The “resultant bifurcation diagram” method and its application to bifurcation behaviors of a symmetric railway bogie system

The concept of symmetric bifurcation for a symmetric wheel-rail system is defined. After that, the time response of the system can be achieved by the numerical integration method, and an unfixed and dynamic Poincaré section and its symmetric section for the symmetric wheel-rail system are established. Then the ??resultant bifurcation diagram?? method is constructed. The method is used to study the symmetric/asymmetric bifurcation behaviors and chaotic motions of a two-axle railway bogie running on an ideal straight and perfect track, and a variety of characteristics and dynamic processes can be obtained in the results. It is indicated that, for the possible sub-critical Hopf bifurcation in the railway bogie system, the stable stationary solutions and the stable periodic solutions coexist. When the speed is in the speed range of Hopf bifurcation point and saddle-node bifurcation point, the coexistence of multiple solutions can cause the oscillating amplitude change for different kinds of disturbance. Furthermore, it is found that there are symmetric motions for lower speeds, and then the system passes to the asymmetric ones for wide ranges of the speed, and returns again to the symmetric motions with narrow speed ranges. The rule of symmetry breaking in the system is through a blue sky catastrophe in the beginning.     

Unsteady and transient flow of compressible fluids in pipelines—a review of theoretical and some experimental studies

The mathematical modelling of highly compressible unsteady flows has been of interest for some years. In order to obtain tractable solutions of the governing equations, investigators have made various simplifying assumptions such as presuming isothermal or isentropic flow of ideal gases, etc. The present review, with dense phase gas tranmission systems of particular interest, briefly develops the basic equations without such assumptions and includes the effects of wall friction and heat transfer. After re-expressing the equations in terms of the measurable quantities of pressure, temperature and velocity, previously published work is reviewed for their solution. Relevant experimental work is somewhat limited but contributions from 20 references are included.     

Flow of oil–water emulsions through a constricted capillary

The flow of oil-in-water emulsions through quartz micro-capillary tubes was analyzed experimentally. The capillaries were used as models of connecting pore-throats between adjacent pore body pairs in high-permeability media. Pressure drop between the inlet and outlet ends of the capillary was recorded as a function of time, for several values of the volumetric flow rate. Several distinct emulsions were prepared using synthetic oils in deionized water, stabilized by a surfactant (Triton X-100). Two oils of different viscosity values were used to prepare the emulsions, while two distinct drop size distributions were obtained by varying the mixing procedure. The average oil drop size varied from smaller to larger than the neck radius. The results are presented in terms of the extra-pressure drop due to the presence of the dispersed phase, i.e. the difference between the measured pressure drop and the one necessary to drive the continuous phase alone at the same flow rate. For emulsions with drops smaller than the capillary throat diameter, the extra-pressure drop does not vary with capillary number and it is a function of the viscosity ratio, dispersed phase concentration and drop size distribution. For emulsions with drops larger than the constriction, the large oil drops may partially block the capillary, leading to a high extra pressure difference at low capillary numbers. Changes in the local fluid mobility by means of pore-throat blockage may help to explain the additional oil recovery observed in laboratory experiments and the sparse data on field trials.     

The Response of a Rayleigh–Liénard Oscillator to a Fundamental Resonance

A Rayleigh–Liénard oscillator excited by a fundamentalresonance is investigated by using an asymptotic perturbation method based on Fourier expansion and time rescaling. Two first-order nonlinear ordinarydifferential equations governing the modulation of the amplitude andthe phase of solutions are derived. These equations are used todetermine steady-state responses and their stability. Excitationamplitude-response and frequency-response curves are shown and checkedby numerical integration. Dulac's criterion, the Poincaré–Bendixsontheorem, and energy considerations are used in order to study the existenceand characteristics of limit cycles of the two modulation equations. Alimit cycle corresponds to a modulated motion for the Rayleigh–Liénardoscillator. For small excitation amplitude, the analytical results arein excellent agreement with the numerical solutions. In certain caseswhen the excitation amplitude is very low, an approximate analyticsolution corresponding to a modulated motion can be obtained andnumerically checked. Moreover, if the excitation amplitude is increased,an infinite-period bifurcation occurs because the modulation periodlengthens and becomes infinite, while the modulation amplitude remainsfinite and suddenly the attractor settles down into a periodic motion.     

The nature of “slow” rarefaction waves accompanying the escape of a boiling liquid

The depressurization of a high-pressure container in the form of a tube closed at both ends and initially filled with water not heated to the boiling point is studied. At the zero time, one of the ends of the tube is opened, and the liquid begins to escape into the atmosphere. Since the atmospheric pressure is less than the saturation pressure of the liquid, its escape is accompanied by boiling. It is known from experiments [1, 2] that after depressurization a rarefaction wave travels into the channel with the speed of sound in the pure liquid. After it has passed, a two-phase mixture is formed in the container. A slow (or secondary) rarefaction wave (Fig. 1a) moves through this mixture with a velocity relative to the tube walls of order 10 m/sec, transforming the two-phase mixture to the equilibrium state. To explain the features of the escape process, we propose a new mathematical model of a boiling liquid that takes into account two mechanisms of vapor formation —boiling at nucleating centers present in the liquid and fragmentation of the formed bubbles. If the second mechanism is to be realized, a certain relationship must be established between the bubble size and the difference of the velocities of the phases. The slow rarefaction wave is described by means of the proposed model.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 61–66, May–June, 1991.     

Exact solution of the capillary diffusion equation for a three‐component mixture

The paper gives an exact solution of the steady system of equations for stable threecomponent diffusion in the entire range of concentrations for a long capillary under a controlled capillary pressure differential. The solution allows one to calculate the distributions of component concentrations and mixture density along the capillary. It is shown that if the diffusion coefficients are markedly different, an extremum of mixture density can arise inside the capillary. In particular, if the density of the mixture in the upper flask is higher than that in the lower flask and the stratification of the system is generally stable, a region with a reverse density gradient that is unstable against gravity convection can appear inside the capillary. A comparison with experimental results shows that the resistance to gravity convection is disturbed when an extremum of mixture density arises in the channel during steady diffusion.     

Comparative investigations of surface instabilities (“sharkskin”) of a linear and a long-chain branched polyethylene

An experimental study of the physical origin and the mechanisms of the sharkskin instability is presented. Extrusion flows through a slit die are studied for two materials: a linear low density polyethylene (LLDPE) which exhibits sharkskin instability for flow rates larger than an onset value and a low density polyethylene (LDPE) which does not show any instability over a broad range of flow rates. By combining laser-Doppler velocimetry (LDV) with rheological measurements in both uniaxial extension and shear, the distributions of tensile and shear stresses in extrusion flows are measured for both materials. The experimentally measured flow fields appear to be qualitatively similar for both the unstable (LLDPE) and stable case (LDPE): around the die exit the flow accelerates near the boundaries and decelerates around the flow axis. The fields of the axial gradients of the axial velocity component are, however, quite different in the two cases. In the unstable case there exists a strongly non-uniform transversal distribution of velocity gradients near the die exit. This non-uniformity of the distribution of gradients is significantly smaller in the stable case. Significant differences in the extensional rheological properties of the two materials are found as well. Due to its branched structure, the LDPE is able to sustain higher tensile stresses prior to failure. Measurements of the distributions of tensile stresses around the die exit reveal a stress boundary layer and a stress imbalance between the boundaries and the bulk. The magnitude of the stress imbalance exceeds the melt strength in the experiments with the LLDPE which causes the failure of the material in the superficial layers and results in the emergence of the sharkskin instability. In the experiments with the LDPE, the magnitude of the stress imbalance remains smaller than the melt strength which explains the lack of an instability. The measured shear stresses around the die exit are significantly smaller than the tensile stresses, suggesting that the shear component of the flow plays no significant role in the emergence of the sharkskin instability.     

Addendum to “Concentration and Lack of Observability of Waves in Highly Heterogeneous Media”

In [1] we introduced a class of 1?d wave equations with rapidly oscillating Hölder continuous coefficients for which the classical boundary observability property fails. We also established that these examples could be used to contradict Strichartz-type inequalities for the wave equation with low regularity coefficients. The object of this addendum is to further analyze this issue. As we will see, the argument in [1] only provides sharp counter-examples to the Strichartz estimates when the coefficient ρ belongs to L ^∞. We carefully analyze these counter-examples for Hölder continuous coefficients. We also give a new application of our construction which shows that some eigenfunction estimates for elliptic operators due to Sogge can fail when coefficients are not smooth enough.     

Creeping motion of a sphere in tubes filled with Herschel–Bulkley fluids

Previous numerical simulations for the flow of Bingham plastics past a sphere contained in cylindrical tubes of different diameter ratios are extended to Herschel–Bulkley fluids with the purpose of comparing them with experiments. The emphasis is on determining the extent and shape of yielded/unyielded regions along with the drag coefficient as a function of the pertinent dimensionless groups. Good overall agreement is obtained between the numerical results and the experimental studies.     

On the stability of the simple shear flow of a Johnson–Segalman fluid

We solve the time-dependent simple shear flow of a Johnson–Segalman fluid with added Newtonian viscosity. We focus on the case where the steady-state shear stress/shear rate curve is not monotonic. We show that, in addition to the standard smooth linear solution for the velocity, there exists, in a certain range of the velocity of the moving plate, an uncountable infinity of steady-state solutions in which the velocity is piecewise linear, the shear stress is constant and the other stress components are characterized by jump discontinuities. The stability of the steady-state solutions is investigated numerically. In agreement with linear stability analysis, it is shown that steady-state solutions are unstable only if the slope of a linear velocity segment is in the negative-slope regime of the shear stress/shear rate curve. The time-dependent solutions are always bounded and converge to a stable steady state. The number of the discontinuity points and the final value of the shear stress depend on the initial perturbation. No regimes of self-sustained oscillations have been found.     

“Everything flows?”: elastic effects on startup flows of yield-stress fluids

It is now 30 years since Barnes and Walters published a provocative paper in which they asserted that the yield stress is an experimental artifact. We now know that the situation is far more complicated than understood at the time, and that the mechanics of the solid material prior to yielding must be considered carefully. In this paper, we examine the response of a well-studied “simple” yield-stress material, namely a Carbopol gel that exhibits no thixotropy, and demonstrate the significance of the pre-yielding behavior through a number of elementary measurements.     

Application of Takagi–Sugeno fuzzy model to a class of chaotic synchronization and anti-synchronization

In this study, we investigate a class of chaotic synchronization and anti-synchronization with stochastic parameters. A controller is composed of a compensation controller and a fuzzy controller which is designed based on fractional stability theory. Three typical examples, including the synchronization between an integer-order Chen system and a fractional-order Lü system, the anti-synchronization of different 4D fractional-order hyperchaotic systems with non-identical orders, and the synchronization between a 3D integer-order chaotic system and a 4D fractional-order hyperchaos system, are presented to illustrate the effectiveness of the controller. The numerical simulation results and theoretical analysis both demonstrate the effectiveness of the proposed approach. Overall, this study presents new insights concerning the concepts of synchronization and anti-synchronization, synchronization and control, the relationship of fractional and integer order nonlinear systems.     

The Richtmyer–Meshkov instability of a 'V' shaped air/helium interface subjected to a weak shock

The Richtmyer–Meshkov instability of a ‘V' shaped air/helium gaseous interface subjected to a weak shock wave is experimentally studied.A soap film technique is adopted to create a ‘V' shaped interface with accurate initial conditions.Five kinds of ‘V' shaped interfaces with different vertex angles are formed to highlight the effects of initial conditions on the flow characteristics.The results show that a spike is generated after the shock impact,and grows constantly with time.As the vertex angle increases,vortices generated on the interface become less noticeable,and the spike develops less pronouncedly.The linear growth rate of interface width after compression phase is estimated by a linear model and a revised linear model,and the latter is proven to be more effective for the interface with high initial amplitudes.The linear growth rate of interface width is,for the first time in a heavy/light interface configuration,found to be a non-monotonous function of the initial perturbation amplitude–wavelength ratio.     

The “Butterfly Effect” in a Porous Slab

It is shown that the governing equation for the stream function of the Darcy free convection boundary layer flows past a vertical surface is invariant under arbitrary translations of the transverse coordinate y. The consequences of this basic symmetry property on the solutions corresponding to a prescribed surface temperature distribution T _w (x) are investigated. It is found that starting with a “primary solution” which describes the temperature boundary layer on an impermeable surface, infinitely many “translated solutions” can be generated which form a continuous group, the “translation group” of the given primary solution. The elements of this group describe free convection boundary layer flows from permeable counterparts of the original surface with a transformed temperature distribution [(T)\tilde]_w ( x ){\tilde {T}_w \left( x \right)}, when simultaneously a suitable lateral suction/injection of the fluid is applied. It turns out in this way that several exact solutions discovered during the latter few decades are in fact not basically new solutions, but translated counterparts of some formerly reported primary solutions. A few specific examples are discussed in detail.     

Experimental study of liquid–vapor phase distribution in a single tube capillary evaporator

An experimental setup was developed to study the vaporization of a liquid inducing capillary pumping. The study was performed on a single square cross-section micro-channel of 500 μm internal edge, heated only on the upper and lower walls to allow visualization through the 2 other faces. The images were obtained by a shadowgraph technique using a digital camera. A program for processing the images was developed to allow the determination of several quantities characterizing the flow, supplemented by information provided by the rest of the instrumentation equipping the experimental device. The results highlight the unstable nature of the phase distribution. Despite these instabilities, films of liquid spread in an averaged manner over a length substantially equal to the length of the zone where the heat flux was applied, independently of the intensity of the heat flux.     

A simple approach to detect the nonlocal effects in the static analysis of Euler–Bernoulli and Timoshenko beams

In this paper, the well-known Mohr analogy is applied to the computation of displacements and rotations of carbon nanotubes, and some simple formula is derived which allows the direct generalization of the Mohr theory to the nonlocal Euler–Bernoulli and Timoshenko beam theories. Finally, some examples show the effectiveness and simplicity of the proposed approach.