Determination of axial forces during the capillary breakup of liquid filaments – the tilted CaBER method

The capillary breakup extensional rheometry (CaBER) is a versatile method to characterize the elongational behavior of low-viscosity fluids. Commonly, data evaluation is based on the assumption of zero normal stress in axial direction ( $\upsigma_{\rm zz}=0$ ). In this paper, we present a simple method to determine the axial force using a CaBER device rotated by 90° and analyzing the deflection of the filament due to gravity. Forces in the range of 0.1–1,000?μN could be assessed. Our study includes experimental investigations of Newtonian fructose solutions and silicon oil mixtures (viscosity range, 0.9–60?Pa s) and weakly viscoelastic polyethylene oxide (PEO, $M_{\rm w}=10^{6}$ ?g/mol) solutions covering a concentration range from c?≈?c* (critical overlap concentration) up to c?>?c _e (entanglement concentration). Papageorgiou’s solution for the stress ratio $\upsigma_{\rm zz}/\upsigma_{\rm rr}$ in Newtonian fluids during capillary thinning is experimentally confirmed, but the widely accepted assumption of vanishing axial stress in weakly viscoelastic fluids is not fulfilled for PEO solutions, if c _e is exceeded.     

Experimental study on the capillary thinning of entangled polymer solutions

The transient elongation behavior of entangled polymer and wormlike micelles (WLM) solutions has been investigated using capillary breakup extensional rheometry (CaBER). The transient force ratio X = 0.713 reveals the existence of an intermediate Newtonian thinning region for polystyrene and WLM solutions prior to the viscoelastic thinning. The exponential decay of X(t) in the first period of thinning defines an elongational relaxation time λ _x which is equal to elongational relaxation time λ _e obtained from exponential diameter decay D(t) indicating that the initial stress decay is controlled by the same molecular relaxation process as the strain hardening observed in the terminal regime of filament thinning. Deviations in true and apparent elongational viscosity are discussed in terms of X(t). A minimum Trouton ratio is observed which decreases exponentially with increasing polymer concentration leveling off at Tr_min = 3 for the solutions exhibiting intermediate Newtonian thinning and Tr_min ≈ 10 otherwise. The relaxation time ratio λ _e/ λ _s, where λ _s is the terminal shear relaxation time, decreases exponentially with increasing polymer concentration and the data for all investigated solutions collapse onto a master curve irrespective of polymer molecular weight or solvent viscosity when plotted versus the reduced concentration c[ η], with [ η] being the intrinsic viscosity. This confirms the strong effect of the nonlinear deformation in CaBER experiments on entangled polymer solutions as suggested earlier. On the other hand, λ _e ≈λ _s is found for all WLM solutions clearly indicating that these nonlinear deformations do not affect the capillary thinning process of these living polymer systems.     

Extensional rheology of concentrated emulsions as probed by capillary breakup elongational rheometry (CaBER)

Elongational flow behavior of w/o emulsions has been investigated using a capillary breakup elongational rheometer (CaBER) equipped with an advanced image processing system allowing for precise assessment of the full filament shape. The transient neck diameter D(t), time evolution of the neck curvature κ(t), the region of deformation l _def and the filament lifetime t _c are extracted in order to characterize non-uniform filament thinning. Effects of disperse volume fraction ?, droplet size d _sv , and continuous phase viscosity η _c on the flow properties have been investigated. At a critical volume fraction ? _c , strong shear thinning, and an apparent shear yield stress τ _y,s occur and shear flow curves are well described by a Herschel–Bulkley model. In CaBER filaments exhibit sharp necking and t _c as well as κ _max ?=?κ (t?=?t _c ) increase, whereas l _def decreases drastically with increasing ?. For ? <?? _c , D(t) data can be described by a power-law model based on a cylindrical filament approximation using the exponent n and consistency index k from shear experiments. For ??≥?? _c , D(t) data are fitted using a one-dimensional Herschel–Bulkley approach, but k and τ _y,s progressively deviate from shear results as ? increases. We attribute this to the failure of the cylindrical filament assumption. Filament lifetime is proportional to η _c at all ?. Above ? _c, κ _max as well as t _c /η _c scale linearly with τ _y,s . The Laplace pressure at the critical stretch ratio ε _c which is needed to induce capillary thinning can be identified as the elongational yield stress τ _y,e , if the experimental parameters are chosen such that the axial curvature of the filament profile can be neglected. This is a unique and robust method to determine this quantity for soft matter with τ _y ?< 1,000 Pa. For the emulsion series investigated here a ratio τ _y,e /τ _y,s = 2.8 ± 0.4 is found independent of ?. This result is captured by a generalized Herschel–Bulkley model including the third invariant of the strain-rate tensor proposed here for the first time, which implies that τ _y,e and τ _y,s are independent material parameters.     

Asymptotic behavior of Toeplitz matrices and determinants

We consider the inverse X _N and determinant D_N(c) of an N×N Toeplitz matrix C_N=[c_i?j] ₀ ^N?1 as N ar∞. Under the condition that there exists a monotonic decreasing summable bound b _n ≧|c _n |+|c _?n |, and that the generating function \(c(\theta ) = \sum\limits_{n = - \infty }^\infty {c_n e^{i{\text{ }}n{\text{ }}\theta } }\) does not vanish, we construct a matrix iterative process which yields (i) explicit asymptotic formulae for the elements of X_N when v(c) = (2π)^?1 [arg{c(2π)}?arg{c(0)}] is zero. Thence we obtain (ii) expressions for the constants, and bounds on the remainder, in the asymptotic formula $$\ln D_N (c) = N{\text{ }}k_0 (c) + E_0 (c) + E_{1,N} (c) + \mathcal{R}_N (c),$$ and (iii) the extension of this formula to the case of general integral v(c). Under certain further conditions the monotonicity of E_1,N+?_N is proved. We discuss various identities for D_N which apply when c(θ) is a rational function of e^iθ and mention a conjecture for D _N when c(θ) has zeros, and is discontinuous with arbitrary v(c).     

Mixed Convection Boundary-Layer Flow Along a Vertical Cylinder Embedded in a Porous Medium Filled by a Nanofluid

The steady mixed convection boundary-layer flow on a vertical circular cylinder embedded in a porous medium filled by a nanofluid is studied for both cases of a heated and a cooled cylinder. The governing system of partial differential equations is reduced to ordinary differential equations by assuming that the surface temperature of the cylinder and the velocity of the external (inviscid) flow vary linearly with the axial distance x measured from the leading edge. Solutions of the resulting ordinary differential equations for the flow and heat transfer characteristics are evaluated numerically for various values of the governing parameters, namely the nanoparticle volume fraction ${\phi}$ , the mixed convection or buoyancy parameter ?? and the curvature parameter ??. Results are presented for the specific case of copper nanoparticles. A critical value ?? _c of ?? with ?? _c <?0 is found, with the values of | ?? _c| increasing as the curvature parameter ?? or nanoparticle volume fraction ${\phi}$ is increased. Dual solutions are seen for all values of ?? >??? _c for both aiding, ?? >?0 and opposing, ?? <?0, flows. Asymptotic solutions are also determined for both the free convection limit ${(\lambda \gg 1)}$ and for large curvature parameter ${(\gamma \gg 1)}$ .     

Rheological modeling of the diffusion process and the interphase of symmetrical bilayers based on PVDF and PMMA with varying molecular weights

The diffusion process in the molten state at a polymer/polymer interface of symmetrical and model bilayers has been investigated using a small-amplitude oscillatory shear measurement. The polymers employed in this study were poly (vinylidene fluoride) (PVDF) and poly (methyl methacrylate) (PMMA) of varying molecular weights and polydispersities. The measurements were conducted in the linear viscoelastic regime (small deformations) so as to decouple the effect of flow from the diffusion. The focus of this paper has been to investigate the effects of healing time, angular frequency (ω), temperature, and molecular weight on the inter-diffusion and the triggered interphase between the neighboring layers. The kinetics of diffusion, based on the evolution of the apparent diffusion coefficient (D _a) versus the healing time, was experimentally obtained. The transition from the non-Fickian to the normal Fickian region for the inter-diffusion at the interface was clearly observed, qualitatively consistent with the reptation model, but it occurred at a critical time greater than the reptation time (τ _rep). In non-Fickian region, effects of frequency and temperature were studied with regard to the ratio of the apparent diffusion coefficient to the self-diffusion coefficient (D _a/D _s). The D _s determined in the Fickian region was found to be consistent with Graessley’s model as well as with the literatures. And the dependence of the D_s on the frequency agreed well with the Doi–Edwards theory, in particular, scaling as $D_{\rm s} \sim \omega^{1/2}$ at ω?>?1/τ _e and $D_{\rm s} \sim \omega^{0}$ at ω?<?1/τ _rep. Our experimental results also confirmed that the dependence of the D _s on the temperature for PMMA and PVDF can be well described by the Arrhenius law. Moreover, blends of PMMAs have been proposed in order to be able to change the $\overline M_\emph{w} $ . The rheological investigations of these corresponding bilayers rendered it possible to monitor the effect of $\overline M_\emph{w} $ on the diffusion process. The obtained results gave $D_{\rm s} \sim \overline M_\emph{w}^{-1}$ , thus corroborating some earlier studies and some experimental results recently reported by Time-Resolved Neutron Reflectivity Measurements. Lastly, the thickness of the interphase and its corresponding viscoelastic properties could be theoretically determined as a function of the healing?time.     

The effect of step-stretch parameters on capillary breakup extensional rheology (CaBER) measurements

Extensional rheometry has only recently been developed into a commercially available tool with the introduction of the capillary breakup extensional rheometer (CaBER). CaBER is currently being used to measure the transient extensional viscosity evolution of mid to low-viscosity viscoelastic fluids. The elegance of capillary breakup extensional experiments lies in the simplicity of the procedure. An initial step-stretch is applied to generate a fluid filament. What follows is a self-driven uniaxial extensional flow in which surface tension is balanced by the extensional stresses resulting from the capillary thinning of the liquid bridge. In this paper, we describe the results from a series of experiments in which the step-stretch parameters of final length, and the extension rate of the stretch were varied and their effects on the measured extensional viscosity and extensional relaxation time were recorded. To focus on the parameter effects, well-characterized surfactant wormlike micelle solutions, polymer solutions, and immiscible polymer blends were used to include a range of characteristic relaxation times and morphologies. Our experimental results demonstrate a strong dependence of extensional rheology on step-stretch conditions for both wormlike micelle solutions and immiscible polymer blends. Both the extensional viscosity and extensional relaxation time of the wormlike micelle solutions were found to decrease with increasing extension rate and strain of the step-stretch. For the case of the immiscible polymer blends, fast step-stretches were found to result in droplet deformation and an overshoot in the extensional viscosity which increased with increasing strain rates. Conversely, the polymer solutions tested were found to be insensitive to step-stretch parameters. In addition, numerical simulations were performed using the appropriate constitutive models to assist in both the interpretation of the CaBER results and the optimization of the experimental protocol. From our results, it is clear that any rheological results obtained using the CaBER technique must be properly considered in the context of the stretch parameters and the effects that preconditioning has on viscoelastic fluids.     

Temperature solutions due to time-dependent moving-line-heat sources

A closed-form model for the computation of temperature distribution in an infinitely extended isotropic body with a time-dependent moving-line-heat sources is discussed. The temperature solutions are presented for the sources of the forms: (i) $\dot Q_1 (t) = \dot Q_0 \exp ( - \lambda t)$ , (ii) $\dot Q_2 (t) = \dot Q_0 (t/t^ \star )\exp ( - \lambda t)$ , and $\dot Q_3 (t) = \dot Q_0 [1 + a\cos (\omega t)]$ , whereλ andω are real parameters andt? characterizes the limiting time. The reduced (or dimensionless) temperature solutions are presented in terms of the generalized representation of an incomplete gamma function Γ (α,x;b) and its decompositionsC _Γ andS _Γ. It is also demonstrated that the present analysis covers the classical temperature solution of a constant strength source under quasi-steady-state situations.     

On Near-Wall Treatment in (U)RANS-Based Closure Models

The present work is concerned with computational evaluation of a recently formulated near-wall relationship providing the value of the dissipation rate ε of the kinetic energy of turbulence k through its exact dependence on the Taylor microscale λ: ε = 10νk/λ ², (Jakirli? and Jovanovi?, J. Fluid Mech. 656:530–539, 2010). Dissipation rate determination benefits from the asymptotic behavior of the Taylor microscale resulting in its linear variation in terms of the wall distance (λ?∝?y) being valid throughout entire viscous sublayer. Accordingly, it can be applied as a unified near-wall treatment in all computational frameworks relying on a RANS-based model of turbulence (including also hybrid LES/RANS schemes) independent of modeling level—both main modeling concepts eddy-viscosity and Reynolds stress models can be employed. Presently, the feasibility of the proposed formulation was demonstrated by applying a conventional near-wall second-moment closure model based on the homogeneous dissipation rate ε _h ( ${\varepsilon_h =\varepsilon -0.5\partial \left( {{\nu \partial k}/ {\partial x_j }} \right)} / {\partial x_j }$ ; Jakirli? and Hanjali?, J. Fluid Mech. 539:139–166, 2002) and its instability-sensitive version, modeled in terms of the inverse turbulent time scale ω _h (ω _h ?=?ε _h /k; Maduta and Jakirli?, 2011), to a fully-developed channel flow with both flat walls and periodic hill-shaped constrictions mounted on the bottom wall in a Reynolds number range. The latter configuration is subjected to boundary layer separation from a continuous curved wall. The influence of the near-wall resolution lowering with respect to the location of the wall-closest computational node, coarsened even up to the viscous sublayer edge situated at $y_P^+ \approx 5$ in equilibrium flows, is analyzed. The results obtained follow closely those pertinent to the conventional near-wall integration with the wall-next node positioned at $y_P^+ \le 0.5$ .     

On the Symmetry of Energy-Minimising Deformations in Nonlinear Elasticity II: Compressible Materials

Consider a homogeneous, isotropic, hyperelastic body occupying the region ${A = \{{\bf x}\in\mathbb{R}^{n}\, : \,a <\,|{\bf x} |\,< b \}}$ in its reference state and subject to radially symmetric displacement, or mixed displacement/traction, boundary conditions. In Part I (Sivaloganathan and Spector in Arch Ration Mech Anal, 2009, in press) the authors restricted their attention to incompressible materials. For a large-class of polyconvex constitutive relations that grow sufficiently rapidly at infinity it was shown that to each nonradial isochoric deformation of A there corresponds a radial isochoric deformation that has strictly less elastic energy than the given deformation. In this paper that analysis is further developed and extended to the compressible case. The key ingredient is a new radial-symmetrisation procedure that is appropriate for problems where the symmetrised mapping must be one-to-one in order to prevent interpenetration of matter. For the pure displacement boundary-value problem, the radial symmetrisation of an orientation-preserving diffeomorphism u : A → A* between spherical shells A and A* is the deformation $${\bf u}^{\rm rad}({\bf x})=\frac{r(R)}{R}{\bf x}, \quad R=|{\bf x}|,\qquad\qquad\qquad\qquad(0.1)$$ that maps each sphere ${S_R\subset\,A}$ , of radius R > 0, centred at the origin into another such sphere ${S_r={\bf u}^{\rm rad}(S_R)\subset\,A^*}$ that encloses the same volume as u(S _R ). Since the volumes enclosed by the surfaces u(S _R ) and u ^rad (S _R ) are equal, the classical isoperimetric inequality implies that ${{{\rm Area}( {\bf u}^{\rm rad} (S_R))\leqq {\rm Area}({\bf u} (S_R))}}$ . The equality of the enclosed volumes together with this reduction in surface area is then shown to give rise to a reduction in total energy for many of the constitutive relations used in nonlinear elasticity. These results are also extended to classes of Sobolev deformations and applied to prove that the radially symmetric solutions to these boundary-value problems are local or global energy minimisers in various classes of (possibly nonsymmetric) deformations of a thick spherical shell.     

Statistical Analysis of Cross Scalar Dissipation Rate Transport in Turbulent Partially Premixed Flames: A Direct Numerical Simulation Study

Statistically planar turbulent partially premixed flames for different initial intensities of decaying turbulence have been simulated for global equivalence ratios <????> = 0.7 and <????> = 1.0 using three-dimensional simplified chemistry based Direct Numerical Simulations (DNS). The simulation parameters are chosen such that the combustion situation belongs to the thin reaction zones regime and a random bi-modal distribution of equivalence ratio ?? is introduced in the unburned gas ahead of the flame to account for mixture inhomogeneity. The DNS data has been used to analyse the statistical behaviour of the transport of the cross-scalar dissipation rate based on the fuel mass fraction Y _F and the mixture fraction ?? fluctuations $\,\tilde{\varepsilon}_{Y\xi}={\overline{\rho D\nabla Y_{F}^{\prime\prime}.\nabla \xi^{\prime\prime}} } \big/ {\bar {\rho }}$ (where $\bar{q}$ , $\tilde{q}={\overline{\rho q} } \big/ {\bar {\rho }}$ and $q^{\prime\prime} =q-\tilde {q}$ are Reynolds average, Favre mean and Favre fluctuation of a general quantity q) in the context of Reynolds Averaged Navier?CStokes simulations where ?? is the gas density and D is the gas diffusivity. The statistical behaviours of the unclosed terms in the $\tilde{\varepsilon }_{Y\xi } $ transport equation originating from turbulent transport T ₁, density variation T ₂, scalar?Cturbulence interaction T ₃, chemical reaction rate T ₄ and the molecular dissipation rate D ₂ have been analysed in detail. It has been observed that the contributions of T ₂, T ₃, T ₄ and D ₂ play important roles in the $\tilde{\varepsilon }_{Y\xi } $ transport for the globally stoichiometric cases, but in the globally fuel-lean cases the contributions of T ₂ and T ₄ become relatively weaker in comparison to the contributions of T ₃ and D ₂. The term T ₁ remains small in comparison to the leading order contributions of T ₃ and D ₂ for all cases, but the contribution of T ₁ plays a more important role in the low Damköhler combustion cases. The term T ₂ behaves as a sink term towards the unburned gas side but becomes a source term towards the burned gas side. The scalar?Cturbulence interaction term T ₃ has been found to be generally positive throughout the flame brush, but in globally stoichiometric cases the contribution of T ₃ becomes negative in regions of intense heat release. The combined contribution of (T ₄ ?C D ₂) remains mostly as a sink in all cases studied here. Models are proposed for the unclosed terms of the $\tilde{\varepsilon }_{Y\xi } $ transport equation in the context of Reynolds Averaged Navier?CStokes simulations, which are shown to satisfactorily predict the corresponding quantities extracted from the DNS data for all cases.     

An Updated Portrait of Transition to Turbulence in Laminar Pipe Flows with Periodic Time Dependence (A Correlation Study)

Transition to turbulence in axially symmetrical laminar pipe flows with periodic time dependence classified as pure oscillating and pulsatile (pulsating) ones is the concern of the paper. The current state of art on the transitional characteristics of pulsatile and oscillating pipe flows is introduced with a particular attention to the utilized terminology and methodology. Transition from laminar to turbulent regime is usually described by the presence of the disturbed flow with small amplitude perturbations followed by the growth of turbulent bursts. The visual treatment of velocity waveforms is therefore a preferred inspection method. The observation of turbulent bursts first in the decelerating phase and covering the whole cycle of oscillation are used to define the critical states of the start and end of transition, respectively. A correlation study referring to the available experimental data of the literature particularly at the start of transition are presented in terms of the governing periodic flow parameters. In this respect critical oscillating and time averaged Reynolds numbers at the start of transition; Re _os,crit and Re _ta,crit are expressed as a major function of Womersley number, $\sqrt {\omega ^\prime } $ defined as dimensionless frequency of oscillation, f. The correlation study indicates that in oscillating flows, an increase in Re _os,crit with increasing magnitudes of $\sqrt {\omega ^\prime } $ is observed in the covered range of $1<\sqrt {\omega ^\prime } <72$ . The proposed equation (Eq. 7), ${\rm{Re}}_{os,crit} ={\rm{Re}}_{os,crit} \left( {\sqrt {\omega ^\prime } } \right)$ , can be utilized to estimate the critical magnitude of $\sqrt {\omega ^\prime }$ at the start of transition with an accuracy of ±12?% in the range of $\sqrt {\omega ^\prime } <41$ . However in pulsatile flows, the influence of $\sqrt {\omega ^\prime }$ on Re _ta,crit seems to be different in the ranges of $\sqrt {\omega ^\prime } <8$ and $\sqrt {\omega ^\prime } >8$ . Furthermore there is rather insufficient experimental data in pulsatile flows considering interactive influences of $\sqrt {\omega ^\prime } $ and velocity amplitude ratio, A ₁. For the purpose, the measurements conducted at the start of transition of a laminar sinusoidal pulsatile pipe flow test case covering the range of 0.21<?A ₁?<0.95 with $\sqrt {\omega ^\prime } <8$ are evaluated. In conformity with the literature, the start of transition corresponds to the observation of first turbulent bursts in the decelerating phase of oscillation. The measured data indicate that increase in $\sqrt {\omega ^\prime } $ is associated with an increase in Re _ta,crit up to $\sqrt {\omega ^\prime } =3.85$ while a decrease in Re _ta,crit is observed with an increase in $\sqrt {\omega ^\prime } $ for $\sqrt {{\omega }'} >3.85$ . Eventually updated portrait is pointing out the need for further measurements on i) the end of transition both in oscillating and pulsatile flows with the ranges of $\sqrt {\omega ^\prime } <8$ and $\sqrt {\omega ^\prime } >8$ , and ii) the interactive influences of $\sqrt {\omega ^\prime } $ and A ₁ on Re _ta,crit in pulsatile flows with the range of $\sqrt {\omega ^\prime } >8$ .     

A-Priori Direct Numerical Simulation Modelling of Co-variance Transport in Turbulent Stratified Flames

Three-dimensional Direct Numerical Simulations of statistically planar turbulent stratified flames at global equivalence ratios <???>?=?0.7 and <???>?=?1.0 have been carried out to analyse the statistical behaviour of the transport of co-variance of the fuel mass fraction Y _F and mixture fraction ξ (i.e. $\widetilde{Y_F^{\prime\prime} \xi ^{\prime\prime}}={\overline {\rho Y_F^{\prime\prime} \xi^{\prime\prime}} } \Big/ {\overline \rho })$ for Reynolds Averaged Navier Stokes simulations where $\overline q $ , $\tilde{q} ={\overline {\rho q} } \big/ {\overline \rho }$ and $q^{\prime\prime}= q-\tilde{q}$ are Reynolds averaged, Favre mean and Favre fluctuation of a general quantity q with ρ being the gas density and the overbar suggesting a Reynolds averaging operation. It has been found that existing algebraic expressions may not capture the statistical behaviour of $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ with sufficient accuracy in low Damköhler number combustion and therefore, a transport equation for $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ may need to be solved. The statistical behaviours of $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ and the unclosed terms of its transport equation (i.e. the terms originating from turbulent transport T ₁ , reaction rate T ₄ and molecular dissipation $\left( {-D_2 } \right))$ have been analysed in detail. The contribution of T ₁ remains important for all cases considered here. The term T ₄ acts as a major contributor in <???>?=?1.0 cases, but plays a relatively less important role in <???>?=?0.7 cases, whereas the term $\left( {-D_2 } \right)$ acts mostly as a leading order sink. Through an a-priori DNS analysis, the performances of the models for T ₁ , T ₄ and $\left( {-D_2 } \right)$ have been addressed in detail. A model has been identified for the turbulent transport term T ₁ which satisfactorily predicts the corresponding term obtained from DNS data. The models for T ₄ , which were originally proposed for high Damköhler number flames, have been modified for low Damköhler combustion. Predictions of the modified models are found to be in good agreement with T ₄ obtained from DNS data. It has been found that existing algebraic models for $D_2 =2\overline {\rho D\nabla Y_F^{\prime\prime} \nabla \xi^{\prime\prime}} $ (where D is the mass diffusivity) are not sufficient for low Damköhler number combustion and therefore, a transport equation may need to be solved for the cross-scalar dissipation rate $\widetilde{\varepsilon }_{Y\xi } ={\overline {\rho D\nabla Y_F^{\prime\prime} \nabla \xi^{\prime\prime}} } \big/ {\overline \rho }$ for the closure of the $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ transport equation.     

Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equations

We consider a family of linearly elastic shells with thickness 2?, clamped along their entire lateral face, all having the same middle surfaceS=φ(?ω) ?R ³, whereω ?R ² is a bounded and connected open set with a Lipschitz-continuous boundaryγ, andφ ∈l ³ ( $\overline \omega$ ;R ³). We make an essential geometrical assumption on the middle surfaceS, which is satisfied ifγ andφ are smooth enough andS is “uniformly elliptic”, in the sense that the two principal radii of curvature are either both>0 at all points ofS, or both<0 at all points ofS. We show that, if the applied body force density isO(1) with respect to?, the fieldtu(?)=(u _i(?)), whereu _i (?) denote the three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elasticity, one “scaled” so as to be defined over the fixed domain Ω=ω×]?1, 1[, converges inH ¹(Ω)×H ¹(Ω)×L ²(Ω) as?→0 to a limitu, which is independent of the transverse variable. Furthermore, the averageξ=1/2ε _?1 ¹ u dx ₃, which belongs to the space $$V_M (\omega ) = H_0^1 (\omega ) \times H_0^1 (\omega ) \times L^2 (\omega ),$$ satisfies the (scaled) two-dimensional equations of a “membrane shell” viz., $$\mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \gamma _{\sigma \tau } (\zeta )\gamma _{\alpha \beta } (\eta ) \sqrt \alpha dy = \mathop \smallint \limits_\omega \left\{ {\mathop \smallint \limits_{ - 1}^1 f^i dx_3 } \right\}\eta _i \sqrt a dy$$ for allη=(η _i) εV _M(ω), where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor of the surfaceS, $$\gamma _{\alpha \beta } (\eta ) = \frac{1}{2}\left( {\partial _{\alpha \eta \beta } + \partial _{\beta \eta \alpha } } \right) - \Gamma _{\alpha \beta }^\sigma \eta _\sigma - b_{\alpha \beta \eta 3} $$ are the components of the linearized change of metric tensor ofS, $\Gamma _{\alpha \beta }^\sigma$ are the Christoffel symbols ofS, $b_{\alpha \beta }$ are the components of the curvature tensor ofS, andf ⁱ are the scaled components of the applied body force. Under the above assumptions, the two-dimensional equations of a “membrane shell” are therefore justified.     

Multiple Blow-Up Phenomena for the Sinh-Poisson Equation

We consider the sinh-Poisson equation $$(P) _ \lambda - \Delta{u} = \lambda \, {\rm sinh} \, u \quad {\rm in} \, \Omega, \quad u = 0 \quad {\rm on} \, \partial\Omega$$ , where Ω is a smooth bounded domain in ${\mathbb{R}^2}$ and λ is a small positive parameter. If ${0 \in \Omega}$ and Ω is symmetric with respect to the origin, for any integer k if λ is small enough, we construct a family of solutions to (P) _λ , which blows up at the origin, whose positive mass is 4πk(k?1) and negative mass is 4πk(k + 1). This gives a complete answer to an open problem formulated by Jost et al. (Calc Var PDE 31(2):263–276, 2008).     

Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations

We consider as in Part I a family of linearly elastic shells of thickness 2?, all having the same middle surfaceS=?(?)?R ³, whereω?R ² is a bounded and connected open set with a Lipschitz-continuous boundary, and?∈l ³ (?;R ³). The shells are clamped on a portion of their lateral face, whose middle line is?(γ ₀), whereγ ₀ is any portion of?ω withlength γ ₀>0. We make an essential geometrical assumption on the middle surfaceS and on the setγ ₀, which states that the space of inextensional displacements $$\begin{gathered} V_F (\omega ) = \{ \eta = (\eta _i ) \in H^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \hfill \\ \eta _i = \partial _v \eta _3 = 0 on \gamma _0 ,\gamma _{\alpha \beta } (\eta ) = 0 in \omega \} , \hfill \\ \end{gathered}$$ where $\gamma _{\alpha \beta }$ (η) are the components of the linearized change is metric tensor ofS, contains non-zero functions. This assumption is satisfied in particular ifS is a portion of cylinder and?(γ ₀) is contained in a generatrix ofS. We show that, if the applied body force density isO(? ²) with respect to?, the fieldu(?)=(u _i (?)), whereu _i (?) denote the three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elasticity, once “scaled” so as to be defined over the fixed domain Ω=ω×]?1, 1[, converges as?→0 inH ¹(Ω) to a limitu, which is independent of the transverse variable. Furthermore, the averageζ=1/2ts _?1 ¹ u dx ₃, which belongs to the spaceV _F (ω), satisfies the (scaled) two-dimensional equations of a “flexural shell”, viz., $$\frac{1}{3}\mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta )\rho _{\alpha \beta } (\eta )\sqrt {a } dy = \mathop \smallint \limits_\omega \left\{ {\mathop \smallint \limits_{ - 1}^1 f^i dx_3 } \right\} \eta _i \sqrt {a } dy$$ for allη=(η _i ) ∈V _F (ω), where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor of the surfaceS, $$\begin{gathered} \rho _{\alpha \beta } (\eta ) = \partial _{\alpha \beta } \eta _3 - \Gamma _{\alpha \beta }^\sigma \partial _\sigma \eta _3 + b_\beta ^\sigma \left( {\partial _\alpha \eta _\sigma - \Gamma _{\alpha \sigma }^\tau \eta _\tau } \right) \hfill \\ + b_\alpha ^\sigma \left( {\partial _\beta \eta _\sigma - \Gamma _{\beta \sigma }^\tau \eta _\tau } \right) + b_\alpha ^\sigma {\text{|}}_\beta \eta _\sigma - c_{\alpha \beta } \eta _3 \hfill \\ \end{gathered} $$ are the components of the linearized change of curvature tensor ofS, $\Gamma _{\alpha \beta }^\sigma$ are the Christoffel symbols ofS, $b_\alpha ^\beta$ are the mixed components of the curvature tensor ofS, andf ⁱ are the scaled components of the applied body force. Under the above assumptions, the two-dimensional equations of a “flexural shell” are therefore justified.     

On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term

We study questions of existence, uniqueness and asymptotic behaviour for the solutions of u(x, t) of the problem $$\begin{gathered} {\text{ }}u_t - \Delta u = \lambda e^u ,{\text{ }}\lambda {\text{ > 0, }}t > 0,{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ (P){\text{ }}u(x,0) = u_0 (x),{\text{ }}x{\text{ }}\varepsilon B, \hfill \\ {\text{ }}u(x,t) = 0{\text{ }}on{\text{ }}\partial B \times (0,\infty ), \hfill \\ \end{gathered} $$ where B is the unit ball $\{ x\varepsilon R^N :|x|{\text{ }} \leqq {\text{ }}1\} {\text{ and }}N \geqq 3$ . Our interest is focused on the parameter λ ₀=2(N?2) for which (P) admits a singular stationary solution of the form $$S(x) = - 2log|x|$$ . We study the dynamical stability or instability of S, which depends on the dimension. In particular, there exists a minimal bounded stationary solution u which is stable if $3 \leqq N \leqq 9$ , while S is unstable. For $N \geqq 10$ there is no bounded minimal solution and S is an attractor from below but not from above. In fact, solutions larger than S cannot exist in any time interval (there is instantaneous blow-up), and this happens for all dimensions.     

Nondispersive solutions to the L 2-critical Half-Wave Equation

We consider the focusing L ²-critical half-wave equation in one space dimension, $$i \partial_t u = D u - |u|^2 u$$ , where D denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold ${M_{*} > 0}$ such that all H ^1/2 solutions with ${\|u\|_{L^2} < M_*}$ extend globally in time, while solutions with ${\|u\|_{L^2} \geq M_*}$ may develop singularities in finite time. In this paper, we first prove the existence of a family of traveling waves with subcritical arbitrarily small mass. We then give a second example of nondispersive dynamics and show the existence of finite-time blowup solutions with minimal mass ${\|u_0\|_{L^2} = M_*}$ . More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy E ₀ > 0 and the linear momentum ${P_0 \in \mathbb{R}}$ . In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for L ²-critical nonlinear PDEs with nonlocal dispersion.     

Asymptotic analysis of linearly elastic shells. III. Justification of Koiter's shell equations

We consider as in Parts I and II a family of linearly elastic shells of thickness 2?, all having the same middle surfaceS=?(?)?R ³, whereω?R ² is a bounded and connected open set with a Lipschitz-continuous boundary, and? ∈ ?³ (?;R ³). The shells are clamped on a portion of their lateral face, whose middle line is?(γ ₀), whereγ ₀ is a portion of?ω withlength γ ₀>0. For all?>0, let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $u_i^\varepsilon g^{i,\varepsilon }$ of the points of the shell, obtained by solving the three-dimensional problem; let $\zeta _i^\varepsilon$ denote the covariant components of the displacement $\zeta _i^\varepsilon$ a ⁱ of the points of the middle surfaceS, obtained by solving the two-dimensional model ofW.T. Koiter, which consists in finding $$\zeta ^\varepsilon = \left( {\zeta _i^\varepsilon } \right) \in V_K (\omega ) = \left\{ {\eta = (\eta _\iota ) \in {\rm H}^1 (\omega ) \times H^1 (\omega ) \times H^2 (\omega ); \eta _i = \partial _v \eta _3 = 0 on \gamma _0 } \right\}$$ such that $$\begin{gathered} \varepsilon \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \gamma _{\sigma \tau } (\zeta ^\varepsilon )\gamma _{\alpha \beta } (\eta )\sqrt a dy + \frac{{\varepsilon ^3 }}{3} \mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \rho _{\sigma \tau } (\zeta ^\varepsilon )\rho _{\alpha \beta } (\eta )\sqrt a dy \hfill \\ = \mathop \smallint \limits_\omega p^{i,\varepsilon } \eta _i \sqrt a dy for all \eta = (\eta _i ) \in V_K (\omega ), \hfill \\ \end{gathered}$$ where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor ofS, $\gamma _{\alpha \beta }$ (η) and $\rho _{\alpha \beta }$ (η) are the components of the linearized change of metric and change of curvature tensors ofS, and $p^{i,\varepsilon }$ are the components of the resultant of the applied forces. Under the same assumptions as in Part I, we show that the fields $\frac{1}{{2_\varepsilon }}\smallint _{ - \varepsilon }^\varepsilon u_i^\varepsilon g^{i,\varepsilon } dx_3^\varepsilon$ and $\zeta _i^\varepsilon$ a ⁱ , both defined on the surfaceS, have the same principal part as? → 0, inH ¹ (ω) for the tangential components, and inL ²(ω) for the normal component; under the same assumptions as in Part II, we show that the same fields again have the same principal part as? → 0, inH ¹ (ω) for all their components. For “membrane” and “flexural” shells, the two-dimensional model ofW.T. Koiter is therefore justified.     

A Global Lower Bound for the Fundamental Solution of Kolmogorov-Fokker-Planck Equations

The main result of this paper is a global lower bound for the fundamental solution Γ of the ultraparabolic differential operator where the a _{i , j} 's and their first derivatives are Hölder continuous functions and 0?p ₀?N. The bound will follow from a local estimate of Γ and a Harnack inequality for non-negative solutions of Lu?=?0, by exploiting the invariance of the Harnack inequality with respect to suitable translation and dilation groups. For non-degenerate parabolic operators, our methods and results generalize those of Aronson &; Serrin [1].