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

Experimental time-resolved study of the interaction between a pulsating injectant and a steady cross-flow: aerodynamics of film cooling

A test rig incorporating the injection from a single cylindrical hole with an inclination of 30° to a thermally uniform mainstream flow was used for determining variations in flow structures due to injectant pulsation. The average blowing ratios ([`(M)] \overline{M} ) were 0.65, 1, and 1.25. The periodic variations in injectant flow were rendered by a loudspeaker-based pulsation system to nondimensionalized excitation frequency (St St ) of 0, 0.2, 0.3, and 0.5. Pulsation resulting in a close-wall orientation of injectant fluid compared with steady blowing bearing outward orientation was only observed in few cases. At [`(M)] \overline{M}  = 0.65, jet fluid remains aligned and covers a significant part of the wall under steady blowing. At higher blowing ratios, pulsation induces large spatial variations in the jet trajectory, collapsing of the jet body, and the shedding of wake structures due to the periodic variation of injection flow rate. It was found that the pulsation improves wall coverage of the injectant fluid under low frequency excitation as the separation of the jet from the wall becomes evident ([`(M)] \overline{M}  = 1 and 1.25).     

Streamwise evolution of an inclined cylinder wake

The streamwise evolution of an inclined circular cylinder wake was investigated by measuring all three velocity and vorticity components using an eight-hotwire vorticity probe in a wind tunnel at a Reynolds number Re_d of 7,200 based on free stream velocity (U _∞) and cylinder diameter (d). The measurements were conducted at four different inclination angles (α), namely 0°, 15°, 30°, and 45° and at three downstream locations, i.e., x/d = 10, 20, and 40 from the cylinder. At x/d = 10, the effects of α on the three coherent vorticity components are negligibly small for α ≤ 15°. When α increases further to 45°, the maximum of coherent spanwise vorticity reduces by about 50%, while that of the streamwise vorticity increases by about 70%. Similar results are found at x/d = 20, indicating the impaired spanwise vortices and the enhancement of the three-dimensionality of the wake with increasing α. The streamwise decay rate of the coherent spanwise vorticity is smaller for a larger α. This is because the streamwise spacing between the spanwise vortices is bigger for a larger α, resulting in a weak interaction between the vortices and hence slower decaying rate in the streamwise direction. For all tested α, the coherent contribution to [`(v<sup>2</sup>)] \overline{{v^{2}}} is remarkable at x/d = 10 and 20 and significantly larger than that to [`(u²)] \overline{{u^{2}}} and [`(w<sup>2</sup>)]. \overline{{w^{2}}}. This contribution to all three Reynolds normal stresses becomes negligibly small at x/d = 40. The coherent contribution to [`(u²)] \overline{{u^{2}}} and [`(v²)] \overline{{v^{2}}} decays slower as moving downstream for a larger α, consistent with the slow decay of the coherent spanwise vorticity for a larger α.     

The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation ${{\Delta^2} u=\frac{\lambda}{(1-u)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation D² u=\fracl(1-u)²{{\Delta^2} u=\frac{\lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball B ì \mathbbR^N{B\subset{\mathbb{R}}^N}, under Dirichlet boundary conditions u=?_n u=0{u=\partial_\nu u=0} on ?B{\partial B}. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u _λ with 0 < u _λ < 1 exists for l ? (0,l^*){\lambda\in (0,\lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution u_l^*{u_{\lambda^*}} is regular (sup_B u_l^* < 1 ){({\rm sup}_B u_{\lambda^*} <1 )} provided N \leqq 8{N \leqq 8} while u_l^*{u_{\lambda^*}} is singular (sup_B u_l^* = 1){({\rm sup}_B u_{\lambda^*} =1)} for N \geqq 9{N \geqq 9}, in which case 1-C₀|x|^4/3 \leqq u_l^* (x) \leqq 1-|x|^4/3{1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}} on the unit ball, where C₀:=(\fracl^*[`(l)])<sup>\frac</sup>13{C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}} and [`(l)]: = \frac89(N-\frac23)(N- \frac83){\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}.     

Mean flow and turbulence measurements of the impingement wall jet on a semi-circular convex surface

The detailed mean flow and turbulence measurements of a turbulent air slot jet impinging on two different semi-circular convex surfaces were investigated in both free jet and impingement wall jet regions at a jet Reynolds number Re_w=12,000, using a hot-wire X-probe anemometer. The parametric effects of dimensionless circumferential distance, S/W=2.79-7.74, slot jet-to-impingement surface distance Y/W=1-13, and surface curvature D/W=10.7 and 16 on the impingement wall jet flow development along a semi-circular convex surface were examined. The results show that the effect of surface curvature D/W increases with increasing S/W. Compared with transverse Reynolds normal stress, [`(v<sup>2</sup> )] /U<sub>m</sub><sup>2</sup> \overline {v^2 } /U_{\rm m}^2 , the streamwise Reynolds normal stress, [`(u² )] /U_m² \overline {u^2 } /U_{\rm m}^2 , is strongly affected by the examined dimensionless parameters of D/W, Y/W and S/W in the near-wall region. It is also evidenced that the Reynolds shear stress, -[`(uv)] /U_m² - \overline {uv} /U_{\rm m}^2 is much more sensitive to surface curvature, D/W.     

A constitutive model for non-Newtonian materials based on the persistence-of-straining tensor

We present a constitutive equation for non-Newtonian materials which is capable of predicting, independently, steady state rheological material functions both in shear and in extension. The basic assumption is that the extra-stress tensor is a function of both the rate-of-strain tensor, D, and the persistence-of-straining tensor, -\boldsymbol{P}=\boldsymbol{D}\overline{\boldsymbol{W}}-\overline{\boldsymbol {W}}\boldsymbol{D}, introduced in Thompson and de Souza Mendes (Int. J. Eng. Sci. 43(1–2):79–105, 2005). The resulting equation falls within the category of constitutive equations of the form t=t(D,[`(W)])\boldsymbol{\tau}=\boldsymbol{\tau}(\boldsymbol {D},\overline{\boldsymbol{W}}), with the advantage of eliminating the undesirable stress jumps that may occur when [`(W)]\overline {\boldsymbol{W}} becomes locally undetermined. We also show that this formulation is not restricted to motions with constant relative principle stretch history (MWCRPSH), in contrast to what is suggested in the literature. The same basis of tensors that comes from representation theorems also arises from an elastic constitutive equation based on the difference between the Jauman and the Harnoy convected time derivatives, in the limit of small values of the Deborah number.     

Lattice Boltzmann simulation of heat transfer and fluid flow in a microchannel with nanofluids

Mathematical modeling is performed to simulate forced convection flow of 47 nm- Al₂O₃/water nanofluids in a microchannel using the lattice Boltzmann method (LBM). Single channel flow and conjugate heat transfer problem are taken into consideration and the heat transfer rate using a nanofluid is examined. Simulations are conducted at low Reynolds numbers (2 ≤ Re ≤ 16). The computed average Nusselt number, which is associated with the thermal conductivity of nanofluid, is in the range of 0.6 ￡ [`(Nu)] ￡ 13 0.6 \le \overline{Nu} \le 13 . Results indicate that the average Nusselt number increases with the increase of Reynolds number and particle volume concentration. The fluid temperature distribution is more uniform with the use of nanofluid than that of pure water. Furthermore, great deviations of computed Nusselt numbers using different models associated with the physical properties of a nanofluid are revealed. The results of LBM agree well with the classical CFD method for predictions of flow and heat transfer in a single channel and a microchannel heat sink concerning the conjugate heat transfer problem, and consequently LBM is robust and promising for practical applications.     

Boundary Regularity in Variational Problems

We prove that, if ${u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N}We prove that, if u : W ì \mathbbRⁿ ? \mathbbR^N{u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N} is a solution to the Dirichlet variational problem

minwòW F(x, w, Dw) dx     subject  to     w o u0  on  ?W,\mathop {\rm min}\limits_{w}\int_{\Omega} F(x, w, Dw)\,{\rm d}x \quad {\rm subject \, to} \quad w \equiv u_0\; {\rm on}\;\partial \Omega,      9. On the Reynolds number scaling of vorticity production at no-slip walls during vortex-wall collisions      G. H. Keetels  W. Kramer  H. J. H. Clercx  G. J. F. van Heijst 《Theoretical and Computational Fluid Dynamics》2011,25(5):293-300 Recently, numerical studies revealed two different scaling regimes of the peak enstrophy Z and palinstrophy P during the collision of a dipole with a no-slip wall [Clercx and van Heijst, Phys. Rev. E 65, 066305, 2002]: Z μ Re0.8{Z\propto{\rm Re}^{0.8}} and P μ Re2.25{P\propto {\rm Re}^{2.25}} for 5 × 102 ≤ Re ≤ 2 × 104 and Z μ Re0.5{Z\propto{\rm Re}^{0.5}} and P μ Re1.5{P\propto{\rm Re}^{1.5}} for Re ≥ 2 × 104 (with Re based on the velocity and size of the dipole). A critical Reynolds number Re c (here, Rec ? 2×104{{\rm Re}_c\approx 2\times 10^4}) is identified below which the interaction time of the dipole with the boundary layer depends on the kinematic viscosity ν. The oscillating plate as a boundary-layer problem can then be used to mimick the vortex-wall interaction and the following scaling relations are obtained: Z μ Re3/4, P μ Re9/4{Z\propto{\rm Re}^{3/4}, P\propto {\rm Re}^{9/4}} , and dP/dt μ Re11/4{\propto {\rm Re}^{11/4}} in agreement with the numerically obtained scaling laws. For Re ≥ Re c the interaction time of the dipole with the boundary layer becomes independent of the kinematic viscosity and, applying flat-plate boundary-layer theory, this yields: Z μ Re1/2{Z\propto{\rm Re}^{1/2}} and P μ Re3/2{P\propto {\rm Re}^{3/2}}.      10. Asymptotics for Turbulent Flame Speeds of the Viscous G-Equation Enhanced by Cellular and Shear Flows      Yu-Yu?Liu  Jack?XinEmail author  Yifeng?Yu 《Archive for Rational Mechanics and Analysis》2011,202(2):461-492 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 T. An important problem in turbulent combustion theory is to study properties of s T, in particular how s T 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 $\bar H(A,d)\leqq C(d) \quad \text{for all}\ d >0 ,$\bar H(A,d)\leqq C(d) \quad \text{for all}\ d >0 ,      11. 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},      12. Normal and cross-flow Reynolds stresses: differences between confined and semi-confined flows      Matthias H. Buschmann  Mohamed Gad-el-Hak 《Experiments in fluids》2010,49(1):213-223 Understanding turbulent wall-bounded flows remains an elusive goal. Most turbulent phenomena are non-linear, complex and have broad range of scales that are difficult to completely resolve. Progress is made only in minute steps and enlightening models are rare. Herein, we undertake the effort to bundle several experimental and numerical databases to overcome some of these difficulties and to learn more about the kinematics of turbulent wall-bounded flows. The general scope of the present work is to quantify the characteristics of wall-normal and spanwise Reynolds stresses, which might be different for confined (e.g., pipe) and semi-confined (e.g., boundary layer) flows. In particular, the peak position of wall-normal stress and a shoulder in spanwise stress never described in detail before are investigated using select experimental and direct numerical simulation databases available in the open literature. It is found that the positions of the á v￠2 ñ + \left\langle {v'{^2} } \right\rangle^{ + } -peak in confined and semi-confined flow differ significantly above δ + ≈ 600. A similar behavior is found for the position of the á u￠v￠ ñ + \left\langle {u'v'} \right\rangle^{ + } -peak. The upper end of the logarithmic region seems to be closely related to the position of the á v￠2 ñ + \left\langle {v'{^2} } \right\rangle^{ + } -peak. The á w￠2 ñ + \left\langle {w'{^2} } \right\rangle^{ + } -shoulder is found to be twice as far from the wall than the á v￠2 ñ + \left\langle {v'{^2} } \right\rangle^{ + } -peak. It covers a significantly large portion of the typical zero-pressure-gradient turbulent boundary layer.      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. Well-posedness and Regularity for the Elasticity Equation with Mixed Boundary Conditions on Polyhedral Domains and Domains with Cracks      Anna L. Mazzucato  Victor Nistor 《Archive for Rational Mechanics and Analysis》2010,195(1):25-73 We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}We prove a regularity result for the anisotropic linear elasticity equationP u : = div ( C ·?u) = f{P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f} , with mixed (displacement and traction) boundary conditions on a curved polyhedral domain W ì \mathbbR3{\Omega \subset \mathbb{R}^3} in weighted Sobolev spaces Km+1a+1(W){\mathcal {K}^{m+1}_{a+1}(\Omega)} , for which the weight is given by the distance to the set of edges. In particular, we show that there is no loss of Kma{\mathcal {K}^{m}_{a}} -regularity. Our curved polyhedral domains are allowed to have cracks. We establish a well-posedness result when there are no neighboring traction boundary conditions and |a| < η, for some small η > 0 that depends on P, on the boundary conditions, and on the domain Ω. Our results extend to other strongly elliptic systems and higher dimensions.      15. On the Regularity Conditions of Suitable Weak Solutions of the 3D Navier–Stokes Equations      Dongho Chae 《Journal of Mathematical Fluid Mechanics》2010,12(2):171-180 Let v and ω be the velocity and the vorticity of the a suitable weak solution of the 3D Navier–Stokes equations in a space-time domain containing z0=(x0, t0)z_{0}=(x_{0}, t_{0}), and let Qz0,r = Bx0,r ×(t0 -r2, t0)Q_{z_{0},r}= B_{x_{0},r} \times (t_{0} -r^{2}, t_{0}) be a parabolic cylinder in the domain. We show that if either $\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r})$\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r}) with $\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r})$\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r}) with \frac3g + \frac2a ￡ 2\frac{3}{\gamma} + \frac{2}{\alpha} \leq 2, where Lγ, αx,t denotes the Serrin type of class, then z0 is a regular point for ν. This refines previous local regularity criteria for the suitable weak solutions.      16. On the Schrödinger–Poisson–Slater System: Behavior of Minimizers, Radial and Nonradial Cases      David Ruiz 《Archive for Rational Mechanics and Analysis》2010,198(1):349-368 This paper is motivated by the study of a version of the so-called Schrödinger–Poisson–Slater problem: $- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,$ where ${u \in H^{1}(\mathbb {R}^3)}This paper is motivated by the study of a version of the so-called Schr?dinger–Poisson–Slater problem: - Du + wu + l( u2 *\frac1|x| ) u=|u|p-2u,- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,      17. Hydromagnetic stagnation point flow of a viscous fluid over?a?stretching or shrinking sheet      Robert A. Van Gorder  K. Vajravelu  I. Pop 《Meccanica》2012,47(1):31-50 We establish the existence and uniqueness results over the semi-infinite interval [0,∞) for a class of nonlinear third order ordinary differential equations of the form lf"￠( h) + f( h)f"( h) - ( f￠( h) )2 - Mf￠( h)    + C(C + M ) = 0,f( 0 ) = s ,       f￠( 0 ) = c,       limh? ￥ f￠( h) = C.\begin{array}{l}f'( \eta) + f( \eta)f'( \eta) - ( f'( \eta) )^{2} - Mf'( \eta)\\[6pt]\quad {}+ C(C + M ) = 0,\\[6pt]f( 0 ) = s ,\qquad f'( 0 ) = \chi ,\qquad \displaystyle\lim\limits_{\eta \to \infty} f'( \eta) = C.\end{array}      18. Symmetries of center singularities of plane vector fields      S. I. Maksymenko 《Nonlinear Oscillations》2010,13(2):196-227 Let D2 ì \mathbbR2 {D^2} \subset {\mathbb{R}^2} be a closed unit 2-disk centered at the origin O ? \mathbbR2 O \in {\mathbb{R}^2} and let F be a smooth vector field such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus, topologically O is a “center” singularity. Let q:D2\{ O } ? ( 0, + ￥ ) \theta :D2\backslash \left\{ O \right\} \to \left( {0, + \infty } \right) be the function associating with each z ≠ O its period with respect to F. In general, such a function cannot be even continuously defined at O. Let also D+ (F) {\mathcal{D}^{+} }(F) be the group of diffeomorphisms of D 2 that preserve orientation and leave invariant each orbit of F. It is proved that θ smoothly extends to all of D 2 if and only if the 1-jet of F at O is a “rotation,” i.e., j1F(O) = - y\frac??x + x\frac??y {j^1}F(O) = - y\frac{\partial }{{\partial x}} + x\frac{\partial }{{\partial y}} . Then D+ (F) {\mathcal{D}^{+} }(F) is homotopy equivalent to a circle.      19. Global Nonexistence Theorems for a Class of Evolution Equations with Dissipation   总被引：4，自引：0，他引：4   Enzo Vitillaro 《Archive for Rational Mechanics and Analysis》1999,149(2):155-182 We study abstract evolution equations with nonlinear damping terms and source terms, including as a particular case a nonlinear wave equation of the type $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea where 0 < T ￡ ￥0\Omega is a bounded regular open subset of \mathbbRn\mathbb{R}^n, n 3 1n\ge 1, b,c > 0b,c>0, p > 2p>2, m > 1m>1. We prove a global nonexistence theorem for positive initial value of the energy when 1 < m < p,    2 < p ￡ \frac2nn-2. 1-Laplacian operator, q > 1q>1.      20. 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.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

On the Reynolds number scaling of vorticity production at no-slip walls during vortex-wall collisions

Recently, numerical studies revealed two different scaling regimes of the peak enstrophy Z and palinstrophy P during the collision of a dipole with a no-slip wall [Clercx and van Heijst, Phys. Rev. E 65, 066305, 2002]: Z μ Re^0.8{Z\propto{\rm Re}^{0.8}} and P μ Re^2.25{P\propto {\rm Re}^{2.25}} for 5 × 10² ≤ Re ≤ 2 × 10⁴ and Z μ Re^0.5{Z\propto{\rm Re}^{0.5}} and P μ Re^1.5{P\propto{\rm Re}^{1.5}} for Re ≥ 2 × 10⁴ (with Re based on the velocity and size of the dipole). A critical Reynolds number Re _c (here, Re_c ? 2×10⁴{{\rm Re}_c\approx 2\times 10^4}) is identified below which the interaction time of the dipole with the boundary layer depends on the kinematic viscosity ν. The oscillating plate as a boundary-layer problem can then be used to mimick the vortex-wall interaction and the following scaling relations are obtained: Z μ Re^3/4, P μ Re^9/4{Z\propto{\rm Re}^{3/4}, P\propto {\rm Re}^{9/4}} , and dP/dt μ Re^11/4{\propto {\rm Re}^{11/4}} in agreement with the numerically obtained scaling laws. For Re ≥ Re _c the interaction time of the dipole with the boundary layer becomes independent of the kinematic viscosity and, applying flat-plate boundary-layer theory, this yields: Z μ Re^1/2{Z\propto{\rm Re}^{1/2}} and P μ Re^3/2{P\propto {\rm Re}^{3/2}}.     

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

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},     

Normal and cross-flow Reynolds stresses: differences between confined and semi-confined flows

Understanding turbulent wall-bounded flows remains an elusive goal. Most turbulent phenomena are non-linear, complex and have broad range of scales that are difficult to completely resolve. Progress is made only in minute steps and enlightening models are rare. Herein, we undertake the effort to bundle several experimental and numerical databases to overcome some of these difficulties and to learn more about the kinematics of turbulent wall-bounded flows. The general scope of the present work is to quantify the characteristics of wall-normal and spanwise Reynolds stresses, which might be different for confined (e.g., pipe) and semi-confined (e.g., boundary layer) flows. In particular, the peak position of wall-normal stress and a shoulder in spanwise stress never described in detail before are investigated using select experimental and direct numerical simulation databases available in the open literature. It is found that the positions of the á v￠² ñ ⁺ \left\langle {v'{^2} } \right\rangle^{ + } -peak in confined and semi-confined flow differ significantly above δ ⁺ ≈ 600. A similar behavior is found for the position of the á u￠v￠ ñ ⁺ \left\langle {u'v'} \right\rangle^{ + } -peak. The upper end of the logarithmic region seems to be closely related to the position of the á v￠² ñ ⁺ \left\langle {v'{^2} } \right\rangle^{ + } -peak. The á w￠² ñ ⁺ \left\langle {w'{^2} } \right\rangle^{ + } -shoulder is found to be twice as far from the wall than the á v￠² ñ ⁺ \left\langle {v'{^2} } \right\rangle^{ + } -peak. It covers a significantly large portion of the typical zero-pressure-gradient turbulent boundary layer.     

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).     

Well-posedness and Regularity for the Elasticity Equation with Mixed Boundary Conditions on Polyhedral Domains and Domains with Cracks

We prove a regularity result for the anisotropic linear elasticity equation ${P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f}We prove a regularity result for the anisotropic linear elasticity equationP u : = div ( C ·?u) = f{P u := {\rm div} \left( \boldmath\mathsf{C} \cdot \nabla u\right) = f} , with mixed (displacement and traction) boundary conditions on a curved polyhedral domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} in weighted Sobolev spaces K^m+1_a+1(W){\mathcal {K}^{m+1}_{a+1}(\Omega)} , for which the weight is given by the distance to the set of edges. In particular, we show that there is no loss of K^m_a{\mathcal {K}^{m}_{a}} -regularity. Our curved polyhedral domains are allowed to have cracks. We establish a well-posedness result when there are no neighboring traction boundary conditions and |a| < η, for some small η > 0 that depends on P, on the boundary conditions, and on the domain Ω. Our results extend to other strongly elliptic systems and higher dimensions.     

On the Regularity Conditions of Suitable Weak Solutions of the 3D Navier–Stokes Equations

Let v and ω be the velocity and the vorticity of the a suitable weak solution of the 3D Navier–Stokes equations in a space-time domain containing z₀=(x₀, t₀)z_{0}=(x_{0}, t_{0}), and let Q_z0,r = B_x0,r ×(t₀ -r², t₀)Q_{z_{0},r}= B_{x_{0},r} \times (t_{0} -r^{2}, t_{0}) be a parabolic cylinder in the domain. We show that if either $\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r})$\nu \times \frac{\omega}{|\omega|} \in L^{\gamma,\alpha}_{x,t}(Q_{z_{0},r}) with $\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r})$\frac{3}{\gamma} + \frac{2}{\alpha} \leq 1, {\rm or} \omega \times \frac{\nu} {|\nu|} \in L^{\gamma,\alpha}_{x,t} (Q_{z_{0},r}) with \frac3g + \frac2a ￡ 2\frac{3}{\gamma} + \frac{2}{\alpha} \leq 2, where L^{γ, α}_x,t denotes the Serrin type of class, then z₀ is a regular point for ν. This refines previous local regularity criteria for the suitable weak solutions.     

On the Schrödinger–Poisson–Slater System: Behavior of Minimizers, Radial and Nonradial Cases

This paper is motivated by the study of a version of the so-called Schrödinger–Poisson–Slater problem: $- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,$ where ${u \in H^{1}(\mathbb {R}^3)}This paper is motivated by the study of a version of the so-called Schr?dinger–Poisson–Slater problem:

- Du + wu + l( u2 *\frac1|x| ) u=|u|p-2u,- \Delta u + \omega u + \lambda \left( u^2 \star \frac{1}{|x|} \right) u=|u|^{p-2}u,      17. Hydromagnetic stagnation point flow of a viscous fluid over?a?stretching or shrinking sheet      Robert A. Van Gorder  K. Vajravelu  I. Pop 《Meccanica》2012,47(1):31-50 We establish the existence and uniqueness results over the semi-infinite interval [0,∞) for a class of nonlinear third order ordinary differential equations of the form lf"￠( h) + f( h)f"( h) - ( f￠( h) )2 - Mf￠( h)    + C(C + M ) = 0,f( 0 ) = s ,       f￠( 0 ) = c,       limh? ￥ f￠( h) = C.\begin{array}{l}f'( \eta) + f( \eta)f'( \eta) - ( f'( \eta) )^{2} - Mf'( \eta)\\[6pt]\quad {}+ C(C + M ) = 0,\\[6pt]f( 0 ) = s ,\qquad f'( 0 ) = \chi ,\qquad \displaystyle\lim\limits_{\eta \to \infty} f'( \eta) = C.\end{array}      18. Symmetries of center singularities of plane vector fields      S. I. Maksymenko 《Nonlinear Oscillations》2010,13(2):196-227 Let D2 ì \mathbbR2 {D^2} \subset {\mathbb{R}^2} be a closed unit 2-disk centered at the origin O ? \mathbbR2 O \in {\mathbb{R}^2} and let F be a smooth vector field such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus, topologically O is a “center” singularity. Let q:D2\{ O } ? ( 0, + ￥ ) \theta :D2\backslash \left\{ O \right\} \to \left( {0, + \infty } \right) be the function associating with each z ≠ O its period with respect to F. In general, such a function cannot be even continuously defined at O. Let also D+ (F) {\mathcal{D}^{+} }(F) be the group of diffeomorphisms of D 2 that preserve orientation and leave invariant each orbit of F. It is proved that θ smoothly extends to all of D 2 if and only if the 1-jet of F at O is a “rotation,” i.e., j1F(O) = - y\frac??x + x\frac??y {j^1}F(O) = - y\frac{\partial }{{\partial x}} + x\frac{\partial }{{\partial y}} . Then D+ (F) {\mathcal{D}^{+} }(F) is homotopy equivalent to a circle.      19. Global Nonexistence Theorems for a Class of Evolution Equations with Dissipation   总被引：4，自引：0，他引：4   Enzo Vitillaro 《Archive for Rational Mechanics and Analysis》1999,149(2):155-182 We study abstract evolution equations with nonlinear damping terms and source terms, including as a particular case a nonlinear wave equation of the type $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea where 0 < T ￡ ￥0\Omega is a bounded regular open subset of \mathbbRn\mathbb{R}^n, n 3 1n\ge 1, b,c > 0b,c>0, p > 2p>2, m > 1m>1. We prove a global nonexistence theorem for positive initial value of the energy when 1 < m < p,    2 < p ￡ \frac2nn-2. 1-Laplacian operator, q > 1q>1.      20. 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.

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Hydromagnetic stagnation point flow of a viscous fluid over?a?stretching or shrinking sheet

We establish the existence and uniqueness results over the semi-infinite interval [0,∞) for a class of nonlinear third order ordinary differential equations of the form

Symmetries of center singularities of plane vector fields

Let D² ì \mathbbR² {D^2} \subset {\mathbb{R}^2} be a closed unit 2-disk centered at the origin O ? \mathbbR² O \in {\mathbb{R}^2} and let F be a smooth vector field such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus, topologically O is a “center” singularity. Let q:D2\{ O } ? ( 0, + ￥ ) \theta :D2\backslash \left\{ O \right\} \to \left( {0, + \infty } \right) be the function associating with each z ≠ O its period with respect to F. In general, such a function cannot be even continuously defined at O. Let also D⁺ (F) {\mathcal{D}^{+} }(F) be the group of diffeomorphisms of D ² that preserve orientation and leave invariant each orbit of F. It is proved that θ smoothly extends to all of D ² if and only if the 1-jet of F at O is a “rotation,” i.e., j¹F(O) = - y\frac??x + x\frac??y {j^1}F(O) = - y\frac{\partial }{{\partial x}} + x\frac{\partial }{{\partial y}} . Then D⁺ (F) {\mathcal{D}^{+} }(F) is homotopy equivalent to a circle.     

Global Nonexistence Theorems for a Class of Evolution Equations with Dissipation

We study abstract evolution equations with nonlinear damping terms and source terms, including as a particular case a nonlinear wave equation of the type $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, &;(t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, &;(t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es&; \ea where 0 < T ￡ ￥0\Omega is a bounded regular open subset of \mathbbRⁿ\mathbb{R}^n, n 3 1n\ge 1, b,c > 0b,c>0, p > 2p>2, m > 1m>1. We prove a global nonexistence theorem for positive initial value of the energy when 1 < m < p,    2 < p ￡ \frac2nn-2. 1-Laplacian operator, q > 1q>1.     

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.