperature distribution is considered. The outstanding characteristics of this self-similar flow which, according to the usual reduction procedure of pseudo-similarity to full similarity, should not exist at all, are analyzed in detail. Thus it is shown that this flow only exists if a lateral suction with a sufficiently large suction parameter _min = 1.079131 is applied. For the threshold value _min the solution is unique but above it multiple solutions are encountered for every given value of .     

An Element Free Galerkin analysis of steady dynamic growth of a mode i crack in elastic–plastic materials

An Element Free Galerkin (EFG) method based formulation for steady dynamic crack growth in elastic–plastic materials is developed. A domain convecting parallel to the steadily moving crack tip is employed. The EFG methodology eliminates the stringent mesh requirements of the Finite Element Method (FEM) for such problems. Both rate-independent materials and rate-dependent materials are considered. The material is characterized by von Mises yielding condition and an associated flow rule. For rate-independent materials, both the influence of crack speeds and that of strain hardening on the mechanics of steady dynamic crack growth are investigated. For rate-dependent materials, only a non-hardening material is considered with emphasis on determining the influence of viscous properties of materials and crack speeds. The influence of strain hardening on steady dynamic crack growth shows the same trends as for steady quasi-static crack growth. The simplifications used in the literature in deriving analytical solutions for high strain-rate crack growth have been examined thoroughly using the numerical results.     

Effect of Free‐Stream Turbulence on the Flow Structure near a Wedge and the Windward Side of an Airfoil

The flow structure behind wire grids is studied for flows with a low subsonic velocity, and the effect of grids on the boundarylayer flow structure is considered. It is shown that the meanvelocity inhomogeneity induced by the grid does not disappear until a distance of 925 calibers downstream of the grid is reached. Liquidcrystal thermography combined with hotwire measurements made it possible to find the source of steady largescale streamwise vortex structures in the boundary layer on a wedge and on an airfoil and to determine the parameters of these structures.