On a Volume‐Constrained Variational Problem

. Existence of minimizers for a volume-constrained energy $ E(u) := \int_{\Omega} W(\nabla u)\, dx Existence of minimizers for a volume-constrained energy E(u) : = ò_W W(?u) dx E(u) := \int_{\Omega} W(\nabla u)\, dx where L^N({u = z_i}) = a_i, i = 1, ?, P, {\cal L}^N(\{u = z_i\}) = \alpha_i, i = 1, \ldots, P, is proved for the case in which z_iz_i are extremal points of a compact, convex set in \Bbb R^d\Bbb R^d and under suitable assumptions on a class of quasiconvex energy densities W. Optimality properties are studied in the scalar-valued problem where d=1d=1, P=2P=2, W(x)=|x|²W(\xi)=|\xi|^2, and the &-limit as the sum of the measures of the 2 phases tends to \L(W)\L(\Omega) is identified. Minimizers are fully characterized when N=1N=1, and candidates for solutions are studied for the circle and the square in the plane.     

On the fundamental linear fractional order differential equation

This paper deals with the rational function approximation of the irrational transfer function G(s) = \fracX(s)E(s) = \frac1[(t₀s)^2m + 2z(t₀s)^m + 1]G(s) = \frac{X(s)}{E(s)} = \frac{1}{[(\tau _{0}s)^{2m} + 2\zeta (\tau _{0}s)^{m} + 1]} of the fundamental linear fractional order differential equation (t₀)^2m\fracd^2mx(t)dt^2m + 2z(t₀)^m\fracd^mx(t)dt^m + x(t) = e(t)(\tau_{0})^{2m}\frac{d^{2m}x(t)}{dt^{2m}} + 2\zeta(\tau_{0})^{m}\frac{d^{m}x(t)}{dt^{m}} + x(t) = e(t), for 0<m<1 and 0<ζ<1. An approximation method by a rational function, in a given frequency band, is presented and the impulse and the step responses of this fractional order system are derived. Illustrative examples are also presented to show the exactitude and the usefulness of the approximation method.     

The Erpenbeck High Frequency Instability Theorem for Zeldovitch–von Neumann–D?ring Detonations

The rigorous study of spectral stability for strong detonations was begun by Erpenbeck (Phys. Fluids 5:604–614 1962). Working with the Zeldovitch–von Neumann–D?ring (ZND) model (more precisely, Erpenbeck worked with an extension of ZND to general chemistry and thermodynamics), which assumes a finite reaction rate but ignores effects such as viscosity corresponding to second order derivatives, he used a normal mode analysis to define a stability function V(t,e){V(\tau,\epsilon)} whose zeros in ${\mathfrak{R}\tau > 0}${\mathfrak{R}\tau > 0} correspond to multidimensional perturbations of a steady detonation profile that grow exponentially in time. Later in a remarkable paper (Erpenbeck in Phys. Fluids 9:1293–1306, 1966; Stability of detonations for disturbances of small transverse wavelength, 1965) he provided strong evidence, by a combination of formal and rigorous arguments, that for certain classes of steady ZND profiles, unstable zeros of V exist for perturbations of sufficiently large transverse wavenumber e{\epsilon} , even when the von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in the sense defined (nearly 20 years later) by Majda. In spite of a great deal of later numerical work devoted to computing the zeros of V(t,e){V(\tau,\epsilon)} , the paper (Erpenbeck in Phys. Fluids 9:1293–1306, 1966) remains one of the few works we know of [another is Erpenbeck (Phys. Fluids 7:684–696, 1964), which considers perturbations for which the ratio of longitudinal over transverse components approaches ∞] that presents a detailed and convincing theoretical argument for detecting them. The analysis in Erpenbeck (Phys. Fluids 9:1293–1306, 1966) points the way toward, but does not constitute, a mathematical proof that such unstable zeros exist. In this paper we identify the mathematical issues left unresolved in Erpenbeck (Phys. Fluids 9:1293–1306, 1966) and provide proofs, together with certain simplifications and extensions, of the main conclusions about stability and instability of detonations contained in that paper. The main mathematical problem, and our principal focus here, is to determine the precise asymptotic behavior as e?￥{\epsilon\to\infty} of solutions to a linear system of ODEs in x, depending on e{\epsilon} and a complex frequency τ as parameters, with turning points x _* on the half-line [0,∞).     

Equilibrium states in homogeneous turbulence with baroclinic instability

The evolution of energies and fluxes in homogeneous turbulence with baroclinic instability is analyzed using the linear theory. The mean flow corresponds to a vertical shear having a uniform mean velocity gradient, ?U _i /?x _j  = S δ _i1 δ _j3, a system rotation about the vertical axis with rate Ω, Ω _i  = Ωδ _i3, and uniform buoyancy gradients in the spanwise ${(\partial B{/}\partial x_2\,{=}\, N_h^2\,{=}\,-2\Omega S)}The evolution of energies and fluxes in homogeneous turbulence with baroclinic instability is analyzed using the linear theory. The mean flow corresponds to a vertical shear having a uniform mean velocity gradient, ∂U _i /∂x _j  = S δ _i1 δ _j3, a system rotation about the vertical axis with rate Ω, Ω _i  = Ωδ _i3, and uniform buoyancy gradients in the spanwise (?B/?x₂ = N_h² = -2WS){(\partial B{/}\partial x_2\,{=}\, N_h^2\,{=}\,-2\Omega S)} and vertical (?B/?x₃ = N_v²){(\partial B{/}\partial x_3\,{=}\,N_v^2)} directions. Computations based on the rapid distortion theory (RDT) are performed for several values of the rotation number R = 2Ω/S and the Richardson number R_i = N_v²/S² < 1{R_i\,{=}\,N_v^2/S^2 <1 }. It is shown that, during an initial phase, the energies and the buoyancy fluxes are sensitive to the effects of pressure and viscosity. At large time, the ratios of energies, as well as the normalized fluxes, evolve to an asymptotically constant value, while the pressure–strain correlation scaled with the product of the turbulent kinetic energy by the shear rate approaches zero. Accordingly, an analytical parametric study based on the “pressure-less” approach (PLA) is also presented. The analytical study indicates that, when R _i  < 1, there is an exponential instability and equilibrium states of turbulence, in agreement with RDT. The energies and the buoyancy fluxes grow exponentially for large times with the same rate (γ in St units). The asymptotic value of the ratios of energies yielded by RDT is well described by its PLA counterpart derived analytically. At R _i  = 0, the asymptotic value of γ increases with increasing R approaching 2 for high rotation rates. At low rotation rates, an important contribution to the kinetic energy comes from the streamwise kinetic energy, whereas, at high rotation rates, the contribution of the vertical kinetic energy is dominant. When 0 < R _i  < 1 and R 1 0{R\ne 0}, the asymptotic value of γ decreases as R _i increases so as it becomes zero at R _i  = 1.     

Slip flow of non-plasticized PVC compounds

Measurements on seven rigid PVC compounds were carried out with a slit rheometer working in combination with an injection moulding machine. Plastication of the compounds occurred in the screw of the plastication unit, which also forced the melt through the die with a controlled forward velocity. The rectangular slit had a length of 90 mm and a widthB of 20 mm. The heightH could be varied between 0.8 and 3.3 mm. Pressures and temperatures were recorded at several positions in and before the die. Measurements were carried out at shear rates from 10 to 2000 s^–1.When the reduced volume output was plotted against the wall shear stress _W , only four compounds showed master curves independent ofH, which is indicative of wall adhesion. In the other cases this plot did not produce such a master curve, but the plot of the mean velocity against _W was independent ofH (slip curve). This indicated that slip flow prevailed with a slip velocityv _G When, in the case of wall slip, the smooth inner surfaces of the die were replaced by surfaces with grooves perpendicular to the direction of flow, slip flow was prevented and the flow curves were shifted to much higher values of _Wc Above a critical value of the wall shear stress ( _Wc ) at which slip flow began, the output became nearly independent of _W . From the measurements made below _Wc a vs. relation for the shear flow could be derived, which was used to calculate the superimposed shear flow . Exact values of the slip velocity were then given by . Wall slip only occurred for compounds with a high shear viscosity, which corresponds to a high molecular weight (K-value).Dedicated to Professor H. Janeschitz-Kriegl on the occasion of his 60th birthday.     

Minimizers of Caffarelli–Kohn–Nirenberg Inequalities with the Singularity on the Boundary

Let Ω be a bounded smooth domain in ${{\bf R}^N, N\geqq 3}Let Ω be a bounded smooth domain in R^N, N\geqq 3{{\bf R}^N, N\geqq 3}, and D_a^1,2(W){D_a^{1,2}(\Omega)} be the completion of C₀^￥(W){C_0^\infty(\Omega)} with respect to the norm:

||u||a2=òW |x|-2a|?u|2dx.||u||_a^2=\int_\Omega |x|^{-2a}|\nabla u|^2{d}x.      7. Time-Stepping Approximation of Rigid-Body Dynamics with Perfect Unilateral Constraints. II: The Partially Elastic Impact Case      L. Paoli 《Archive for Rational Mechanics and Analysis》2010,198(2):505-568 As in Paoli (Arch Rational Mech Anal, 2010), we consider a discrete mechanical system with a non-trivial mass matrix subjected to perfect unilateral constraints described by geometrical inequalities ${f_{\alpha}(q) \geqq 0, \alpha \in \{1, \ldots, \nu \}\, (\nu \geqq 1)}We consider a discrete mechanical system with a non-trivial mass matrix, subjected to perfect unilateral constraints described by the geometrical inequalities fa (q) \geqq 0, a ? {1, ..., n} (n\geqq 1){f_{\alpha} (q) \geqq 0, \alpha \in \{1, \dots, \nu\} (\nu \geqq 1)}. We assume that the transmission of the velocities at impact is governed by Newton’s Law with a coefficient of restitution e = 0 (so that the impact is inelastic). We propose a time-discretization of the second order differential inclusion describing the dynamics, which generalizes the scheme proposed in Paoli (J Differ Equ 211:247–281, 2005) and, for any admissible data, we prove the convergence of approximate motions to a solution of the initial-value problem.      8. Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems      Jaume Llibre  Clàudia Valls 《Journal of Dynamics and Differential Equations》2010,22(4):657-675 We classify new classes of centers and of isochronous centers for polynomial differential systems in \mathbb R2{\mathbb R^2} of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as [(z)\dot] = (l+i) z + (z[`(z)])\fracd-7-2j2 (A z5+j[`(z)]2+j + B z4+j[`(z)]3+j + C z3+j[`(z)]4+j+D[`(z)]7+2j ),\dot z = (\lambda+i) z + (z \overline z)^{\frac{d-7-2j}2} \left(A z^{5+j} \overline z^{2+j} + B z^{4+j} \overline z^{3+j} + C z^{3+j} \overline z^{4+j}+D \overline z^{7+2j} \right),      9. On the Dynamics of Solitons in the Nonlinear Schr?dinger Equation      Vieri Benci  Marco Ghimenti  Anna Maria Micheletti 《Archive for Rational Mechanics and Analysis》2012,205(2):467-492 We study the behavior of the soliton solutions of the equation i\frac?y?t = - \frac12m Dy+ \frac12We￠(y) + V(x)y,i\frac{\partial\psi}{{\partial}t} = - \frac{1}{2m} \Delta\psi + \frac{1}{2}W_{\varepsilon}^{\prime}(\psi) + V(x){\psi},      10. On some nearly viscometric flows of viscoelastic liquids      Professor A. Siginer 《Rheologica Acta》1991,30(5):447-473 Superposition of oscillatory shear imposed from the boundary and through pressure gradient oscillations and simple shear is investigated. The integral fluid with fading memory shows flow enhancement effects due to the nonlinear structure. Closed-form expressions for the change in the mass transport rate are given at the lowest significant order in the perturbation algorithm. The elasticity of the liquid plays as important a role in determining the enhancement as does the shear dependent viscosity. Coupling of shear thinning and elasticity may produce sharp increases in the flow rate. The interaction of oscillatory shear components may generate a steady flow, either longitudinal or orthogonal, resulting in increases in flow rates akin to resonance, and due to frequency cancellation, even in the absence of a mean gradient. An algorithm to determine the constitutive functions of the integral fluid of order three is outlined.Nomenclature A n Rivlin-Ericksen tensor of order . - A k Non-oscillatory component of the first order linear viscoelastic oscillatory velocity field induced by the kth wave in the pressure gradient - d Half the gap between the plates - e x, e z Unit vectors in the longitudinal and orthogonal directions, respectively - G(s) Relaxation modulus - G History of the deformation - Stress response functional - I() Enhancement defined as the ratio of the frequency dependent part of the discharge to the frequencyindependent part of it at the third order - I *() Enhancement defined as the ratio of the increase in discharge due to oscillations to the total discharge without the oscillations - k Power index in the relaxation modulus G(s) - k i –1 Relaxation times in the Maxwell representation of the quadratic shear relaxation modulus (s 1, s 2) - m i –1, n i –1 Relaxation times in the Maxwell representations of the constitutive functions 1(s 1,s 2,s 3) and 4 (s 1, s 2,s 3), respectively - P Constant longitudinal pressure gradient - p Pressure field - mx ,(3) nz ,(3) Mean volume transport rates at the third order in the longitudinal and orthogonal directions, respectively - 0,(3), 1,(3) Frequency independent and dependent volume transport rates, respectively, at the third order - s = t- Difference between present and past times t and       11. A terminally anchored polymer chain in shear flow: Self-consistent velocity and segment density profiles      Dr. R. S. Parnas  Y. Cohen 《Rheologica Acta》1994,33(6):485-505 The behavior of a terminally anchored freely-jointed bead-rod chain, subjected to solvent shear flow, was investigated via Brownian dynamics simulations. Previous calculations have been improved by computing the segment density and fluid velocity profiles self-consistently. The segment density distributions, components of the radius of gyration, and chain attachment shear and normal stresses were found to be sensitive to low values of shear rate. Additionally, it was found that the thickness of a model polymer layer was a strong function of the shear rate, and that the functional dependence on shear rate changed dramatically as the chain length increased. For the longest chains studied, the thickness of the model polymer layer first increased as the shear rate increased, passed through a maximum, and then decreased at high shear rates, in accordance with experimental results in theta solvents. These results suggest that a dilute or semi-dilute layer model may explain hydrodynamic behavior previously thought to be due to the entanglements that occur in dense surface bound polymer layers.Nomenclature a i acceleration of bead i - b radius of the beads - d length of the rods connecting the chain beads - d i vector from bead i to bead i + 1 - F i external force applied to bead i - F i b external force on bead i due to Brownian motion of surrounding fluid - F i h external force on bead i due to viscous drag - F i s external force on bead i due to surface interactions - f Stokes drag coefficient - Boltzmann's constant - L h effective hydrodynamic thickness - m i mass of bead i - N number of beads on a model chain - n number of chains anchored to the surface per unit surface area - P segment density distribution P pressure - Q flow in a tube with no surface bound polymer layer - Q a flow in a tube with a surface bound polymer layer - R g vector representation of the radius of gyration - R tube radius - r radial coordinate in the tube geometry - S ij pair hydrodynamic interaction tensor for beads i and j - T i internal chain force in rod i connecting beads i and i + 1 - T X component of the surface attachment force in the direction of the fluid flow - T y component of the surface attachment force perpendicular to the surface - T temperature - v i velocity of the center of mass of bead i - V if average fluid velocity at the location of bead i - v if 0 fluid velocity in the absence of a polymer chain - v if perturbation to the fluid velocity due to hydrodynamic interactions - V b bead volume = 4 b 3/3 - scalar fluid speed in the axial direction down the tube - x axial coordinate in the tube geometry Greek symbols w apparent shear rate - fluid viscosity - polymer layer permeability - volume fraction of space occupied by chain beads - (w)a chain attachment stress perpendicular to the surface - (w)a chain attachment stress in the plane of the surface and in the direction of fluid flow      12. Three views of viscoelasticity for Cox–Merz materials      H. Henning Winter 《Rheologica Acta》2009,48(3):241-243 A slight rearrangement of the classical Cox and Merz rule suggests that the shear stress value of steady shear flow, , and complex modulus value of small amplitude oscillatory shear, G ∗ (ω) = (G′2 + G″2)1/2, are equivalent in many respects. Small changes of material structure, which express themselves most sensitively in the steady shear stress, τ, show equally pronounced in linear viscoelastic data when plotting these with G ∗  as one of the variables. An example is given to demonstrate this phenomenon: viscosity data that cover about three decades in frequency get stretched out over about nine decades in G ∗  while maintaining steep gradients in a transition region. This suggests a more effective way of exploiting the Cox–Merz rule when it is valid and exploring reasons for lack of validity when it is not. The τ −G ∗  equivalence could also further the understanding of the steady shear normal stress function as proposed by Laun.      13. Hydrodynamic Limit of Gradient Exclusion Processes with Conductances      Tertuliano Franco  Claudio Landim 《Archive for Rational Mechanics and Analysis》2010,195(2):409-439 Fix a strictly increasing right continuous with left limits function ${W: \mathbb{R} \to \mathbb{R}}Fix a strictly increasing right continuous with left limits function W: \mathbbR ? \mathbbR{W: \mathbb{R} \to \mathbb{R}} and a smooth function F: [l,r] ? \mathbb R{\Phi : [l,r] \to \mathbb R}, defined on some interval [l, r] of \mathbb R{\mathbb R}, such that 0 < b\leqq F￠\leqq b-1{0 < b\leqq \Phi'\leqq b^{-1}}. On the diffusive time scale, the evolution of the empirical density of exclusion processes with conductances given by W is described by the unique weak solution of the non-linear differential equation ?t r = (d/dx)(d/dW) F(r){\partial_t \rho = ({\rm d}/{\rm d}x)({\rm d}/{\rm d}W) \Phi(\rho)}. We also present some properties of the operator (d/dx)(d/dW).      14. Fully developed flow of a viscoelastic film down a vertical cylindrical or planar wall      M. Pavlidis  Y. Dimakopoulos  J. Tsamopoulos 《Rheologica Acta》2009,48(9):1031-1048 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.      15. Boundary Regularity of Shear Thickening Flows      Hugo Beir?o da Veiga  Petr Kaplicky  Michael R??i?ka 《Journal of Mathematical Fluid Mechanics》2011,13(3):387-404 This article is concerned with the global regularity of weak solutions to systems describing the flow of shear thickening fluids under the homogeneous Dirichlet boundary condition. The extra stress tensor is given by a power law ansatz with shear exponent p≥ 2. We show that, if the data of the problem are smooth enough, the solution u of the steady generalized Stokes problem belongs to W1,(np+2-p)/(n-2)(W){W^{1,(np+2-p)/(n-2)}(\Omega)} . We use the method of tangential translations and reconstruct the regularity in the normal direction from the system, together with anisotropic embedding theorem. Corresponding results for the steady and unsteady generalized Navier–Stokes problem are also formulated.      16. Positive Least Energy Solutions and Phase Separation for Coupled Schr?dinger Equations with Critical Exponent      Zhijie Chen  Wenming Zou 《Archive for Rational Mechanics and Analysis》2012,205(2):515-551 In this paper we study the following coupled Schr?dinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem: {ll-Du +l1 u = m1 u3+buv2,     x ? W,-Dv +l2 v = m2 v3+bvu2,     x ? W,u\geqq 0, v\geqq 0 in W,    u=v=0     on ?W.\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right.      17. Radial heat transfer in fixed-bed packing with small tube/particle diameter ratios      A. Grah  U. Nowak  M. Schreier  R. Adler 《Heat and Mass Transfer》2009,45(4):417-425       18. Potential Flow Past a Slightly Deformed Porous Circular Cylinder      Anindita Bhattacharya  G. P. Raja Sekhar 《Transport in Porous Media》2010,81(3):367-389 A uniform potential flow past a porous circular cylinder with a core of different permeability is discussed. The porous circular cylinder is slightly deformed whose radius is r=r1(1+ecosm q){r=r_1(1+\epsilon \cos m \theta)} , where | e | << 1{\mid\epsilon\mid\ll 1} and m is a positive integer. Here r, θ are the polar coordinates and r 1 is the characteristic radius of the cylinder. The drag force exerted by the exterior flow on the surface of the cylinder is calculated and it depends on the thickness of the porous material and on the permeabilities of the two porous regions. As special cases, porous cylinder with hollow core, rigid core, and deformed cylinder is discussed.      19. A Discussion of the Effect of Tortuosity on the Capillary Imbibition in Porous Media   总被引：2，自引：0，他引：2   Jianchao Cai  Boming Yu 《Transport in Porous Media》2011,89(2):251-263 In the past decades, there was considerable controversy over the Lucas–Washburn (LW) equation widely applied in capillary imbibition kinetics. Many experimental results showed that the time exponent of the LW equation is less than 0.5. Based on the tortuous capillary model and fractal geometry, the effect of tortuosity on the capillary imbibition in wetting porous media is discussed in this article. The average height growth of wetting liquid in porous media driven by capillary force following the [`(L)] s(t) ~ t1/2DT{\overline L _{\rm {s}}(t)\sim t^{1/{2D_{\rm {T}}}}} law is obtained (here D T is the fractal dimension for tortuosity, which represents the heterogeneity of flow in porous media). The LW law turns out to be the special case when the straight capillary tube (D T = 1) is assumed. The predictions by the present model for the time exponent for capillary imbibition in porous media are compared with available experimental data, and the present model can reproduce approximately the global trend of variation of the time exponent with porosity changing.      20. The flow field around a micropillar confined in a microchannel      《International Journal of Heat and Fluid Flow》2012 The flow field over a low aspect ratio (AR) circular pillar (L/D = 1.5) in a microchannel was studied experimentally. Microparticle image velocimetry (μPIV) was employed to quantify flow parameters such as flow field, spanwise vorticity, and turbulent kinetic energy (TKE) in the microchannel. Flow regimes of cylinder-diameter-based Reynolds number at 100 ⩽ ReD ⩽ 700 (i.e., steady, transition from quasi-steady to unsteady, and unsteady flow) were elucidated at the microscale. In addition, active flow control (AFC), via a steady control jet (issued from the pillar itself in the downstream direction), was implemented to induce favorable disturbances to the flow in order to alter the flow field, promote turbulence, and increase mixing. Together with passive flow control (i.e., a circular pillar), turbulent kinetic energy was significantly increased in a controllable manner throughout the flow field.

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Time-Stepping Approximation of Rigid-Body Dynamics with Perfect Unilateral Constraints. II: The Partially Elastic Impact Case

As in Paoli (Arch Rational Mech Anal, 2010), we consider a discrete mechanical system with a non-trivial mass matrix subjected to perfect unilateral constraints described by geometrical inequalities ${f_{\alpha}(q) \geqq 0, \alpha \in \{1, \ldots, \nu \}\, (\nu \geqq 1)}We consider a discrete mechanical system with a non-trivial mass matrix, subjected to perfect unilateral constraints described by the geometrical inequalities f_a (q) \geqq 0, a ? {1, ..., n} (n\geqq 1){f_{\alpha} (q) \geqq 0, \alpha \in \{1, \dots, \nu\} (\nu \geqq 1)}. We assume that the transmission of the velocities at impact is governed by Newton’s Law with a coefficient of restitution e = 0 (so that the impact is inelastic). We propose a time-discretization of the second order differential inclusion describing the dynamics, which generalizes the scheme proposed in Paoli (J Differ Equ 211:247–281, 2005) and, for any admissible data, we prove the convergence of approximate motions to a solution of the initial-value problem.     

Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems

We classify new classes of centers and of isochronous centers for polynomial differential systems in \mathbb R²{\mathbb R^2} of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as

On the Dynamics of Solitons in the Nonlinear Schr?dinger Equation

We study the behavior of the soliton solutions of the equation i\frac?y?t = - \frac12m Dy+ \frac12W_e^￠(y) + V(x)y,i\frac{\partial\psi}{{\partial}t} = - \frac{1}{2m} \Delta\psi + \frac{1}{2}W_{\varepsilon}^{\prime}(\psi) + V(x){\psi},     

On some nearly viscometric flows of viscoelastic liquids

Superposition of oscillatory shear imposed from the boundary and through pressure gradient oscillations and simple shear is investigated. The integral fluid with fading memory shows flow enhancement effects due to the nonlinear structure. Closed-form expressions for the change in the mass transport rate are given at the lowest significant order in the perturbation algorithm. The elasticity of the liquid plays as important a role in determining the enhancement as does the shear dependent viscosity. Coupling of shear thinning and elasticity may produce sharp increases in the flow rate. The interaction of oscillatory shear components may generate a steady flow, either longitudinal or orthogonal, resulting in increases in flow rates akin to resonance, and due to frequency cancellation, even in the absence of a mean gradient. An algorithm to determine the constitutive functions of the integral fluid of order three is outlined.Nomenclature A _n Rivlin-Ericksen tensor of order . - A _k Non-oscillatory component of the first order linear viscoelastic oscillatory velocity field induced by the kth wave in the pressure gradient - d Half the gap between the plates - e _x, e _z Unit vectors in the longitudinal and orthogonal directions, respectively - G(s) Relaxation modulus - G History of the deformation - Stress response functional - I() Enhancement defined as the ratio of the frequency dependent part of the discharge to the frequencyindependent part of it at the third order - I ^*() Enhancement defined as the ratio of the increase in discharge due to oscillations to the total discharge without the oscillations - k Power index in the relaxation modulus G(s) - k _i ^–1 Relaxation times in the Maxwell representation of the quadratic shear relaxation modulus (s ₁, s ₂) - m _i ^–1, n _i ^–1 Relaxation times in the Maxwell representations of the constitutive functions ₁(s ₁,s ₂,s ₃) and ₄ (s ₁, s ₂,s ₃), respectively - P Constant longitudinal pressure gradient - p Pressure field - _mx ,⁽³⁾ _nz ,⁽³⁾ Mean volume transport rates at the third order in the longitudinal and orthogonal directions, respectively - ₀,⁽³⁾, ₁,⁽³⁾ Frequency independent and dependent volume transport rates, respectively, at the third order - s = t- Difference between present and past times t and      

A terminally anchored polymer chain in shear flow: Self-consistent velocity and segment density profiles

The behavior of a terminally anchored freely-jointed bead-rod chain, subjected to solvent shear flow, was investigated via Brownian dynamics simulations. Previous calculations have been improved by computing the segment density and fluid velocity profiles self-consistently. The segment density distributions, components of the radius of gyration, and chain attachment shear and normal stresses were found to be sensitive to low values of shear rate. Additionally, it was found that the thickness of a model polymer layer was a strong function of the shear rate, and that the functional dependence on shear rate changed dramatically as the chain length increased. For the longest chains studied, the thickness of the model polymer layer first increased as the shear rate increased, passed through a maximum, and then decreased at high shear rates, in accordance with experimental results in theta solvents. These results suggest that a dilute or semi-dilute layer model may explain hydrodynamic behavior previously thought to be due to the entanglements that occur in dense surface bound polymer layers.Nomenclature a _i acceleration of bead i - b radius of the beads - d length of the rods connecting the chain beads - d _i vector from bead i to bead i + 1 - F _i external force applied to bead i - F _i ^b external force on bead i due to Brownian motion of surrounding fluid - F _i ^h external force on bead i due to viscous drag - F _i ^s external force on bead i due to surface interactions - f Stokes drag coefficient - Boltzmann's constant - L _h effective hydrodynamic thickness - m _i mass of bead i - N number of beads on a model chain - n number of chains anchored to the surface per unit surface area - P segment density distribution P pressure - Q flow in a tube with no surface bound polymer layer - Q _a flow in a tube with a surface bound polymer layer - R _g vector representation of the radius of gyration - R tube radius - r radial coordinate in the tube geometry - S _ij pair hydrodynamic interaction tensor for beads i and j - T _i internal chain force in rod i connecting beads i and i + 1 - T _X component of the surface attachment force in the direction of the fluid flow - T _y component of the surface attachment force perpendicular to the surface - T temperature - v _i velocity of the center of mass of bead i - V _if average fluid velocity at the location of bead i - v _if ⁰ fluid velocity in the absence of a polymer chain - v _if perturbation to the fluid velocity due to hydrodynamic interactions - V _b bead volume = 4 b ³/3 - scalar fluid speed in the axial direction down the tube - x axial coordinate in the tube geometry Greek symbols _w apparent shear rate - fluid viscosity - polymer layer permeability - volume fraction of space occupied by chain beads - (_w)_a chain attachment stress perpendicular to the surface - (_w)_a chain attachment stress in the plane of the surface and in the direction of fluid flow     

Three views of viscoelasticity for Cox–Merz materials

A slight rearrangement of the classical Cox and Merz rule suggests that the shear stress value of steady shear flow, , and complex modulus value of small amplitude oscillatory shear, G^∗(ω) = (G^′2 + G^″2)^1/2, are equivalent in many respects. Small changes of material structure, which express themselves most sensitively in the steady shear stress, τ, show equally pronounced in linear viscoelastic data when plotting these with G^∗ as one of the variables. An example is given to demonstrate this phenomenon: viscosity data that cover about three decades in frequency get stretched out over about nine decades in G^∗ while maintaining steep gradients in a transition region. This suggests a more effective way of exploiting the Cox–Merz rule when it is valid and exploring reasons for lack of validity when it is not. The τ −G^∗ equivalence could also further the understanding of the steady shear normal stress function as proposed by Laun.     

Hydrodynamic Limit of Gradient Exclusion Processes with Conductances

Fix a strictly increasing right continuous with left limits function ${W: \mathbb{R} \to \mathbb{R}}Fix a strictly increasing right continuous with left limits function W: \mathbbR ? \mathbbR{W: \mathbb{R} \to \mathbb{R}} and a smooth function F: [l,r] ? \mathbb R{\Phi : [l,r] \to \mathbb R}, defined on some interval [l, r] of \mathbb R{\mathbb R}, such that 0 < b\leqq F￠\leqq b^-1{0 < b\leqq \Phi'\leqq b^{-1}}. On the diffusive time scale, the evolution of the empirical density of exclusion processes with conductances given by W is described by the unique weak solution of the non-linear differential equation ?_t r = (d/dx)(d/dW) F(r){\partial_t \rho = ({\rm d}/{\rm d}x)({\rm d}/{\rm d}W) \Phi(\rho)}. We also present some properties of the operator (d/dx)(d/dW).     

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.     

Boundary Regularity of Shear Thickening Flows

This article is concerned with the global regularity of weak solutions to systems describing the flow of shear thickening fluids under the homogeneous Dirichlet boundary condition. The extra stress tensor is given by a power law ansatz with shear exponent p≥ 2. We show that, if the data of the problem are smooth enough, the solution u of the steady generalized Stokes problem belongs to W^{1,(np+2-p)/(n-2)}(W){W^{1,(np+2-p)/(n-2)}(\Omega)} . We use the method of tangential translations and reconstruct the regularity in the normal direction from the system, together with anisotropic embedding theorem. Corresponding results for the steady and unsteady generalized Navier–Stokes problem are also formulated.     

Positive Least Energy Solutions and Phase Separation for Coupled Schr?dinger Equations with Critical Exponent

In this paper we study the following coupled Schr?dinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem: {ll-Du +l₁ u = m₁ u³+buv²,     x ? W,-Dv +l₂ v = m₂ v³+bvu²,     x ? W,u\geqq 0, v\geqq 0 in W,    u=v=0     on ?W.\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right.     

Potential Flow Past a Slightly Deformed Porous Circular Cylinder

A uniform potential flow past a porous circular cylinder with a core of different permeability is discussed. The porous circular cylinder is slightly deformed whose radius is r=r₁(1+ecosm q){r=r_1(1+\epsilon \cos m \theta)} , where | e | << 1{\mid\epsilon\mid\ll 1} and m is a positive integer. Here r, θ are the polar coordinates and r ₁ is the characteristic radius of the cylinder. The drag force exerted by the exterior flow on the surface of the cylinder is calculated and it depends on the thickness of the porous material and on the permeabilities of the two porous regions. As special cases, porous cylinder with hollow core, rigid core, and deformed cylinder is discussed.     

A Discussion of the Effect of Tortuosity on the Capillary Imbibition in Porous Media

In the past decades, there was considerable controversy over the Lucas–Washburn (LW) equation widely applied in capillary imbibition kinetics. Many experimental results showed that the time exponent of the LW equation is less than 0.5. Based on the tortuous capillary model and fractal geometry, the effect of tortuosity on the capillary imbibition in wetting porous media is discussed in this article. The average height growth of wetting liquid in porous media driven by capillary force following the [`(L)] _s(t) ~ t^1/2D_T{\overline L _{\rm {s}}(t)\sim t^{1/{2D_{\rm {T}}}}} law is obtained (here D _T is the fractal dimension for tortuosity, which represents the heterogeneity of flow in porous media). The LW law turns out to be the special case when the straight capillary tube (D _T = 1) is assumed. The predictions by the present model for the time exponent for capillary imbibition in porous media are compared with available experimental data, and the present model can reproduce approximately the global trend of variation of the time exponent with porosity changing.     

The flow field around a micropillar confined in a microchannel

The flow field over a low aspect ratio (AR) circular pillar (L/D = 1.5) in a microchannel was studied experimentally. Microparticle image velocimetry (μPIV) was employed to quantify flow parameters such as flow field, spanwise vorticity, and turbulent kinetic energy (TKE) in the microchannel. Flow regimes of cylinder-diameter-based Reynolds number at 100 ⩽ Re_D ⩽ 700 (i.e., steady, transition from quasi-steady to unsteady, and unsteady flow) were elucidated at the microscale. In addition, active flow control (AFC), via a steady control jet (issued from the pillar itself in the downstream direction), was implemented to induce favorable disturbances to the flow in order to alter the flow field, promote turbulence, and increase mixing. Together with passive flow control (i.e., a circular pillar), turbulent kinetic energy was significantly increased in a controllable manner throughout the flow field.