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.     

Natural Convection Induced by a Centrifugal Force Field in a Horizontal Annular Porous Layer Saturated with a Binary Fluid

The Darcy Model with the Boussinesq approximation is used to study natural convection in a horizontal annular porous layer filled with a binary fluid, under the influence of a centrifugal force field. Neumann boundary conditions for temperature and concentration are applied on the inner and outer boundary of the enclosure. The governing parameters for the problem are the Rayleigh number, Ra, the Lewis number, Le, the buoyancy ratio, j{\varphi } , the radius ratio of the cavity, R, the normalized porosity, e{\varepsilon } , and parameter a defining double-diffusive convection (a = 0) or Soret induced convection (a = 1). For convection in a thin annular layer (R → 1), analytical solutions for the stream function, temperature and concentration fields are obtained using a concentric flow approximation and an integral form of the energy equation. The critical Rayleigh number for the onset of supercritical convection is predicted explicitly by the present model. Also, results are obtained from the analytical model for finite amplitude convection for which the flow and heat and mass transfer are presented in terms of the governing parameters of the problem. Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters. A good agreement is observed between the analytical model and the numerical simulations.     

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).     

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

Spectral Stability of Small-Amplitude Viscous Shock Waves in Several Space Dimensions

A planar viscous shock profile of a hyperbolic–parabolic system of conservation laws is a steady solution in a moving coordinate frame. The asymptotic stability of viscous profiles and the related vanishing-viscosity limit are delicate questions already in the well understood case of one space dimension and even more so in the case of several space dimensions. It is a natural idea to study the stability of viscous profiles by analyzing the spectrum of the linearization about the profile. The Evans function method provides a geometric dynamical-systems framework to study the eigenvalue problem. In this approach eigenvalues correspond to zeros of an essentially analytic function E(rl,rw){\mathcal{E}(\rho\lambda,\rho\omega)} which detects nontrivial intersections of the so-called stable and unstable spaces, that is, spaces of solutions that decay on one (“−∞”) or the other side (“ + ∞”) of the shock wave, respectively. In a series of pioneering papers, Kevin Zumbrun and collaborators have established in various contexts that spectral stability, that is, the non-vanishing of E(rl,rw){\mathcal{E}(\rho\lambda,\rho\omega)} and the non-vanishing of the Lopatinski–Kreiss–Majda function Δ(λ,ω), imply nonlinear stability of viscous shock profiles in several space dimensions. In this paper we show that these conditions hold true for small amplitude extreme shocks under natural assumptions. This is done by exploiting the slow-fast nature of the small-amplitude limit, which was used in a previous paper by the authors to prove spectral stability of small-amplitude shock waves in one space dimension. Geometric singular perturbation methods are applied to decompose the stable and unstable spaces into subbundles with good control over their limiting behavior. Three qualitatively different regimes are distinguished that relate the small strength e{\epsilon} of the shock wave to appropriate ranges of values of the spectral parameters (ρλ, ρ ω). Various rescalings are used to overcome apparent degeneracies in the problem caused by loss of hyperbolicity or lack of transversality.     

Natural frequencies of thick, complete, circular rings with an elliptical or circular cross-section from a three-dimensional theory

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (circumferentially closed), circular rings with an elliptical or circular cross-section. Displacement components u_r, u_θ, and u_z in the radial, circumferential, and axial directions, respectively, are taken to be periodic in θ and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the circular rings are formulated, and upper-bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the rings. Novel numerical results are presented for the circular rings having an elliptical cross-section based upon 3D theory. Comparisons are also made between the frequencies from the present 3D Ritz method and ones obtained from thin and thick ring theories, experiments, and other 3D methods.     

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 {SL(2, \mathbb R)} Valued Smooth Proximal Cocycles and Cocycles with Positive Lyapunov Exponents Over Irrational Rotation Flows

Consider the class of C ^r -smooth SL(2, \mathbb R){SL(2, \mathbb R)} valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition: |a-\fracp_nq_n| ￡ C exp (-q^r+1+k_n)\left|\alpha-\frac{p_n}{q_n}\right| \leq C {\rm exp} (-q^{r+1+\kappa}_{n}), where C >  0 and 0 < k < 1{0 < \kappa <1 } are some constants and \fracP_nq_n{\frac{P_n}{q_n}} is some sequence of irreducible fractions.     

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.     

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,∞).     

Maximal L p – L q -Estimates for the Stokes Equation: a Short Proof of Solonnikov’s Theorem

Introducing a new localization method involving Bogovskiĭ's operator we give a short and new proof for maximal L^p – L^q-estimates for the solution of the Stokes equation. Moreover, it is shown that, up to constants, the Stokes operator is an R{\mathcal{R}}-sectorial operator in L^p_s(W)L^{p}_{\sigma}(\Omega), 1 < p < ￥1 < p < \infty, of R{\mathcal{R}}-angle 0, for bounded or exterior domains of Ω.     

Complete Orbits for Twist Maps on the Plane: Extensions and Applications

Twist maps (θ ₁, r ₁) = f (θ, r) on the plane are considered which do not exhibit any kind of periodicity in their dependence on θ. Some general results are obtained which typically yield the existence of infinitely many complete and bounded orbits. Examples that can be treated with this theory include oscillators of the type [(x)\ddot]+V￠(x)=p(t){\ddot{x}+V'(x)=p(t)} under appropriate hypotheses, the bouncing ball system, and the standard map.     

Lower Bound for the Energy of Bloch Walls in Micromagnetics

We study a two-dimensional nonconvex and nonlocal energy in micromagnetics defined over S ²-valued vector fields. This energy depends on two small parameters, β and e{\varepsilon} , penalizing the divergence of the vector field and its vertical component, respectively. Our objective is to analyze the asymptotic regime b << e << 1{\beta \ll \varepsilon \ll 1} through the method of Γ-convergence. Finite energy configurations tend to become divergence-free and in-plane in the magnetic sample except in some small regions of typical width e{\varepsilon} (called Bloch walls) where the magnetization connects two directions on S ². We are interested in quantifying the limit energy of the transition layers in terms of the jump size between these directions. For one-dimensional transition layers, we show by Γ-convergence analysis that the exact line density of the energy is quadratic in the jump size. We expect the same behaviour for the two-dimensional model. In order to prove that, we investigate the concept of entropies. In the prototype case of a periodic strip, we establish a quadratic lower bound for the energy with a non-optimal constant. Then we introduce and study a special class of Lipschitz entropies and obtain lower bounds coinciding with the one-dimensional Γ-limit in some particular cases. Finally, we show that entropies are not appropriate in general for proving the expected sharp lower bound.     

Energy Transport in Stochastically Perturbed Lattice Dynamics

We consider lattice dynamics with a small stochastic perturbation of order ${\varepsilon}We consider lattice dynamics with a small stochastic perturbation of order e{\varepsilon} and prove that for a space–time scale of order e^-1{\varepsilon^{-1}} the local spectral density (Wigner function) evolves according to a linear transport equation describing inelastic collisions. For an energy and momentum conserving chain, the transport equation predicts a slow decay, as 1/?t{1/\sqrt t} , for the energy current correlation in equilibrium. This is in agreement with previous studies using a different method.     

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.     

Sensitivity Study of Simulation Parameters Controlling CO<Subscript>2</Subscript> Trapping Mechanisms in Saline Formations

The primary purpose of this study is to understand quantitative characteristics of mobile, residual, and dissolved CO₂ trapping mechanisms within ranges of systematic variations in different geologic and hydrologic parameters. For this purpose, we conducted an extensive suite of numerical simulations to evaluate the sensitivities included in these parameters. We generated two-dimensional numerical models representing subsurface porous media with various permutations of vertical and horizontal permeability (k _v and k _h), porosity (f{\phi}), maximum residual CO₂ saturation (S_gr^max{S_{\rm gr}^{\max}}), and brine density (ρ _br). Simulation results indicate that residual CO₂ trapping increases proportionally to k_v, k_h, S_gr^max{k_{\rm v}, k_{\rm h}, S_{\rm gr}^{\max}} and ρ _br but is inversely proportional to f.{\phi.} In addition, the amount of dissolution-trapped CO₂ increases with k _v and k _h, but does not vary with f{\phi } , and decreases with S_gr^max{S_{\rm gr}^{\max}} and ρ _br. Additionally, the distance of buoyancy-driven CO₂ migration increases proportionally to k _v and ρ _br only and is inversely proportional to k_h, f{k_{\rm h}, \phi } , and S_gr^max{S_{\rm gr}^{\max}} . These complex behaviors occur because the chosen sensitivity parameters perturb the distances of vertical and horizontal CO₂ plume migration, pore volume size, and fraction of trapped CO₂ in both pores and formation fluids. Finally, in an effort to characterize complex relationships among residual CO₂ trapping and buoyancy-driven CO₂ migration, we quantified three characteristic zones. Zone I, expressing the variations of S_gr^max{S_{\rm gr}^{\max}} and k _h, represents the optimized conditions for geologic CO₂ sequestration. Zone II, showing the variation of f{\phi} , would be preferred for secure CO₂ sequestration since CO₂ has less potential to escape from the target formation. In zone III, both residual CO₂ trapping and buoyancy-driven migration distance increase with k _v and ρ _br.     

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}}.     

Stationary Flows of Shear Thickening Fluids in 2D

We investigate the steady flow of a shear thickening generalized Newtonian fluid under homogeneous boundary conditions on a domain in \mathbbR²{\mathbb{R}^{2}}. We assume that the stress tensor is generated by a potential of the form H = h (|e(u)|){H = h (|\varepsilon (u)|)}, e(u){\varepsilon (u)} denoting the symmetric part of the velocity gradient. We prove the existence of strong solutions for a large class of functions h having the property that h′ (t)/t increases (shear thickening case).     

On Saint-Venant's Principle for a Homogeneous Elastic Arch-Like Region

In this paper, we consider a two-dimensional homogeneous isotropic elastic material state in the arch-like region a ≤ r ≤ b, 0 ≤ θ ≤ α, where (r, θ) denote plane polar coordinates. We assume that three of the edges r = a, r = b, θ = α are traction-free, while the edge θ = 0 is subjected to an (in plane) self-equilibrated load. We define an appropriate measure for the Airy stress function φ and then we establish a clear relationship with the Saint-Venant's principle on such regions. We introduce a cross-sectional integral function I(θ) which is shown to be a convex function and satisfies a second-order differential inequality. Consequently, we establish a version of the Saint-Venant principle for such a curvilinear strip, without requiring of any condition upon the dimensions of the arch-like region.     

Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions

In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem, neither a maximum principle nor a comparison principle or—equivalently—a positivity preserving property is available. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being reasonably written as a system of second order boundary value problems. It is shown that, on the other hand, for bounded smooth domains W ì \mathbbRⁿ{\Omega \subset\mathbb{R}^n} , the negative part of the corresponding Green’s function is “small” when compared with its singular positive part, provided n\geqq 3{n\geqq 3} . Moreover, the biharmonic Green’s function in balls B ì \mathbbRⁿ{B\subset\mathbb{R}^n} under Dirichlet (that is, clamped) boundary conditions is known explicitly and is positive. It has been known for some time that positivity is preserved under small regular perturbations of the domain, if n = 2. In the present paper, such a stability result is proved for n\geqq 3{n\geqq 3} .