On Extension and Torsion of Strain-Stiffening Rubber-Like Elastic Circular Cylinders

This paper is concerned with investigation of the effects of strain-stiffening on the response of solid circular cylinders in the combined deformation of torsion superimposed on axial extension. The cylinders are composed of incompressible isotropic nonlinearly elastic materials. Our primary focus is on materials that undergo severe strain-stiffening in the stress-stretch response. In particular, we consider two particular phenomenological constitutive models for such materials that reflect limiting chain extensibility at the molecular level. The axial stretch γ and twist that can be sustained in cylinders composed of such materials are shown to be constrained in a coupled fashion. It is shown that, in the absence of an additional axial force, a transition value γ=γ _t of the axial stretch exists such that for γ<γ _t , the stretched cylinder tends to elongate on twisting whereas for γ>γ _t , the stretched cylinder tends to shorten on twisting. These results are in sharp contrast with those for classical models such as the Mooney-Rivlin (and neo-Hookean) models that predict that the stretched circular cylinder always tends to further elongate on twisting. We also obtain results for materials modeled by the well-known exponential strain-energy widely used in biomechanics applications. This model reflects a strain-stiffening that is less abrupt than that for the limiting chain extensibility models. Surprisingly, it turns out that the results in this case are somewhat more complicated. For a fixed stiffening parameter, provided that the stretch is sufficiently small, the stretched bar always tends to elongate on twisting in the absence of an additional axial force. However, for sufficiently large stretch, the cylinder tends to shorten on undergoing sufficiently small twist but then tends to elongate on further twisting. These results are of interest in view of the widespread use of exponential models in the context of the mechanics of soft biological tissues. The special case of pure torsion is also briefly considered. In this case, the resultant axial force required to maintain pure torsion is compressive for all the models discussed here. In the absence of such a force, the bar would elongate on twisting reflecting the celebrated Poynting effect.      

On the behavior of solutions of systems of linear functional differential equations with constant coefficients and linearly transformed argument in the neighborhood of a singular point

We establish new properties of C ¹(0, +∞)-solutions of systems of linear functional differential equations x′(t) = Ax(t) + Bx(qt) + Cx′(qt) in the neighborhood of the singular point t = 0.     

Mixed Convection Boundary Layer Flow from a Horizontal Circular Cylinder Embedded in a Porous Medium Filled with a Nanofluid

Steady mixed convection boundary layer flow from an isothermal horizontal circular cylinder embedded in a porous medium filled with a nanofluid has been studied for both cases of a heated and cooled cylinder. The resulting system of nonlinear partial differential equations is solved numerically using an implicit finite-difference scheme. The solutions for the flow and heat transfer characteristics are evaluated numerically for various values of the governing parameters, namely the nanoparticle volume fraction φ and the mixed convection parameter λ. Three different types of nanoparticles are considered, namely Cu, Al₂O₃ and TiO₂. It is found that for each particular nanoparticle, as the nanoparticle volume fraction φ increases, the magnitude of the skin friction coefficient decreases, and this leads to an increase in the value of the mixed convection parameter λ which first produces no separation. On the other hand, it is also found that of all the three types of nanoparticles considered, for any fixed values of φ and λ, the nanoparticle Cu gives the largest values of the skin friction coefficient followed by TiO₂ and Al₂O₃. Finally, it is worth mentioning that heating the cylinder (λ > 0) delays separation of the boundary layer and if the cylinder is hot enough (large values of λ > 0), then it is suppressed completely. On the other hand, cooling the cylinder (λ < 0) brings the boundary layer separation point nearer to the lower stagnation point and for a sufficiently cold cylinder (large values of λ < 0) there will not be a boundary layer on the cylinder.     

Minimizing Sequences Selected via Singular Perturbations, and their Pattern Formation

We study the Cahn-Hilliard energy E _ɛ(u) over the unit square under the constraint of a constant mass m with (ɛ > 0) and without ɛ= 0) interfacial energy. Minimizers of E ₀(u) have no preferred pattern and we select patterns via sequences of conditionally critical points of E _ɛ(u) converging to minimizers as ɛ tends to zero. Those critical points are not minimizers if the singular limit has no minimal interface. We obtain them by a global bifurcation analysis of the Euler-Lagrange equations for E _ɛ(u) where the mass m is the bifurcation parameter. We make use of the symmetry of the unit square, and the elliptic maximum principle, in turn, implies that the location of maxima and minima is fixed for all solutions on global branches. This property is used to guarantee the existence of a singular limit and to verify the Weierstrass-Erdmann corner condition which proves its minimizing property. Accepted January 21, 2000?Published online November 24, 2000     

The application of Moire interferometry in the measurement of displacement field and strain field at notch-tip and crack-tip

This paper presents the application of Moire interferometry in measuring the displacement and strain field at notch-tip and crack-tip before and after crack propagation. The experiment is carried out using a three point bending beam with a notch. TheN _x andN _y fringe patterns representing displacement field, and the ΔN _x/Δx and ΔN _γ/Δγ fringe patterns representing the strain field are obtained. The sensitivity of the measured displacement is 0.417 μm per fringe order. The displacement and strain distribution along the sectionx=0 have been worked out according toN _x andN _γ fringe patterns. The project supported by Chinese Academy of Sciences and National Natural Science Foundation of China     

Shock wave triple-point morphology

The existence and characteristics of shock wave triple points are examined. The analysis utilizes a single flow plane for the three shocks and is local to the triple point. It applies when the flow is unsteady, three-dimensional, and the upstream flow is nonuniform. Under more restrictive conditions, a relation is also derived for the ratio of the curvature of the Mach stem to that of the reflected shock. For given values of the ratio of specific heats, γ, and the upstream Mach number, M ₁, a solution window is established. A parametric set of solutions is generated within the window for γ = 1, 1.4, and 5/3 and for 16 values of M ₁ ranging from solution onset to M ₁ = 6.A solution can be one of three types, these stem from the velocity tangency condition along the slip stream. Topics are addressed such as solution multiplicity, shock wave and slip stream orientation, the nature of the reflected wave (weak, strong, inverted, normal), the nature of the Mach stem (weak, strong, normal), and differences due to changes in γ and M ₁.     

Momentum and heat transfer from an asymmetrically confined circular cylinder in a plane channel

Unsteady momentum and heat transfer from an asymmetrically confined circular cylinder in a plane channel is numerically investigated using FLUENT for the ranges of Reynolds numbers as 10≤Re≤500, of the blockage ratio as 0.1≤β≤0.4, and of the gap ratio as 0.125≤γ≤1 for a constant value of the Prandtl number of 0.744. The transition of the flow from steady to unsteady (characterized by critical Re) is determined as a function of γ and β. The effect of γ on the mean drag and lift coefficients, Strouhal number (St), and Nusselt number (Nu _w ) is studied. Critical Re was found to increase with decreasing γ for all values of β. and St were found to increase with decreasing values of γ for fixed β and Re. The effect of decrease in γ on was found to be negligible for all blockage ratios investigated.     

Phase Transition with the Line‐Tension Effect

We make the connection between the geometric model for capillarity with line tension and the Cahn‐Hilliard model of two‐phase fluids. To this aim we consider the energies where u is a scalar density function and W and V are double‐well potentials. We show that the behaviour of F _ε in the limit ε→0 and λ→∞ depends on the limit of ε log λ. If this limit is finite and strictly positive, then the singular limit of the energies F _ε leads to a coupled problem of bulk and surface phase transitions, and under certain assumptions agrees with the relaxation of the capillary energy with line tension. These results were announced in [ABS1] and [ABS2]. (Accepted November 5, 1997)     

A Collocation-Based Algorithm for Computing the Pull-In Range of a Class of Phase-Locked Loops

The pull-in range (ω_p) of a phase-locked loop (PLL) is defined as the maximum value of loop detuning ω_0s for which pull-in occurs from anywhere on the PLL's phase plane. That is, pull-in is guaranteed from anywhere on the phase plane if ω_0s < ω_p. Simple approximation is available for computing ω_p for the high gain PLL where saddle-node bifurcation occurs at ω_0s = ω_p. Unlike the high gain case, a simple approximation for ω_p is not available for the low gain case where bifurcation from a separatrix cycle occurs at ω_0s = ω_p. The vector field model for a class of second-order PLLs is shown to have rotational properties, which imply the existence of a separatrix cycle. The external stability of this separatrix cycle is an indicator of the type of bifurcation (saddle-node or separatrix cycle) which terminates the limit cycle associated with the PLL's stable false lock state and the PLL pulls-in (i.e. achieve phase lock). A formula is given for determining the separatrix cycle's stability, which indicates that these paratrix cycle is externally stable for small values of closed loop gain. A collocation-based algorithm is presented for computing the PLL's separatrix cycle and the value of pull-in range frequency ω_0s = ω_p at which a stable separatrix cycle exists.     

Qualitative Behavior of Local Minimizers of Singular Perturbed Variational Problems

We consider a non-convex variational problem (P) and the corresponding singular perturbed problem (P _ε ). The qualitative behavior of stable critical points of (P _ε ) depending on ε and a lower order term is discussed and we prove compactness of a sequence of stable critical points as ε ↘ 0. Moreover we show whether this limit is the global minimizer of (P). Furthermore uniform convergence is considered as well as the convergence rate depending on ε.      

Synchronized analysis of an unsteady laminar flow downstream of a circular cylinder centred between two parallel walls using PIV and mass transfer probes

Experiments are carried out in the wake of a cylinder of d _c  = 10 mm diameter placed symmetrically between two parallel walls with a blockage ratio r = 1/3 and a Reynolds number varying between 75 ≤ Re ≤ 277. Particle image velocimetry is exerted to obtain the instantaneous velocity components in the cylinder wake. A snapshot proper orthogonal decomposition (POD) is also applied to these PIV results in order to extract the dominant modes through the implementation of an inhomogeneous filtering of these different snapshots, apart from an interpolation to estimate the wall shear rate at the lower wall downstream the cylinder. Mass transfer circular probes are placed at the lower wall downstream this obstacle so as to further determine the time evolution of the wall shear rate, by bringing the inverse method to bear on the convective-diffusion equation. Comparisons between the two synchronized techniques demonstrate that electrochemical method can give more accurate information about the coherent structures present in the flow and about the interaction of the von Kármán vortices with the walls of the tunnel as well. The comparison between the two measurement techniques in the flow regions concerns the spatiotemporal evolutions of the wall shear rate obtained from PIV measurements and the wall shear rate using mass transfer probes. Discrepancy between the PIV measurements and the electrochemical ones near the wall, where the secondary vortices P ₁′ are generated at wall, are caused by a PIV bias and a limitations of the singular mass transfer probes.     

ROSSBY WAVES WITH THE CHANGE OF β

In this paper, the change of the Rossby parameter β with latitude is considered and the parameter γ≡-dβ/dy=2sinφ/a² is introduced and the β-plane approximation is extended into f=f₀+β₀y-γ₀y²/2 which includes the parameter γ. Such approximation closes further to practice especially in the high latitude regions.We give emphasis to the research of the effect of the parameter γ on the Rossby waves. It is seen that the effect of the parameter γ is remarkable in the high latitude regions. It can produce the Rossby waves caused by the pure parameter γ. And the phase speed formula of Rossby waves with the change of ft is generally given, which is degenerated into the well-known Rossby formula when γ₀=0. The researches also point out that when the change of β is regarded, even if the basic current u is a linear function of y the unstable modes can also take place. However,the parameter γ usually plays a stable part in the Rossby waves and it does affect the longitudinal scale and the structure of constant phase lines(trough-ridge lines)of Rossby waves and slow down the growing or decaying of Rossby waves.     

Simple shear in nonlinear Cosserat elasticity: bifurcation and induced microstructure

We discuss the simple shear problem for a geometrically exact Cosserat model. In contrast to linear Cosserat elasticity, where the unique solution is available in closed form we exhibit a multitude of solutions to the nonlinear problem, even if the two fields of deformations φ and microrotations remain homogeneous. This motivates a search for new conditions on the microrotations which single out a unique, physically reasonable, response. The influence of material parameters, notably the Cosserat couple modulus μ _c and the internal length scale L _c on the response is also studied. For small Cosserat couple modulus μ _c  > 0 we observe a pitchfork bifurcation of the homogeneous response and for vanishing internal length L _c  = 0 and zero Cosserat couple modulus μ _c  = 0 the Cosserat model may show highly oscillating “microstructure” solutions which are energetically better than the homogeneous response. Thus, the large scale nonlinear Cosserat limit is not necessarily a classical limit.      

On the asymptotic properties of solutions of linear functional differential equations with linearly transformed argument

We establish new properties of solutions of the functional differential equation {fx153-01} in the neighborhood of the singular point t = +∞. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 147–150, April–June, 2008.     

Fully developed flow of a viscoelastic film down a vertical cylindrical or planar wall

The one-dimensional, gravity-driven film flow of a linear (l) or exponential (e) Phan-Thien and Tanner (PTT) liquid, flowing either on the outer or on the inner surface of a vertical cylinder or over a planar wall, is analyzed. Numerical solution of the governing equations is generally possible. Analytical solutions are derived only for: (1) l-PTT model in cylindrical and planar geometries in the absence of solvent, b o [(h)\tilde]_s/([(h)\tilde]_s +[(h)\tilde]_p)=0\beta\equiv {\tilde{\eta}_s}/\left({\tilde{\eta}_s +\tilde{\eta}_p}\right)=0, where [(h)\tilde]_p\widetilde{\eta}_p and [(h)\tilde]_s\widetilde{\eta}_s are the zero-shear polymer and solvent viscosities, respectively, and the affinity parameter set at ξ = 0; (2) l-PTT or e-PTT model in a planar geometry when β = 0 and x 1 0\xi \ne 0; (3) e-PTT model in planar geometry when β = 0 and ξ = 0. The effect of fluid properties, cylinder radius, [(R)\tilde]\tilde{R}, and flow rate on the velocity profile, the stress components, and the film thickness, [(H)\tilde]\tilde{H}, is determined. On the other hand, the relevant dimensionless numbers, which are the Deborah, De=[(l)\tilde][(U)\tilde]/[(H)\tilde]De={\tilde{\lambda}\tilde{U}}/{\tilde{H}}, and Stokes, St=[(r)\tilde][(g)\tilde][(H)\tilde]²/([(h)\tilde]_p +[(h)\tilde]_s )[(U)\tilde]St=\tilde{\rho}\tilde{g}\tilde{\rm H}^{2}/\left({\tilde{\eta}_p +\tilde{\eta}_s} \right)\tilde{U}, numbers, depend on [(H)\tilde]\tilde{H} and the average film velocity, [(U)\tilde]\widetilde{U}. This makes necessary a trial and error procedure to obtain [(H)\tilde]\tilde{H} a posteriori. We find that increasing De, ξ, or the extensibility parameter ε increases shear thinning resulting in a smaller St. The Stokes number decreases as [(R)\tilde]/[(H)\tilde]{\tilde{R}}/{\tilde{H}} decreases down to zero for a film on the outer cylindrical surface, while it asymptotes to very large values when [(R)\tilde]/[(H)\tilde]{\tilde{R}}/{\tilde{H}} decreases down to unity for a film on the inner surface. When x 1 0\xi \ne 0, an upper limit in De exists above which a solution cannot be computed. This critical value increases with ε and decreases with ξ.     

Laminarisation and re-transition of a turbulent boundary layer subjected to favourable pressure gradient

 Experimental results are reported for the response of an initially turbulent boundary layer (Re _θ≈1700) to a favourable pressure gradient with a peak value of K≡(−υ/ρU ³ _E ) dp/dx equal to 4.4×10^-6. In the near-wall region of the boundary layer (y/δ<0.1) the turbulence intensity u′ scales roughly with the free-stream velocity U _E until close to the location where K is a maximum whereas in the outer region u′ remains essentially frozen. Once the pressure gradient is relaxed, the turbulence level increases throughout the boundary layer until K falls to zero when the near wall u′ levels show a significant decrease. The intermittency γ is the clearest indicator of a fundamental change in the turbulence structure: once K exceeds 3×10^-6, the value of γ in the immediate vicinity of the wall γ _s falls rapidly from unity, reaches zero at the location where K again falls below 3×10^-6 and then rises back to unity. Although γ is practically zero throughout the boundary layer in the vicinity of γ _s =0, the turbulence level remains high. The explanation for what appears to be a contradiction is that the turbulent frequencies are too low to induce turbulent mixing. The mean velocity profile changes shape abruptly where K exceeds 3×10^-6. Values for the skin friction coefficient, based upon hot-film measurements, peak at the same location as K and fall to a minimum close to the location where K drops back to zero. Received: 28 January 1998/Accepted: 8 April 1998     

A Sharp-Interface Limit for a Two-Well Problem in Geometrically Linear Elasticity

In the theory of solid-solid phase transitions the deformation of an elastic body is determined via a functional containing a nonconvex energy density and a singular perturbation. We study Frame indifference, within a linearized setting, requires that W depends only on the symmetric part of ∇u. The potential W is nonnegative and vanishes on two wells, i.e., for d = 2, on two lines in the space of matrices. We determine, for d = 2, the Gamma limit I₀ = Γ− lim _ɛ→0 I_ɛ. The limit I₀[u] is finite only for deformations u that fulfill W(∇u)=0 almost everywhere and have sharp interfaces where ∇u has jumps. For these u, I₀[u] equals the line integral over the interfaces of a surface energy density.     

Positive Solutions of Boundary Value Problems for Second-Order Singular Nonlinear Differential Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒ,ｗｅｓｈａｌｌｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｓｉｎｇｕｌａｒｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｓ (ＢＶＰ)ｕ″ ｇ(ｔ)ｆ(ｕ) =0 ,　　 0 <ｔ<1 ,αｕ(0 ) -βｕ′(0 ) =0 ,　　γｕ(1 ) δｕ′(1 ) =0 ,(1 )ｗｈｅｒｅα ,β,γ ,δ≥ 0 ,ρ:=βγ αγ αδ>0 ,ｆ∈Ｃ([0 ,∞ ) ,[0 ,∞ ) ) ,ｇｍａｙｂｅｓｉｎｇｕｌａｒａｔｔ=0ａｎｄ/ｏｒｔ=1 .Ｔｈｉｓｐｒｏｂｌｅｍａｒｉｓｅｓｎａｔｕｒａｌｌｙｉｎｔｈｅｓｔｕｄｙｏｆｒａｄｉａｌｌｙｓｙｍｍｅｔ…     

Scherk-Type Capillary Graphs

This paper concerns the regularity of a capillary graph (the meniscus profile of liquid in a cylindrical tube) over a corner domain of angle α. By giving an explicit construction of minimal surface solutions previously shown to exist (Indiana Univ. Math. J. 50 (2001), no. 1, 411–441) we clarify two outstanding questions. Solutions are constructed in the case α = π/2 for contact angle data (γ₁, γ₂) = (γ, π − γ) with 0 < γ < π. The solutions given with |γ − π/2| < π/4 are the first known solutions that are not C² up to the corner. This shows that the best known regularity (C^{1, ∈}) is the best possible in some cases. Specific dependence of the H?lder exponent on the contact angle for our examples is given. Solutions with γ = π/4 have continuous, but horizontal, normal vector at the corners in accordance with results of Tam (Pacific J. Math. 124 (1986), 469–482). It is shown that our examples are C^{0, β} up to and including the corner for any β < 1. Solutions with |γ − π/2| > π/4 have a jump discontinuity at the corner. This kind of behavior was suggested by numerical work of Concus and Finn (Microgravity sci. technol. VII/2 (1994), 152–155) and Mittelmann and Zhu (Microgravity sci. technol. IX/1 (1996), 22–27). Our explicit construction, however, allows us to investigate the solutions quantitatively. For example, the trace of these solutions, excluding the jump discontinuity, is C^2/3.     

Recoverable strains and retardation times of monodisperse suspensions of silicon dioxide spheres in poly(dimethylsiloxane)

Steady-state viscosities η, steady-state recoverable strains γ _rs and characteristic retardation time τ _1/2 were measured for suspensions of monodisperse silicon dioxide (SiO₂) spheres in poly(dimethylsiloxane) (PDMS) with various volume fractions Φ of the suspended spheres at various creep stresses σ ₀. Two different regions are found in plots of η/η _m vs γ _rs, where η/η _m denotes the relative viscosity of the suspensions. In one region, η/η _m is proportional to γ _rs, while γ _rs is independent of η/η _m in the other region. In both regions, τ _1/2 is the functions of the shear strain rate in the steady-state of creep test independently of Φ. The origin of the elasticity is related to the ‘maximally distorted’ cages recovered owing to the repulsive interaction between the SiO₂ spheres and recovery of the cages in the shear-induced clusters of the suspended spheres.