Simulations of Three-Dimensional Turbulent Mixing for Schmidt Numbers of the Order 1000

We report basic results from new numerical simulations of passive scalar mixing at Schmidt numbers (Sc) of the order of 1000 in isotropic turbulence. The required high grid-resolution is made possible by simulating turbulence at very low Reynolds numbers, which nevertheless possesses universality in dissipative scales of motion. The results obtained are qualitatively consistent with those based on another study (Yeung et al., Phys. Fluids 14 (2002) 4178-4191) with a less extended Schmidt number range and a higher Reynolds number. In the stationary state maintained by a uniform mean scalar gradient, the scalar variance increases slightly with Sc but scalar dissipation is nearly constant. As the Schmidt number increases, there is an increasing trend towards k ^?1 scaling predicted by Batchelor (Batchelor, J. Fluid Mech. 5 (1959) 113-133) for the viscous-convective range of the scalar spectrum; the scalar gradient skewness approaches zero; and the intermittency measured by the scalar gradient flatness approaches its asymptotic state. However, the value of Sc needed for the asymptotic behavior to emerge appears to increase with decreasing Reynolds number of the turbulence. In the viscous-diffusive range, the scalar spectrum is in better agreement with Kraichnan's (Kraichnan., Phys. Fluids 11 (1968) 945-953) result than with Batchelor's.     

Existence of Time-Periodic Solutions for a Magnetoelastic System in Bounded Domains

We investigate the existence of periodic solutions for a semilinear (nonlinearly coupled) magnetoelastic system in bounded, simply connected, three-dimensional domains with boundaries of class C ². The mathematical model includes a nonlinear mechanical dissipation like ρ(u′)=|u′| ^p u′ and a periodic forcing function of period T. We prove the existence of T-periodic weak solutions when p∈[3,4] (p=0 being a simpler case). In the corresponding two-dimensional case, the existence result holds under the assumption that p≥2.     

Analytical Validation of a Continuum Model for Epitaxial Growth with Elasticity on Vicinal Surfaces

Within the context of heteroepitaxial growth of a film onto a substrate, terraces and steps self-organize according to misfit elasticity forces. Discrete models of this behavior were developed by Duport et al. (J Phys I 5:1317–1350, 1995) and Tersoff et al. (Phys Rev Lett 75:2730–2733, 1995). A continuum limit of these was in turn derived by Xiang (SIAM J Appl Math 63:241–258, 2002) (see also the work of Xiang and Weinan Phys Rev B 69:035409-1–035409-16, 2004; Xu and Xiang SIAM J Appl Math 69:1393–1414, 2009). In this paper we formulate a notion of weak solution to Xiang’s continuum model in terms of a variational inequality that is satisfied by strong solutions. Then we prove the existence of a weak solution.     

Existence and Stability of Solutions for Steady Flows of Fibre Suspension Flows

We establish existence, uniqueness, convergence and stability of solutions to the equations of steady flows of fibre suspension flows. The existence of a unique steady solution is proven by using an iterative scheme. One of the restrictions imposed on the data confirms a well known fact proven in Galdi and Reddy (J Non-Newtonian Fluid Mech 83:205–230, 1999), Munganga and Reddy (Math Models Methods Appl Sci 12:1177–1203, 2002) and Munganga et al. (J Non-Newtonian fluid Mech 92:135–150, 2000) that the particle number N _p must be less than 35/2. Exact solutions are calculated for Couette and Poiseuille flows. Solutions of Poiseuille flows are shown to be more accurate than those of Couette flow when a time perturbation is considered.     

A gradient based iterative algorithm for solving structural dynamics model updating problems

The procedure of updating an existing but inaccurate model is an essential step toward establishing an effective model. Updating damping and stiffness matrices simultaneously with measured modal data can be mathematically formulated as following two problems. Problem 1: Let M _a ∈SR ^n×n be the analytical mass matrix, and Λ=diag{λ ₁,…,λ _p }∈C ^p×p , X=[x ₁,…,x _p ]∈C ^n×p be the measured eigenvalue and eigenvector matrices, where rank(X)=p, p<n and both Λ and X are closed under complex conjugation in the sense that $\lambda_{2j} = \bar{\lambda}_{2j-1} \in\nobreak{\mathbf{C}} $ , $x_{2j} = \bar{x}_{2j-1} \in{\mathbf{C}}^{n} $ for j=1,…,l, and λ _k ∈R, x _k ∈R ⁿ for k=2l+1,…,p. Find real-valued symmetric matrices D and K such that M _a XΛ ²+DXΛ+KX=0. Problem 2: Let D _a ,K _a ∈SR ^n×n be the analytical damping and stiffness matrices. Find $(\hat{D}, \hat{K}) \in\mathbf{S}_{\mathbf{E}}$ such that $\| \hat{D}-D_{a} \|^{2}+\| \hat{K}-K_{a} \|^{2}= \min_{(D,K) \in \mathbf{S}_{\mathbf{E}}}(\| D-D_{a} \|^{2} +\|K-K_{a} \|^{2})$ , where S _E is the solution set of Problem 1 and ∥?∥ is the Frobenius norm. In this paper, a gradient based iterative (GI) algorithm is constructed to solve Problems 1 and 2. A sufficient condition for the convergence of the iterative method is derived and the range of the convergence factor is given to guarantee that the iterative solutions consistently converge to the unique minimum Frobenius norm symmetric solution of Problem 2 when a suitable initial symmetric matrix pair is chosen. The algorithm proposed requires less storage capacity than the existing numerical ones and is numerically reliable as only matrix manipulation is required. Two numerical examples show that the introduced iterative algorithm is quite efficient.     

Regularity for Energy-Minimizing Area-Preserving Deformations

In this paper we establish the square integrability of the nonnegative hydrostatic pressure p, that emerges in the minimization problem $$\inf_{\mathcal{K}}\int_{\varOmega}|\nabla \textbf {v}|^2, \quad\varOmega\subset \mathbb {R}^2 $$ as the Lagrange multiplier corresponding to the incompressibility constraint det?v=1 a.e. in Ω. Our method employs the Euler-Lagrange equation for the mollified Cauchy stress C satisfied in the image domain Ω ^?=u(Ω). This allows to construct a convex function ψ, defined in the image domain, such that the measure of the normal mapping of ψ controls the L ² norm of the pressure. As a by-product we conclude that $\textbf {u}\in C^{\frac{1}{2}}_{\textrm {loc}}(\varOmega)$ if the dual pressure (introduced in Karakhanyan, Manuscr. Math. 138:463, 2012) is nonnegative.     

The Rot-Div System in Exterior Domains

The goal of this paper is to reconsider the classical elliptic system rot v =  f, div v =  g in simply connected domains with bounded connected boundaries (bounded and exterior sets). The main result shows solvability of the problem in the maximal regularity regime in the L _p -framework taking into account the optimal/minimal requirements on the smoothness of the boundary. A generalization for the Besov spaces is studied, too, for \({{\bf f} \in \dot B^s_{p,q}(\Omega)}\) for \({-1+\frac 1p < s < \frac 1p}\) . As a limit case we prove the result for \({{\bf f} \in \dot B^0_{3,1}(\Omega)}\) , provided the boundary is merely in \({B^{2-1/3}_{3,1}}\) . The dimension three is distinguished due to the physical interpretation of the system. In other words we revised and extended the classical results of Friedrichs (Commun Pure Appl Math 8;551–590, 1955) and Solonnikov (Zap Nauch Sem LOMI 21:112–158, 1971).     

Non-piezoactivity in piezoacoustics: basic properties and topological features

The existence conditions of zero electric fields E and zero electric displacements D are studied for bulk acoustic waves in piezoelectric crystals. General equations are derived for lines of zero electric fields, E(m)=0, and for specific points m ₀ of vanishing electric displacements, D(m ₀)=0, on the unit sphere of propagation directions m ²=1. The obtained equations are solved for a series of examples of particular crystal symmetry. It is shown that the vectors D _α (m) being generally orthogonal to the wave normal m are characterized by definite orientational singularities in the vicinity of m ₀ and can be described by the Poincaré indices n=0, ±1 or ±2. The algebraic expressions for the indices n are found both for unrestricted anisotropy and for a series of particular cases.     

In the Chapman-Enskog Expansion¶the Burnett Coefficients Satisfy¶the Universal Relation ω3+ω4+θ3= 0

The Chapman–Enskog expansion when applied to a gas of spherical molecules yields formal expressions for the stress deviator P and energy-flux vector q, P=μP ⁽¹⁾+μ² P ⁽²⁾+…, q=μq ⁽¹⁾+μ² q ⁽²⁾+…. The Burnett terms P ⁽²⁾, q ⁽²⁾ depend on 11 coefficients ω _i , 1≦i≦6, θ&; _i , 1≦i≦ 5. This paper shows that ω₃+ω₄+θ₃= 0.     

A Refinement of the Local Serrin-Type Regularity Criterion for a Suitable Weak Solution to the Navier–Stokes Equations

We formulate a new criterion for regularity of a suitable weak solution v to the Navier–Stokes equations at the space-time point (x ₀, t ₀). The criterion imposes a Serrin-type integrability condition on v only in a backward neighbourhood of (x ₀, t ₀), intersected with the exterior of a certain space-time paraboloid with vertex at point (x ₀, t ₀). We make no special assumptions on the solution in the interior of the paraboloid.     

Some Indentities for the Gradient of the Principal Invariants, Traces and Determinants via Grassmann Calculus

Let A be a second order tensor in a finite dimensional space. In this work we determine the gradient of the principal invariants of A and obtain some trace and determinant identities using only some standard rigorous statements concerning Grassmann calculus. We recover some of the results of Dui et al. (J. Elast. 75:193–196, 2004) and of Truesdell and Noll (The Non-linear Field Theories of Mechanics, Springer, Berlin, 2002) and solve an old problem proposed in SIAM Review concerning a determinant identity from a new perspective in a concise and simple manner.     

Compatibility Equations of Nonlinear Elasticity for Non-Simply-Connected Bodies

Compatibility equations of elasticity are almost 150 years old. Interestingly, they do not seem to have been rigorously studied, to date, for non-simply-connected bodies. In this paper we derive necessary and sufficient compatibility equations of nonlinear elasticity for arbitrary non-simply-connected bodies when the ambient space is Euclidean. For a non-simply-connected body, a measure of strain may not be compatible, even if the standard compatibility equations (“bulk” compatibility equations) are satisfied. It turns out that there may be topological obstructions to compatibility; this paper aims to understand them for both deformation gradient F and the right Cauchy-Green strain C = F ^T F. We show that the necessary and sufficient conditions for compatibility of deformation gradient F are the vanishing of its exterior derivative and all its periods, that is, its integral over generators of the first homology group of the material manifold. We will show that not every non-null-homotopic path requires supplementary compatibility equations for F and linearized strain e. We then find both necessary and sufficient compatibility conditions for the right Cauchy-Green strain tensor C for arbitrary non-simply-connected bodies when the material and ambient space manifolds have the same dimensions. We discuss the well-known necessary compatibility equations in the linearized setting and the Cesàro-Volterra path integral. We then obtain the sufficient conditions of compatibility for the linearized strain when the body is not simply-connected. To summarize, the question of compatibility reduces to two issues: i) an integrability condition, which is d(F dX) = 0 for the deformation gradient and a curvature vanishing condition for C, and ii) a topological condition. For F dx this is a homological condition because the equation one is trying to solve takes the form dφ = F dX. For C, however, parallel transport is involved, which means that one needs to solve an equation of the form dR/ ds = RK, where R takes values in the orthogonal group. This is, therefore, a question about an orthogonal representation of the fundamental group, which, as the orthogonal group is not commutative, cannot, in general, be reduced to a homological question.     

On the use of interaction tensors to describe and predict rod interactions in rod suspensions

Recently, Férec et al. (2009a) proposed a model for nondilute rod-like suspensions, where particle interactions are taking into account via a micromechanical approach. The derived governing equation used the well-known second- and fourth-order orientation tensors (a ₂ and a ₄ ) and novel second- and fourth-order interaction tensors (b ₂ and b ₄ ). To completely close the model, it is necessary to express a ₄ , b ₂ , and b ₄ in terms of a ₂ . This paper gives the general framework to elaborate these new relations. Firstly, approximations for b ₂ are developed based on linear combinations of a ₂ and a ₄ . Moreover, a new closure approximation is also derived for b ₄ , based on the orthotropic fitted closure approach. Unknown parameters are determined by a least-square fitting technique with assumed exact solutions constructed from the probability distribution function (PDF). As numerical solutions for the PDF are difficult to obtain given the nonlinearity of the problem, a combination of steady state solutions is used to generate PDF designed to cover uniformly the entire domain of possible orientations. All these proposed approximations are tested against the particle-based simulations in a variety of flow fields. Improvements of the different approximations are observed, and the couple iORW-CO4P3 gives efficient results.     

On the Brézis–Nirenberg Problem

We study the following Brézis–Nirenberg problem (Comm Pure Appl Math 36:437–477, 1983): $-\Delta u=\lambda u+ |u|^{2^\ast-2}u, \quad u\in H_0^1(\Omega),$ where Ω is a bounded smooth domain of R ^N (N ≧ 7) and 2* is the critical Sobolev exponent. We show that, for each fixed λ > 0, this problem has infinitely many sign-changing solutions. In particular, if λ ≧ λ₁, the Brézis–Nirenberg problem has and only has infinitely many sign-changing solutions except zero. The main tool is the estimates of Morse indices of nodal solutions.     

Classification of the structure of positive radial solutions to\Delta u + K(|x|)u^p = 0inR n

Positive radial solutions to the semilinear elliptic equation \(\Delta u + K(|x|)u^p = 0\) inR ⁿ are studied, wherep > 1,n > 2 andK ≧ 0. It is shown that, under a general condition onK(r) andp, the structure of positive radial solutions becomes one of three types. We give sharp criteria to classify the type of the structure, and apply the result to the conformal scalar curvature equation.     

Optimal Regularity and Nondegeneracy of a Free Boundary Problem Related to the Fractional Laplacian

We discuss the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian. This work is related to, but addresses a different problem from, recent work of Caffarelli et al. (J Eur Math Soc (JEMS) 12(5):1151–1179, 2010). A variant of the boundary Harnack inequality is also proved, where it is no longer required that the function be zero along the boundary.     

On the Stokes and Oseen Problems with Singular Data

We consider the steady Stokes and Oseen problems in bounded and exterior domains of ${\mathbb{R}^n}$ of class C ^k-1,1 (n = 2, 3; k ≥ 2). We prove existence and uniqueness of a very weak solution for boundary data a in ${W^{2-k-1/q,q} (\partial\Omega)}$ . If ${\Omega}$ is of class ${C^\infty}$ , we can assume a to be a distribution on ${\partial\Omega}$ .     

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

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

Decay estimates for some semilinear damped hyperbolic problems

Let Ω be a bounded open domain in R ⁿ , g ∶ R → R a non-decreasing continuous function such that g(0)=0 and h ε L _loc ¹ (R⁺; L ²(Ω)). Under suitable assumptions on g and h, the rate of decay of the difference of two solutions is studied for some abstract evolution equations of the general form u ^′′ + Lu + g(u ^′) = h(t,x) as t → + ∞. The results, obtained by use of differential inequalities, can be applied to the case of the semilinear wave equation $$u_u - \Delta u + g{\text{(}}u_t {\text{) = }}h{\text{ in }}R^ + \times \Omega ,{\text{ }}u = {\text{0 on }}R^ + \times \partial \Omega$$ in R ⁺×Ω, u=0 on R ⁺×?Ω. For instance if \(g(s) = c\left| s \right|^{p - 1} s + d\left| s \right|^{q - 1} s\) with c, d>0 and 1 < p≦q, (n?2)q≦n+2, then if \(h \in L^\infty (R + ;L^2 (\Omega ))\) , all solutions are bounded in the energy space for t≧0 and if u, v are two such solutions, the energy norm of u(t) ? v(t) decays like t ^?1/p?1 as t → + ∞.