Operator method for solving the plane elasticity problem for a strip with periodic cuts

In the present paper, we use the conformal mapping z/c = ζ?2a sin ζ (a, c?const, ζ = u + iv) of the strip {|v| ≤ v ₀, |u| < ∞} onto the domain D, which is a strip with symmetric periodic cuts. For the domain D, in the orthogonal system of isometric coordinates u, v, we solve the plane elasticity problem. We seek the biharmonic function in the form F = C ψ ₀ + S ψ*₀ + x(C ψ ₁ ? S ψ ₂) + y(C ψ ₂ + S ψ ₁), where C(v) and S(v) are the operator functions described in [1] and ψ ₀(u), …, ψ ₂(u) are the desired functions. The boundary conditions for the function F posed for v = ±v ₀ are equivalent to two operator equations for ψ ₁(u) and ψ ₂(u) and to two ordinary differential equations of first order for ψ ₀(u) and ψ*₀(u) [2]. By finding the functions ψ _j (u) in the form of trigonometric series with indeterminate coefficients and by solving the operator equations, we obtain infinite systems of linear equations for the unknown coefficients. We present an efficient method for solving these systems, which is based on studying stable recursive relations. In the present paper, we give an example of analysis of a specific strip (a = 1/4, v ₀ = 1) loaded on the boundary v = v ₀ by a normal load of intensity p. We find the particular solutions corresponding to the extension of the strip by the longitudinal force X ^∞ and to the transverse and pure bending of the strip due to the transverse force Y ^∞ and the constant moment M ^∞, respectively. We also present the graphs of normal and tangential stresses in the transverse cross-section x = 0 and study the stress concentration effect near the cut bottom.     

Effective viscosity of a dilute emulsion of spherical drops containing soluble surfactant

The effective viscosity of a dilute emulsion of spherical drops containing a soluble surfactant is calculated under a linear creeping flow. It is assumed that convection of surfactant is small relative to diffusion, and thus the Peclet number, Pe, is small. We calculate the effective viscosity of the emulsion to \(\mathcal {O}(Pe\phi \mu )\), where ? is the small volume fraction of the dispersed drops and μ is the viscosity of the surfactant-free suspending fluid. This \(\mathcal {O}(Pe\phi \mu )\) contribution is a sensitive function of the bulk and interfacial surfactant transport. Specifically, soluble surfactant molecules diffuse from the bulk to the interface and then adsorb to the interface. The ratio of the time scale for bulk diffusion to the time scale for adsorption to the interface is quantified by a Damkohler number, Da. The adsorption of surfactant to the interface may cause a significant decrease in the bulk concentration, which is known as depletion. The impact of depletion is characterized by two parameters: h, which is a dimensionless depletion depth; and k, which is the ratio of the desorption time scale to the adsorption time scale. We analytically determine how the \(\mathcal {O}(Pe\phi \mu )\) contribution to the effective viscosity depends on h, k, and Da. Surprisingly, for certain regimes in the h ? k ? Da parameter space, we predict the effective viscosity of the emulsion to be greater than Einstein’s result for the viscosity of a suspension of rigid spheres. Large Marangoni stresses driven by convective transport of soluble surfactant molecules are responsible for this result.     

Linearized Stability for Semiflows Generated by a Class of Neutral Equations,with Applications to State-Dependent Delays

We prove a principle of linearized stability for semiflows generated by neutral functional differential equations of the form x′(t) = g(? x _t , x _t ). The state space is a closed subset in a manifold of C ²-functions. Applications include equations with state-dependent delay, as for example x′(t) = a x′(t + d(x(t))) + f (x(t + r(x(t)))) with \({a\in\mathbb{R}, d:\mathbb{R}\to(-h,0), f:\mathbb{R}\to\mathbb{R}, r:\mathbb{R}\to[-h,0]}\).     

Regularity of Solutions to the Navier-Stokes Equations for Compressible Barotropic Flows on a Polygon

The Navier-Stokes system for a steady-state barotropic nonlinear compressible viscous flow, with an inflow boundary condition, is studied on a polygon D. A unique existence for the solution of the system is established. It is shown that the lowest order corner singularity of the nonlinear system is the same as that of the Laplacian in suitable L ^q spaces. Let ω be the interior angle of a vertex P of D. If \(\) and \(\), then the velocity u is split into singular and regular parts near the vertex P. If α < 2 and \(\) or if α > 2 and 2 < q < ∞&;, it is shown that u∈ (H ^{2, q} (D))².     

Mixing and Bimolecular Reaction Kinetics in a Plane Poisseulle Flow

Mixing and a nonlinear bimolecular chemical reaction (reactant A + reactant B → product; reaction rate r?=?κc ₁ c ₂) in laminar shear flow are investigated. It is found that asymptotically the dominant balance between the rates of production and dissipation of the mean-squared concentration fluctuations \((\sigma_{c_1 }^2 ,\sigma_{c_2 }^2)\) and cross-covariance of concentration fluctuations \((\overline {c_1 c_2 })\) occurs under nonreactive and reactive conditions. Longitudinal dispersion of the cross-sectional averages (C ₁, C ₂), and variances and the cross-covariance of reactant concentrations can be asymptotically quantified by the classic Taylor dispersion coefficient (D) even under reactive conditions. The characteristic time-scale (τ) over which molecular diffusion dissipates concentration variance and the cross-covariance of reactant concentrations is also shown to be the same under nonreactive and reactive conditions. A variational estimate of τ is shown to be close to the values inferred from detailed numerical simulation. The production-dissipation balance implies that the cross-sectional averaged reaction rate follows \(\overline r =\kappa_{eff} C_1 C_2 \) and \(\kappa _{eff} \approx \kappa \left[ {1+2D\tau \left( {{\partial \ln C_1 } \mathord{\left/ {\vphantom {{\partial \ln C_1 } {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}} \right)\left( {{\partial \ln C_2 } \mathord{\left/ {\vphantom {{\partial \ln C_2 } {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}} \right)} \right]\). The effective reaction rate parameter (κ _eff ) is higher than that of well-mixed batch test reaction rate constant (κ) for initially overlapping species and κ _eff is smaller than κ for initially non-overlapping species.     

Miscible Thermo-Viscous Fingering Instability in Porous Media. Part 1: Linear Stability Analysis

The development of the thermo-viscous fingering instability of miscible displacements in homogeneous porous media is examined. In this first part of the study dealing with stability analysis, the basic equations and the parameters governing the problem in a rectilinear geometry are developed. An exponential dependence of viscosity on temperature and concentration is represented by two parameters, thermal mobility ratio β _T and a solutal mobility ratio β _C , respectively. Other parameters involved are the Lewis number Le and a thermal-lag coefficient λ. The governing equations are linearized and solved to obtain instability characteristics using either a quasi-steady-state approximation (QSSA) or initial value calculations (IVC). Exact analytical solutions are also obtained for very weakly diffusing systems. Using the QSSA approach, it was found that an increase in thermal mobility ratio β _T is seen to enhance the instability for fixed β _C , Le and λ. For fixed β _C and β _T , a decrease in the thermal-lag coefficient and/or an increase in the Lewis number always decrease the instability. Moreover, strong thermal diffusion at large Le as well as enhanced redistribution of heat between the solid and fluid phases at small λ is seen to alleviate the destabilizing effects of positive β _T . Consequently, the instability gets strictly dominated by the solutal front. The linear stability analysis using IVC approach leads to conclusions similar to the QSSA approach except for the case of large Le and unity λ flow where the instability is seen to get even less pronounced than in the case of a reference isothermal flow of the same β _C , but β _T  = 0. At practically, small value of λ, however, the instability ultimately approaches that due to β _C only.     

Regularity of Optimal Maps on the Sphere: the Quadratic Cost and the Reflector Antenna

Building on the results of Ma et al. (in Arch. Rational Mech. Anal. 177(2), 151–183 (2005)), and of the author Loeper (in Acta Math., to appear), we study two problems of optimal transportation on the sphere: the first corresponds to the cost function d ²(x, y), where d(·, ·) is the Riemannian distance of the round sphere; the second corresponds to the cost function ?log |x ? y|, known as the reflector antenna problem. We show that in both cases, the cost-sectional curvature is uniformly positive, and establish the geometrical properties so that the results of Loeper (in Acta Math., to appear) and Ma et al. (in Arch. Rational Mech. Anal. 177(2), 151–183 (2005)) can apply: global smooth solutions exist for arbitrary smooth positive data and optimal maps are Hölder continuous under weak assumptions on the data.     

Evaluation of the Spectral Element Dynamic Model for Large-Eddy Simulation on Unstructured,Deformed Meshes

In this paper, the Spectral-Element Dynamic Model (SEDM), suited for Large-Eddy Simulation (LES) using Discontinuous Finite Element Methods (DFEM), is assessed using unstructured meshes. Five test cases of increasing complexity are considered, namely, the Taylor-Green vortex at Re =?5000, the turbulent channel flow at Re_τ =?587, the circular cylinder in cross-flow at Re_D =?3900, the square cylinder in cross-flow at Re_D =?22400 and the channel with periodic constrictions at Re_h =?10595. Various discretization parameters such as the grid spacing, polynomial degree and numerical flux are assessed and very accurate results are reported in all cases. This consistency in the results demonstrates the versatility of the SEDM approach and its ability to gage the actual resolution and quality of the mesh and, accordingly, to introduce an amount of sub-grid dissipation which is adapted to the spatial discretization considered.     

Experimental Study on Three-Dimensional Deformation Field of Portevin–Le Chatelier Effect Using Digital Image Correlation

In view of its high precision and high efficiency, three-dimensional digital image correlation (3D-DIC) is widely used to accurately measure full-field deformation. A spatiotemporal experimental study using 3D-DIC to explore the Portevin–Le Chatelier (PLC) deformational behavior, provides a new insight into the whole 3D deformation field, including the out-of-plane displacement, and in particular the relationship between the serrations and the strain field in the deformation bands corresponding to individual serrations. Specimens 1, 2 and 3 mm thick of 5456 Al-based alloy were tested in uniaxial tension at room temperature at strain rates from 1.8 × 10^?4 to 9.1 × 10^?3s^?1. The spatial and temporal characteristics of the strain localization were quantitatively analyzed. The out-of-plane displacement increment field (w) of the localized bands was observed by 3D-DIC, and found to be related to the specimen thickness and the in-plane strain increment. The largest displacement increments were respectively 15, 10 and 5 μm for 3, 2 and 1 mm specimens at maximum strain increment of about 12000 με. The elastic shrinkage outside the deformation bands was found to be an essential characteristic of the PLC effect. The width of the PLC band (w_band) increased with increasing thickness; the angle of the PLC band (??_band) was not affected by either specimen thickness or serration amplitude. Temporally, the serrations in the plots both of in-plane strain and out-of-plane displacement vs. time coincided throughout the entire loading procedure. When PLC banding occurred, the serration amplitude within the bands was found to be proportional to the maximum strain increment in the direction of the applied tensile force (??_max).     

New Periodic Solutions for Newtonian n-Body Problems with Dihedral Group Symmetry and Topological Constraints

In this paper, we prove the existence of a family of new non-collision periodic solutions for the classical Newtonian n-body problems. In our assumption, the \({n=2l \geqq 4}\) particles are invariant under the dihedral rotation group D_l in \({\mathbb{R}^3}\) such that, at each instant, the n particles form two twisted l-regular polygons. Our approach is the variational minimizing method and we show that the minimizers are collision-free by level estimates and local deformations.     

Perspectives on the Phenomenology and Modeling of Boundary Layer Transition

The characteristics of the coherent structures in a strongly decelerated large-velocity-defect boundary layer are analysed by direct numerical simulation. The simulated boundary layer starts as a zero-pressure-gradient boundary layer, decelerates under a strong adverse pressure gradient, and separates near the end of the domain, in the form of a very thin separation bubble. The Reynolds number at separation is R e _?? =3912 and the shape factor H=3.43. The three-dimensional spatial correlations of (u, u) and (u, v) are investigated and compared to those of a zero-pressure-gradient boundary layer and another strongly decelerated boundary layer. These velocity pairs lose coherence in the streamwise and spanwise directions as the velocity defect increases. In the outer region, the shape of the correlations suggest that large-scale u structures are less streamwise elongated and more inclined with respect to the wall in large-defect boundary layers. The three-dimensional properties of sweeps and ejections are characterized for the first time in both the zero-pressure-gradient and adverse-pressure-gradient boundary layers, following the method of Lozano-Durán et al. (J. Fluid Mech. 694, 100–130, [2012]). Although longer sweeps and ejections are found in the zero-pressure-gradient boundary layer, with ejections reaching streamwise lengths of 5 boundary layer thicknesses, the sweeps and ejections tend to be bigger in the adverse-pressure-gradient boundary layer. Moreover, small near-wall sweeps and ejections are much less numerous in the large-defect boundary layer. Large sweeps and ejections that reach the wall region (wall-attached) are also less numerous, less streamwise elongated and they occupy less space than in the zero-pressure-gradient boundary layer.     

Revisiting History Effects in Adverse-Pressure-Gradient Turbulent Boundary Layers

The goal of this study is to present a first step towards establishing criteria aimed at assessing whether a particular adverse-pressure-gradient (APG) turbulent boundary layer (TBL) can be considered well-behaved, i.e., whether it is independent of the inflow conditions and is exempt of numerical or experimental artifacts. To this end, we analyzed several high-quality datasets, including in-house numerical databases of APG TBLs developing over flat-plates and the suction side of a wing section, and five studies available in the literature. Due to the impact of the flow history on the particular state of the boundary layer, we developed three criteria of convergence to well-behaved conditions, to be used depending on the particular case under study. (i) In the first criterion, we develop empirical correlations defining the R e _?? -evolution of the skin-friction coefficient and the shape factor in APG TBLs with constant values of the Clauser pressure-gradient parameter β = 1 and 2 (note that β = δ ^?/τ _w dP _e /dx, where δ ^? is the displacement thickness, τ _w the wall-shear stress and dP _e /dx the streamwise pressure gradient). (ii) In the second one, we propose a predictive method to obtain the skin-friction curve corresponding to an APG TBL subjected to any streamwise evolution of β, based only on data from zero-pressure-gradient TBLs. (iii) The third method relies on the diagnostic-plot concept modified with the shape factor, which scales APG TBLs subjected to a wide range of pressure-gradient conditions. These three criteria allow to ensure the correct flow development of a particular TBL, and thus to separate history and pressure-gradient effects in the analysis.     

LES-based Study of the Roughness Effects on the Wake of a Circular Cylinder from Subcritical to Transcritical Reynolds Numbers

This paper investigates the effects of surface roughness on the flow past a circular cylinder at subcritical to transcritical Reynolds numbers. Large eddy simulations of the flow for sand grain roughness of size k/D = 0.02 are performed (D is the cylinder diameter). Results show that surface roughness triggers the transition to turbulence in the boundary layer at all Reynolds numbers, thus leading to an early separation caused by the increased momentum deficit, especially at transcritical Reynolds numbers. Even at subcritical Reynolds numbers, boundary layer instabilities are triggered in the roughness sublayer and eventually lead to the transition to turbulence. The early separation at transcritical Reynolds numbers leads to a wake topology similar to that of the subcritical regime, resulting in an increased drag coefficient and lower Strouhal number. Turbulent statistics in the wake are also affected by roughness; the Reynolds stresses are larger due to the increased turbulent kinetic energy production in the boundary layer and separated shear layers close to the cylinder shoulders.     

Consequences of the Translation Invariance on the Darcy Free Convection Flow Past a Vertical Surface

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 \({\tilde {T}_w \left( x \right)}\), 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.     

On the formation of hidden chaotic attractors and nested invariant tori in the Sprott A system

We consider the well-known Sprott A system, which depends on a single real parameter a and, for \(a=1\), was shown to present a hidden chaotic attractor. We study the formation of hidden chaotic attractors as well as the formation of nested invariant tori in this system, performing a bifurcation analysis by varying the parameter a. We prove that, for \(a=0\), the Sprott A system has a line of equilibria in the z-axis, the phase space is foliated by concentric invariant spheres with two equilibrium points located at the south and north poles, and each one of these spheres is filled by heteroclinic orbits of south pole–north pole type. For \(a\ne 0\), the spheres are no longer invariant algebraic surfaces and the heteroclinic orbits are destroyed. We do a detailed numerical study for \(a>0\) small, showing that small nested invariant tori and a limit set, which encompasses these tori and is the \(\alpha \)- and \(\omega \)-limit set of almost all orbits in the phase space, are formed in a neighborhood of the origin. As the parameter a increases, this limit set evolves into a hidden chaotic attractor, which coexists with the nested invariant tori. In particular, we find hidden chaotic attractors for \(a<1\). Furthermore, we make a global analysis of Sprott A system, including the dynamics at infinity via the Poincaré compactification, showing that for \(a>0\), the only orbit which escapes to infinity is the one contained in the z-axis and all other orbits are either homoclinic to a limit set (or to a hidden chaotic attractor, depending on the value of a), or contained on an invariant torus, depending on the initial condition considered.     

An iterative updating method for undamped structural systems

Finite element model updating is a procedure to minimize the differences between analytical and experimental results and can be mathematically reduced to solving the following problem. Problem P: Let M _a ∈SR ^n×n and K _a ∈SR ^n×n be the analytical mass and stiffness matrices and Λ=diag{λ ₁,…,λ _p }∈R ^p×p and X=[x ₁,…,x _p ]∈R ^n×p be the measured eigenvalue and eigenvector matrices, respectively. Find \((\hat{M}, \hat{K}) \in \mathcal{S}_{MK}\) such that \(\| \hat{M}-M_{a} \|^{2}+\| \hat{K}-K_{a}\|^{2}= \min_{(M,K) \in {\mathcal{S}}_{MK}} (\| M-M_{a} \|^{2}+\|K-K_{a}\|^{2})\), where \(\mathcal{S}_{MK}=\{(M,K)| X^{T}MX=I_{p}, MX \varLambda=K X \}\) and ∥?∥ is the Frobenius norm. This paper presents an iterative method to solve Problem P. By the method, the optimal approximation solution \((\hat{M}, \hat{K})\) of Problem P can be obtained within finite iteration steps in the absence of roundoff errors by choosing a special kind of initial matrix pair. A numerical example shows that the introduced iterative algorithm is quite efficient.     

SBV Regularity for Hamilton–Jacobi Equations in $${{\mathbb R}^n}$$

In this paper we study the regularity of viscosity solutions to the following Hamilton–Jacobi equations

$\partial_{t}u+H(D_{x}u)=0\quad\hbox{in }\Omega\subset{\mathbb R}\times{\mathbb R}^{n}.$

In particular, under the assumption that the Hamiltonian \({H\in C^2({\mathbb R}^n)}\) is uniformly convex, we prove that D _x u and ? _t u belong to the class SBV _loc (Ω).

     

Potentials for tangent tensor fields on spheroids

In three-dimensional Euclidean space let S be a closed simply connected, smooth surface (spheroid). Let \(\hat n\) be the outward unit normal to S, ▽ _S the surface gradient on S, I _S the metric tensor on S, g_ij the four covariant components of I _S (i,j = 1, 2), h _ij the four covariant components of -\(\hat n\)xI _S , and D _i covariant differentiation on S. It is well known that for any tangent vector field u on S there exist scalars ? and ψ on S, unique to within additive constants, such that \(u = \nabla _s \varphi - \hat n \times \nabla _s \psi \); the covariant components of u are \(u_i = D_i \varphi + h_i^j D_j \psi \). This theorem is very useful in the study of vector fields in spherical coordinates. The present paper gives an analogous theorem for real second-order tangent tensor fields F on S: for any such F there exist scalar fields H, L, M, N such that the covariant components of F are

$$F_{ij} = H h{}_{ij} + Lg_{ij} + E_{ij} (M,N),$$     

Effect of vibration on the creep of tensile-test specimens of the aluminum alloy Al−Mg−Si (PA-4)

In this work are presented the results of research on vibrational creep at normal temperature of an aluminum alloy containing magnesium and silicon (designated PA-4). A uni-directional positive load is applied to a tensile-test specimen, such that the stress intensity in the specimen is of the type\(\sigma (t) = \sigma _m (1 + Asin\omega t)\) whereσ _m is the mean (static) stress intensity andA =σ _a /σ _m is the ratio of the vibratory-stress intensity to the mean intensity. The results are given in the form of families of curves of plastic (i.e., permanent) deformation for various values ofA, namely,A=0.0000, 0.0065, 0.05500, 0.1000 and 0.2000. Taking the creep limit for plastic strain as ε _p = 1.8 percent, equations for this creep limit were deduced from experimental data. The following conclusions are drawn from these investigations:

1. 1.
   Vibrations of small-stress-amplitude ratioA encourage creep, particularly with more lengthy tests.     
   
   

Wall slip and multi-tier yielding in capillary suspensions

Impact of wall slip on the yield stress measurement is examined for capillary suspensions consisting of cocoa powder as the dispersed phase, vegetable oil as the continuous primary fluid, and water as the secondary fluid using smooth and serrated parallel plates. Using dynamic oscillatory measurements, we investigated the yielding behavior of this ternary solid-fluid-fluid system with varying particle volume fraction, ?, from 0.45 to 0.65 and varying water volume fraction, ?_w, from 0.02 to 0.08. Yield stress is defined as the maximum in the elastic stress (G^′γ), which is obtained by plotting the product of elastic modulus (G^′) and strain amplitude (γ) as a function of applied strain amplitude. With serrated plates, which offer minimal slippage, capillary suspensions with ? ≥?0.45 and a fixed ?_w =?0.06 showed a two-step yielding behavior as indicated by two peaks in the plots of elastic stress as a function of strain amplitude. On the other hand with smooth plates, the capillary suspensions showed strong evidence of wall slip as evident by the presence of three distinct peaks and lowered first yield stresses for all ? and ?_w. These results can be interpreted based on the fact that a particle-depleted layer, which is known to be responsible for slip, is present in the vicinity of the smooth surfaces. The slip layer presents itself as an additional “pseudo-microstructure” (characteristic length scale) besides the two microstructures, aqueous bridges and solid particle agglomerates, that may occur in the system. With serrated plates, both the yield stresses (σ₁σ₂) and storage moduli plateau at lower strain (before the first yield point) and at higher strain (before the second yield point) (G\(^{\prime }_{p1}\), G\(^{\prime }_{p2}\)) were found to increase with ? (at a fixed ?_w =?0.06) following power-law dependences. Similarly with increasing ?_w (0.02 – 0.08) at a fixed ? =?0.62, the system behaved as a solid-like material in a jammed state with particles strongly held together as manifested by rapidly increasing σ₁ and σ₂. The usage of smooth surfaces primarily affected σ₁ which was reflected by an approximately 70–90% decrement in the measured σ₁ for all values of ?. By contrast, σ₂ and G\(^{\prime }_{p2}\) were found to be unaffected as shown by close agreement of values obtained using serrated geometry due to vanishing slip layers at higher strain amplitudes.