Singular periodic impulse problems

We establish an existence principle for the impulsive periodic boundary-value problem {fx029-01}, where g ∈ C(0, ∞) can have a strong singularity at the origin. Furthermore, we assume that 0 < t ₁ < … < t _m < T, e ∈ L ₁[0, T], c ∈ ℝ, J _i and M _i , i = 1, 2, …, m, are continuous mappings of G[0, T] × G[0, T] into ℝ, and G[0, T] denotes the space of functions regulated on [0, T]. The presented principle is based on an averaging procedure similar to that introduced by Manásevich and Mawhin for singular periodic problems with p-Laplacian. Published in Neliniini Kolyvannya, Vol. 11, No. 1, pp. 32–44, January–March, 2007.     

Steady Flow of a Navier–Stokes Liquid Past an Elastic Body

We perform a mathematical analysis of the steady flow of a viscous liquid, L{\mathcal{L}} , past a three-dimensional elastic body, B{\mathcal{B}} . We assume that L{\mathcal{L}} fills the whole space exterior to B{\mathcal{B}} , and that its motion is governed by the Navier–Stokes equations corresponding to non-zero velocity at infinity, v _∞. As for B{\mathcal{B}} , we suppose that it is a St. Venant–Kirchhoff material, held in equilibrium either by keeping an interior portion of it attached to a rigid body or by means of appropriate control body force and surface traction. We treat the problem as a coupled steady state fluid-structure problem with the surface of B{\mathcal{B}} as a free boundary. Our main goal is to show existence and uniqueness for the coupled system liquid-body, for sufficiently small |v _∞|. This goal is reached by a fixed point approach based upon a suitable reformulation of the Navier–Stokes equation in the reference configuration, along with appropriate a priori estimates of solutions to the corresponding Oseen linearization and to the elasticity equations.     

The Regularity of Weak Solutions of the 3D Navier–Stokes Equations in {B^{-1}_{\infty,\infty}}

We show that if a Leray–Hopf solution u of the three-dimensional Navier–Stokes equation belongs to C((0,T]; B^-1_￥,￥){C((0,T]; B^{-1}_{\infty,\infty})} or its jumps in the B^-1_￥,￥{B^{-1}_{\infty,\infty}}-norm do not exceed a constant multiple of viscosity, then u is regular for (0, T]. Our method uses frequency local estimates on the nonlinear term, and yields an extension of the classical Ladyzhenskaya–Prodi–Serrin criterion.     

Large Time Well-posedness of the Three-Dimensional Capillary-Gravity Waves in the Long Wave Regime

In the regime of weakly transverse long waves, given long-wave initial data, we prove that a non-dimensionalized water wave system in an infinite strip under the influence of gravity and surface tension on the upper free interface has a unique solution on [0, T/ e{0, T/ \varepsilon} ] for some e{\varepsilon} independent of constant T. We shall prove in the subsequent paper (Ming et al., The long wave approximation to the three-dimensional capillary gravity waves, 2011) that on the same time interval, these solutions can be accurately approximated by sums of solutions of two decoupled Kadomtsev–Petviashvili (KP) equations.     

Long-Time Convergence of an Adaptive Biasing Force Method: The Bi-Channel Case

We present convergence results for an adaptive algorithm to compute free energies, namely the adaptive biasing force (ABF) method (Darve and Pohorille in J Chem Phys 115(20):9169–9183, 2001; Hénin and Chipot in J Chem Phys 121:2904, 2004). The free energy is the effective potential associated to a so-called reaction coordinate ξ(q), where q = (q ₁, … , q _3N ) is the position vector of an N-particle system. Computing free energy differences remains an important challenge in molecular dynamics due to the presence of metastable regions in the potential energy surface. The ABF method uses an on-the-fly estimate of the free energy to bias dynamics and overcome metastability. Using entropy arguments and logarithmic Sobolev inequalities, previous results have shown that the rate of convergence of the ABF method is limited by the metastable features of the canonical measures conditioned to being at fixed values of ξ (Lelièvre et al. in Nonlinearity 21(6):1155–1181, 2008). In this paper, we present an improvement on the existing results in the presence of such metastabilities, which is a generic case encountered in practice. More precisely, we study the so-called bi-channel case, where two channels along the reaction coordinate direction exist between an initial and final state, the channels being separated from each other by a region of very low probability. With hypotheses made on ‘channel-dependent’ conditional measures, we show on a bi-channel model, which we introduce, that the convergence of the ABF method is, in fact, not limited by metastabilities in directions orthogonal to ξ under two crucial assumptions: (i) exchange between the two channels is possible for some values of ξ and (ii) the free energy is a good bias in each channel. This theoretical result supports recent numerical experiments (Minoukadeh et al. in J Chem Theory Comput 6:1008–1017, 2010), where the efficiency of the ABF approach is demonstrated for such a multiple-channel situation.     

On a Free Convection Problem Over a Vertical Flat Surface in a Porous Medium

The problem of the free convection boundary-layer flow over a semi-infinite vertical flat surface in a porous medium is considered, in which the surface temperature has a constant value T₁ at the leading edge, where T₁ is above the ambient temperature, and takes a value T₂ at a given distance L along the surface, varying linearly between these two values and remaining constant afterwards. Numerical solutions of the boundary-layer equations are obtained as well as solutions valid for both small and large distance along the surface. Results are presented for the three cases, when the temperature T₂ is greater, equal or less than the ambient temperature T_∞. In the first case, T₂ > T_∞, a boundary-layer flow develops along the surface starting with a flow associated with the temperature difference T₁ − T_∞ at the leading edge and approaching a flow associated with the temperature difference T₂ − T_∞ at large distances. In the second case, T₂ = T_∞, the convective flow set up on the initial part of the surface drives a wall jet in the region where the surface temperature is the same as ambient. In the final case, T₂ < T_∞, a singularity develops in the numerical solution at the point where the surface temperature becomes T_∞. The nature of this singularity is discussed.     

Asymptotics for Turbulent Flame Speeds of the Viscous G-Equation Enhanced by Cellular and Shear Flows

G-equations are well-known front propagation models in turbulent combustion which describe the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are Hamilton–Jacobi equations with convex (L ¹ type) but non-coercive Hamiltonians. Viscous G-equations arise from either numerical approximations or regularizations by small diffusion. The nonlinear eigenvalue [`(H)]{\bar H} from the cell problem of the viscous G-equation can be viewed as an approximation of the inviscid turbulent flame speed s <sub>T</sub>. An important problem in turbulent combustion theory is to study properties of s <sub>T</sub>, in particular how s <sub>T</sub> depends on the flow amplitude A. In this paper, we study the behavior of [`(H)]=[`(H)](A,d){\bar H=\bar H(A,d)} as A → + ∞ at any fixed diffusion constant d > 0. For cellular flow, we show that

Dirichlet problem with Φ-Laplacian and mixed singularities

Assume that A₁, A₂ ⊂ ℝ are closed intervals containing 0, ϕ is an increasing odd homeomorphism with ϕ (ℝ) = ℝ, and T ∈ (0, ∞). We study a singular Dirichlet problem of the form {fx080-01} and prove the existence of its smooth solution satisfying the conditions {fx080-02}, where ƒ satisfies the Carathéodory conditions on the set (0, T) × D and can have time singularities at t = 0 and t = T and space singularities at x = 0, y = 0. Published in Neliniini Kolyvannya, Vol. 11, No. 1, pp. 81–95, January–March, 2007.     

Heat transfer by laminar flow in a vertical and horizontal pipe with twisted-tape inserts

 This article presents the results of laboratory research on heat exchange while heating water in horizontal and vertical tubes with twisted-tape inserts. The scope of the research: 70 ≤ Re ≤ 4000 3.6 ≤ Pr ≤ 5.9 8.6 ≤ Gz ≤ 540 The research was held for three cases: – horizontal experimental tube – vertical experimental tube, the direction of flow according to the free convection vector – vertical experimental tube, the direction of flow not in accordance with the free convection vector For such cases the correlation equation was defined Nu_T=f(Gz; y), Nu = f(Gz) and the proportion Nu_T/Nu was analysed. Received on 30 March 2000     

Influence of thermally induced chemorheological changes on the torsion of elastomeric circular cylinders

When an elastomeric material is deformed and subjected to temperatures above some characteristic value T _cr (near 100^∘C for natural rubber), its macromolecular structure undergoes time and temperature-dependent chemical changes. The process continues until the temperature decreases below T _cr. Compared to the virgin material, the new material system has modified properties (reduced stiffness) and permanent set on removal of the applied load.A new constitutive theory is used to study the influence of the changes of macromolecular structure on the torsion of an initially homogenous elastomeric cylinder. The cylinder is held at its initial length and given a fixed twist while at a temperature below T _cr. The twist is then held fixed and the temperature of the outer radial surface is increased above T _cr for a period of time and then returned to its original value. Assuming radial heat conduction, each material element undergoes a different chemical change. After enough time has elapsed such that the temperature field is again uniform and at its initial value, the cylinder properties are now inhomogeneous. Expressions for the time variation of the twisting moment and axial force are determined, and related to assumptions about material properties. Assuming the elastomeric networks to act as Mooney-Rivlin materials, expressions are developed for the permanent twist on release of torque, residual stress, and the new torsional stiffness in terms of the kinetics of the chemical changes.     

Molecular dynamics simulation of nonodroplets with the modified Lennard-Jones potential function

An Argon droplet in contact with a Platinum surface was simulated by molecular dynamics method. Argon molecules were modeled with modified Lennard-Jones potential function and the Platinum surface was represented by three layers of molecules. The system temperature is fixed at saturation temperature from 72 to 108 K. An Argon droplet in contact with a Platinum surface is also simulated, at different solid–fluid combinations ) \left( {{\frac{{\varepsilon_{sf} }}{\varepsilon }}} \right) (at fixed s_(sf = 0.9s \sigma_{sf} = 0.9\sigma ) from 0.4 to 0.8, and at a temperature of 84 K. It is concluded that the contact angle decreases by increasing system temperature and increases when solid–fluid interaction energy parameter decreases. When the temperature is high enough, the contact angle drops to zero.     

A Navier–Stokes Approximation of the 3D Euler Equation with the Zero Flux on the Boundary

Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T ₀ > 0, ν ₀ > 0 and a unique continuous family of strong solutions u ^ν (0 ≤ ν < ν ₀) of the Euler or Navier–Stokes initial-boundary value problem on the time interval (0, T ₀). In addition to the condition of the zero flux, the solutions of the Navier–Stokes equation satisfy certain natural boundary conditions imposed on curl u ^ν and curl ² u ^ν .      

Thermocapillary convection in an annular cylindrical container

The liquid–gas interface of a liquid in an annular container is subjected to a temperature gradient. Since shear stress on the free liquid surface depends on the temperature it shall create a variable shear stress on the surface which in turn yields by viscous traction a thermocapillary convection in the bulk of the liquid. For constant temperature T ₂ > T ₁ on the outer cylindrical wall and T ₁ on the inner wall a steady convection shall emerge. The morphology of the ensuing flow is investigated as a function of the diameter ratio and the liquid height ratio and exhibits the growth of bottom rim vortex rings, which unite to single vortex rings of opposite flow direction to the original single strong vortex ring above it. The results also show how such a system depends on the magnitude of the diameter ratio and the liquid height ratio. The evaluation of the results exhibits the evolution of vortex rings.     

Experimental free coating flows at high capillary and Reynolds number

 Experimental studies are carried out to enhance the fundamental understanding of coating processes over a broad parametric range. Experiments herein identify the phenomena leading to the formation of an asymptotic meniscus profile, which eventually develops a cusp at the interface. The non-dimensional parameters that describe these phenomena are identified. In addition, flow visualization is carried out to reveal the entire flow structure using a visible laser. Two phenomena of free coating are identified depending on a parameter called the property number(Po). When Po is larger than about 0.5, the non-dimensional final film thickness (T ₀) becomes constant beyond the capillary number(Ca) of about unity. When Po is less than about 0.1, T ₀ depends on Ca and the Reynolds number(Re) but it becomes constant beyond the Weber number(=Ca Re) of about 0.2. In both cases T ₀ becomes constant as the effect of surface tension on the meniscus becomes relatively unimportant. The cusp formation is due to the effect of inertia(Re). The effect of applicator dimensions on T ₀ is also investigated for large Re flows. Received: 12 May 1998/Accepted: 19 January 1999     

Lagrangian Time-Scales in Homogeneous Non-Gaussian Turbulence

Lagrangian time-scales in homogeneous non-Gaussian turbulence were studied using a one-dimensional Lagrangian Stochastic Model. The existence of two time-scales τ _L and T _L , one typical of the inertial subrange and the other which is an integral property, is outlined. Variations of the ratio T _L /τ _L in the plane skewness-flatness (S, F) are shown and a connection with the statistical constraint F ≥S ² + 1 is evidenced. The Lagrangian autocorrelation function ρ(t) of particle velocity was computed for some values of (S, F). It is shown that for small times, say t < T _L , the influence of non-Gaussianity is negligible and ρ(t) presents the same behaviour as in the Gaussian case regardless of variations in (S, F).As the time increases, departures from Gaussianity are observed and autocorrelation turns out to be always larger than in the Gaussiancase. This is supported by some considerations in terms of information entropy, which is shown to decrease with increasing departures from Gaussianity. Spectral analysis of Lagrangian velocity shows that non-Gaussianity is relevant only to large scales of the stochastic process and that the expected inertial subrange decay ω⁻² is attained by spectra of all simulations, except for one case in which the model probability density function is bimodal, due to the vicinity to the statistical limit. This revised version was published online in July 2006 with corrections to the Cover Date.     

Effects of magnetic field,porosity, and wall properties for anisotropically elastic multi-stenosis arteries on blood flow characteristics

A mathematical model for blood flow through an elastic artery with multistenosis under the effect of a magnetic field in a porous medium is presented. The considered arterial segment is simulated by an anisotropically elastic cylindrical tube filled with a viscous incompressible electrically conducting fluid representing blood. An artery with mild local narrowing in its lumen forming a stenosis is analyzed. The effects of arterial wall parameters represent viscoelastic stresses along the longitudinal and circumferential directions T _t and T _θ , respectively. The degree of anisotropy of the vessel wall γ, total mass of the vessel, and surrounding tissues M and contributions of the viscous and elastic constraints to the total tethering C and K respectively on resistance impedance, wall shear stress distribution, and radial and axial velocities are illustrated. Also, the effects of the stenosis shape m, the constant of permeability X, the Hartmann number H _α and the maximum height of the stenosis size δ on the fluid flow characteristics are investigated. The results show that the flow is appreciably influenced by surrounding connective tissues of the arterial wall motion, and the degree of anisotropy of the vessel wall plays an important role in determining the material of the artery. Further, the wall shear stress distribution increases with increasing T _t and γ while decreases with increasing T _θ , M, C, and K. Transmission of the wall shear stress distribution and resistance impedance at the wall surface through a tethered tube are substantially lower than those through a free tube, while the shearing stress distribution at the stenosis throat has inverse characteristic through totally tethered and free tubes. The trapping bolus increases in size toward the line center of the tube as the permeability constant X increases and decreases with the Hartmann number Ha increased. Finally, the trapping bolus appears, gradually in the case of non-symmetric stenosis, and disappears in the case of symmetric stenosis. The size of trapped bolus for the stream lines in a free isotropic tube (i.e., a tube initially unstressed) is smaller than those in a tethered tube.     

Free volume dependence of polymer viscosity

The Simha–Somcynsky (S–S) equation of state (eos) was used to compute the free volume parameter, h, from the pressure–volume–temperature (PVT) dependencies of eight molten polymers. The predicted by eos variation of h with T and P was confirmed by the positron annihilation lifetime spectroscopy; good agreement was found for h(P = constant) = h(T) as well as for h(T = constant) = h(P). Capillary shear viscosity (η) data of the same polymers (measured at three temperatures and six pressures up to 700 bars), were plotted as logη vs 1/h, the latter computed for T and P at which η was measured. In previous works, such a plot for solvents and silicone oils resulted in a “master curve” for the liquid, in a wide range of T and P. However, for molten polymers, no superposition of data onto a “master curve” could be found. The superposition could be obtained allowing the characteristic pressure reducing parameter, P*, to vary. The necessity for using a “rheological” characteristic pressure reducing parameter, P*_R = κP*, with κ = 1 to 2.1 indicates that the free volume parameter extracted from the thermodynamic equilibrium data may not fully describe the dynamic behavior. After eliminating possibility of other sources for the deviation, the most likely culprit seems to be the presence of structures in polymer melts at temperatures above the glass transition, T _g. For example, it was observed that for amorphous polymers at T ≅ 1.52T _g the factor κ = 1, and the deviation vanish.     

Finite difference analysis of laminar free convection flow past a non isothermal vertical cone

A finite-difference analysis for the transient free convection flow of an incompressible viscous fluid past a vertical cone with variable wall surface temperature T _w^′ (x) = T _∞^′ + a x ⁿ varying as power function of distance from the apex (x = 0) is presented here. The dimensionless governing equations of the flow that are unsteady, coupled and non-linear partial differential equations are solved by an efficient, accurate and unconditionally stable finite difference scheme of Crank-Nicolson type. The velocity and temperature fields have been studied for various parameters such as Prandtl number and n (exponent in power law variation in surface temperature). The local as well as average skin-friction and Nusselt number are also presented and analyzed graphically. The present results are compared with available results in literature and are found to be in good agreement.     

Heat transfer in the flow of a viscoelastic fluid over a stretching surface

An analysis is made of heat transfer in the boundary layer of a viscoelastic fluid flowing over a stretching surface. The velocity of the surface varies linearly with the distance x from a fixed point and the surface is held at a uniform temperature T _w higher than the temperature T _∞ of the ambient fluid. An exact analytical solution for the temperature distribution is found by solving the energy equation after taking into account strain energy stored in the fluid (due to its elastic property) and viscous dissipation. It is shown that the temperature profiles are nonsimilar in marked contrast with the case when these profiles are found to be similar in the absence of viscous dissipation and strain energy. It is also found that temperature at a point increases due to the combined influence of these two effects in comparison with its corresponding value in the absence of these two effects. A novel result of this analysis is that for small values of x, heat flows from the surface to the fluid while for moderate and large values of x, heat flows from the fluid to the surface even when T _w >T _∞. Temperature distribution and the surface heat flux are determined for various values of the Prandtl number P, the elastic parameter K ₁ and the viscous dissipation parameter a. Numerical solutions are also obtained through a fourth-order accurate compact finite difference scheme. Received on 14 October 1997     

Euler–Poisson Systems as Action-Minimizing Paths in the Wasserstein Space

This paper uses a variational approach to establish existence of solutions (σ _t , v _t ) for the 1-d Euler–Poisson system by minimizing an action. We assume that the initial and terminal points σ ₀, σ _T are prescribed in , the set of Borel probability measures on the real line, of finite second-order moments. We show existence of a unique minimizer of the action when the time interval [0,T] satisfies T < π. These solutions conserve the Hamiltonian and they yield a path t → σ _t in . When σ _t  = δ _y(t) is a Dirac mass, the Euler–Poisson system reduces to . The kinetic version of the Euler–Poisson, that is the Vlasov–Poisson system was studied in Ambrosio and Gangbo (Comm Pure Appl Math, to appear) as a Hamiltonian system. WG gratefully acknowledges the support provided by NSF grants DMS-02-00267, DMS-03-54729 and DMS-06-00791. TN gratefully acknowledges the postdoctoral support provided by NSF grants DMS-03- 54729 and the School of Mathematics. AT gratefully acknowledges the support provided by the School of Mathematics.