Eulerian–Lagrangian bridge for the energy and dissipation spectra in isotropic turbulence

We study, numerically and analytically, the relationship between the Eulerian spectrum of kinetic energy, E _E(k, t), in isotropic turbulence and the corresponding Lagrangian frequency energy spectrum, E _L(ω, t), for which we derive an evolution equation. Our DNS results show that not only E _L(ω, t) but also the Lagrangian frequency spectrum of the dissipation rate ${\varepsilon_{\rm L} (\omega, t)}$ has its maximum at low frequencies (about the turnover frequency of energy-containing eddies) and decays exponentially at large frequencies ω (about a half of the Kolmogorov microscale frequency) for both stationary and decaying isotropic turbulence. Our main analytical result is the derivation of equations that bridge the Eulerian and Lagrangian spectra and allow the determination of the Lagrangian spectrum, E _L (ω) for a given Eulerian spectrum, E _E (k), as well as the Lagrangian dissipation, ${\varepsilon_{\rm L}(\omega)}$ , for a given Eulerian counterpart, ${\varepsilon_{\rm E} (k)=2\nu k^2 E_{\rm E}(k)}$ . These equations were derived from the Navier–Stokes equations in the sweeping-free coordinate system (intermediate between the Eulerian and Lagrangian frameworks) which eliminates the effect of the kinematic sweeping of the small eddies by the larger eddies. We show that both analytical relationships between E _L (ω) and E _E (k) and between ${\varepsilon_{\rm L} (\omega)}$ and ${\varepsilon_{\rm E} (k)}$ are in very good quantitative agreement with our DNS results and explain how ${\varepsilon_{\rm L} (\omega, t)}$ has its maximum at low frequencies and decays exponentially at large frequencies.     

An experimental study on turbulence modification in the near-wall boundary layer of a dilute gas-particle channel flow

Turbulence modifications of a dilute gas-particle flow are experimentally investigated in the lower boundary layer of a horizontal channel by means of a simultaneous two-phase PIV measurement technique. The measurements are conducted in the near-wall region with y ⁺?<?250 at Re _τ (based on the wall friction velocity u _τ and half channel height h)?=?430. High spatial resolution and small interrogation window are used to minimize the PIV measurement uncertainty due to the velocity gradient near the wall. Polythene beads with the diameter of 60?μm (d _p ⁺ ?=?1.71, normalized by the fluid kinematic viscosity ν and u _τ) are used as dispersed phase, and three low mass loading ratios (Φ _m ) ranging from 10^?4 to 10^?3 are tested. It is found that the addition of the particles noticeably modifies the mean velocity and turbulent intensities of the gas-phase, as well as the turbulence coherent structures, even at Φ _m ?=?0.025?%. Particle inertia changes the viscous sublayer of the gas turbulence with a smaller thickness and a larger streamwise velocity gradient, which increases the peak value of the streamwise fluctuation velocity ( $ u_{\text{rms}}^{ + } $ ) of the gas-phase with its location shifting to the wall. Particle sedimentation increases the roughness of the bottom wall, which significantly increases the wall-normal fluctuation velocity ( $ v_{\text{rms}}^{ + } $ ) and Reynolds shear stress ( $ - \langle u^{ \prime } v^{\prime } \rangle^{ + } $ ) of the gas-phase in the inner region of the boundary layer (y ⁺?<?10). Under effect of particle–wall collision, the Q2 events (ejections) of the gas-phase are slightly increased by particles, while the Q4 events (sweeps) are obviously decreased. The spatial scale of the coherent structures near the wall shrinks remarkably with the presence of the particles, which may be attributed to the intensified crossing-trajectory effects due to particle saltation near the bottom wall. Meanwhile, the $ v_{\text{rms}}^{ + } $ and $ - \langle u^{ \prime } v^{\prime } \rangle^{ + } $ of the gas-phase are significantly reduced in the outer region of the boundary layer (y ⁺?>?20).     

Dynamical Structure of Some Nonlinear Degenerate Diffusion Equations

We consider degenerate reaction diffusion equations of the form u _t ?=???u ^m ?+?f(x, u), where f(x, u) ~ a(x)u ^p with 1??? p m. We assume that a(x)?>?0 at least in some part of the spatial domain, so that ${u \equiv 0}$ is an unstable stationary solution. We prove that the unstable manifold of the solution ${u \equiv 0 }$ has infinite Hausdorff dimension, even if the spatial domain is bounded. This is in marked contrast with the case of non-degenerate semilinear equations. The above result follows by first showing the existence of a solution that tends to 0 as ${t\to -\infty}$ while its support shrinks to an arbitrarily chosen point x* in the region where a(x)?>?0, then superimposing such solutions, to form a family of solutions of arbitrarily large number of free parameters. The construction of such solutions will be done by modifying self-similar solutions for the case where a is a constant.     

Dynamics as a mechanism preventing the formation of finer and finer microstructure

We study the dynamics of pattern formation in the one-dimensional partial differential equation $$u_u - (W'(u_x ))_x - u_{xxt} + u = 0{\text{ (}}u = u(x,t),{\text{ }}x \in (0,1),{\text{ }}t > 0)$$ proposed recently by Ball, Holmes, James, Pego & Swart [BHJPS] as a mathematical “cartoon” for the dynamic formation of microstructures observed in various crystalline solids. Here W is a double-well potential like 1/4((u _x )² ?1)². What makes this equation interesting and unusual is that it possesses as a Lyapunov function a free energy (consisting of kinetic energy plus a nonconvex “elastic” energy, but no interfacial energy contribution) which does not attain a minimum but favours the formation of finer and finer phase mixtures: $$E[u,u_t ] = \int\limits_0^1 {(\frac{{u_t^2 }}{2} + W(u_x ) + \frac{{u^2 }}{2})dx.}$$ Our analysis of the dynamics confirms the following surprising and striking difference between statics and dynamics, conjectured in [BHJPS] on the basis of numerical simulations of Swart & Holmes [SH]:

?While minimizing the above energy predicts infinitely fine patterns (mathematically: weak but not strong convergence of all minimizing sequences (u _nv_n) of E[u,v] in the Sobolev space W ^{1 p}(0, 1)×L²(0,1)), solutions to the evolution equation of ball et al. typically develop patterns of small but finite length scale (mathematically: strong convergence in W ₁ ^p(0,1)×L²(0,1) of all solutions (u(t),u_t(t)) with low initial energy as time t → ∞).

Moreover, in order to understand the finer details of why the dynamics fails to mimic the behaviour of minimizing sequences and how solutions select their limiting pattern, we present a detailed analysis of the evolution of a restricted class of initial data — those where the strain field u _x has a transition layer structure; our analysis includes proofs that

?at low energy, the number of phases is in fact exactly preserved, that is, there is no nucleation or coarsening

?transition layers lock in and steepen exponentially fast, converging to discontinuous stationary sharp interfaces as time t → ∞

?the limiting patterns — while not minimizing energy globally — are ‘relative minimizers’ in the weak sense of the calculus of variations, that is, minimizers among all patterns which share the same strain interface positions.

     

Phragmén-Lindelöf theorems for nonlinear elliptic equations

We obtain theorems of Phragmén-Lindelöf type for the following classes of elliptic partial differential inequalities in an arbitrary unbounded domain \(\Omega \subseteq \mathbb{R}^n ,{\text{ }}n \geqq 2\) (A.1) $$\sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial x_i }}\left( {a_{ij} 9(x)\frac{{\partial u}}{{\partial xj}}} \right)} + \sum\limits_{i = 1}^n {b_i (x,{\text{ }}u,{\text{ }}\nabla u)\frac{\partial }{{\partial x_i }}} \geqq f(x,{\text{ }}u)$$ where a _ij are elliptic in Ω and b _i ε L^∞(Ω) and where also a _ij are uniformly elliptic and Holder continuous at infinity and b _i = O(|x|⁺¹) as x → ∞; (A.2) $${\text{(A}}{\text{.2) }}\sum\limits_{i,j = 1}^n {a_{ij} (x,{\text{ }}u,{\text{ }}\nabla u)\frac{{\partial ^2 u}}{{\partial x_i \partial x_j }}} + \sum\limits_{i = 1}^n {b_i (x,{\text{ }}u,{\text{ }}\nabla u)\frac{\partial }{{\partial x_i }}} \geqq f(x,{\text{ }}u)$$ where a_ijare uniformly elliptic in Ω and b _iε L^∞(Ω); and finally (A.3) $${\text{div(}}\nabla u^p \nabla u {\text{)}} \geqq f{\text{(}}u{\text{), }}p > - 1,$$ where the operator on the left is the so-called P-Laplacian. The function f is always supposed positive and continuous. Moreover u is assumed throughout to be in the natural weak Sobolev space corresponding to the particular inequality under consideration, namely u ε. W _loc ^1,2 (Ω) ∩L _loc ^t8 (Ω) for (A.I), W _loc ^2,n(Ω) for (A.2), and W _loc ^1,p+2 (Ω) ∩ L _loc ^t8 (Ω) for (A.3). As a consequence of our results we obtain both non-existence and Liouville theorems, as well as existence theorems for (A.1).     

Singular Perturbations, Transversality, and Sil'nikov Saddle-Focus Homoclinic Orbits

We consider the singularly perturbed system $\dot x$ =εf(x,y,ε,λ), $\dot y$ =g(x,y,ε,λ). We assume that for small (ε,λ), (0,0) is a hyperbolic equilibrium on the normally hyperbolic centre manifold y=0 and that y ₀(t) is a homoclinic solution of $\dot y$ =g(0,y,0,0). Under an additional condition, we show that there is a curve in the (ε,λ) parameter space on which the perturbed system has a homoclinic orbit also. We investigate the transversality properties of this orbit and use our results to give examples of 4 dimensional systems with Sil'nikov saddle-focus homoclinic orbits.     

Eventual C∞-regularity and concavity for flows in one-dimensional porous media

We study the regularity and the asymptotic behavior of the solutions of the initial value problem for the porous medium equation $$\begin{gathered} {\text{ }}u_t = \left( {u^m } \right)_{xx} {\text{ in }}Q = \mathbb{R} \times \left( {{\text{0,}}\infty } \right){\text{,}} \hfill \\ u\left( {x{\text{,0}}} \right) = u_{\text{0}} \left( x \right){\text{ for }}x \in \mathbb{R}{\text{,}} \hfill \\ \end{gathered}$$ with m > 1 and, u ₀a continuous, nonnegative function. It is well known that, across a moving interface x=ζ(t) of the solution u(x, t), the derivatives v tand v _x of the pressure v = (m/(m?1)) u ^m?1 have jump discontinuities. We prove that each moving part of the interface is a C ^∞curve and that v is C ^∞on each side of the moving interface (and up to it). We also prove that for solutions with compact support the pressure becomes a concave function of x after a finite time. This fact implies sharp convergence rates for the solution and the interfaces as t→∞.     

Moiré interferometry with ±45-deg gratings

Specimen gratings with rulings oriented +45 deg and ?45 deg to they axis can be used to determine theu andv displacement fields, i.e., the displacements in thex andy directions. The fringe patterns are identical to those obtained with (X) and (Y) specimen gratings. The analysis leading to this conclusion is presented. A vector quantity related to the fringe gradient in the interference pattern is defined and found effective for visualizing and describing the image-forming rays. Advantages relate to construction of apparatus for moiré interferometry.     

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 → + ∞.     

Nonhomogeneous Dirichlet Problems for Degenerate Parabolic-Hyperbolic Equations

The aim of the paper is to give a formulation for the initial boundary value problem of parabolic-hyperbolic type in the case of nonhomogeneous boundary data a ₀. Here u=u(x,t)∈?, with (x,t)∈Q=Ω× (0,T), where Ω is a bounded domain in ? ^N with smooth boundary and T>0. The function b is assumed to be nondecreasing (allowing degeneration zones where b is constant), Φ is locally Lipschitz continuous and g∈L ^∞(Ω× (0,b)). After introducing the definition of an entropy solution to the above problem (in the spirit of Otto [14]), we prove uniqueness of the solution in the proposed setting. Moreover we prove that the entropy solution previously defined can be obtained as the limit of solutions of regularized equations of nondegenerate parabolic type (specifically the diffusion function b is approximated by functions b ^? that are strictly increasing). The approach proposed for the hyperbolic-parabolic problem can be used to prove similar results for the class of hyperbolic-elliptic boundary value problems of the form again in the case of nonconstant boundary data a ₀.     

Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equations

We consider a family of linearly elastic shells with thickness 2?, clamped along their entire lateral face, all having the same middle surfaceS=φ(?ω) ?R ³, whereω ?R ² is a bounded and connected open set with a Lipschitz-continuous boundaryγ, andφ ∈l ³ ( $\overline \omega$ ;R ³). We make an essential geometrical assumption on the middle surfaceS, which is satisfied ifγ andφ are smooth enough andS is “uniformly elliptic”, in the sense that the two principal radii of curvature are either both>0 at all points ofS, or both<0 at all points ofS. We show that, if the applied body force density isO(1) with respect to?, the fieldtu(?)=(u _i(?)), whereu _i (?) denote the three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elasticity, one “scaled” so as to be defined over the fixed domain Ω=ω×]?1, 1[, converges inH ¹(Ω)×H ¹(Ω)×L ²(Ω) as?→0 to a limitu, which is independent of the transverse variable. Furthermore, the averageξ=1/2ε _?1 ¹ u dx ₃, which belongs to the space $$V_M (\omega ) = H_0^1 (\omega ) \times H_0^1 (\omega ) \times L^2 (\omega ),$$ satisfies the (scaled) two-dimensional equations of a “membrane shell” viz., $$\mathop \smallint \limits_\omega a^{\alpha \beta \sigma \tau } \gamma _{\sigma \tau } (\zeta )\gamma _{\alpha \beta } (\eta ) \sqrt \alpha dy = \mathop \smallint \limits_\omega \left\{ {\mathop \smallint \limits_{ - 1}^1 f^i dx_3 } \right\}\eta _i \sqrt a dy$$ for allη=(η _i) εV _M(ω), where $a^{\alpha \beta \sigma \tau }$ are the components of the two-dimensional elasticity tensor of the surfaceS, $$\gamma _{\alpha \beta } (\eta ) = \frac{1}{2}\left( {\partial _{\alpha \eta \beta } + \partial _{\beta \eta \alpha } } \right) - \Gamma _{\alpha \beta }^\sigma \eta _\sigma - b_{\alpha \beta \eta 3} $$ are the components of the linearized change of metric tensor ofS, $\Gamma _{\alpha \beta }^\sigma$ are the Christoffel symbols ofS, $b_{\alpha \beta }$ are the components of the curvature tensor ofS, andf ⁱ are the scaled components of the applied body force. Under the above assumptions, the two-dimensional equations of a “membrane shell” are therefore justified.     

Spectral analysis of the Pekeris operator in the theory of acoustic wave propagation in shallow water

ThePekeris differential operator is defined by $$Au = - c^2 (x_n )\rho (x_n )\nabla \cdot \left( {\frac{1}{{\rho (x_n )}}\nabla u} \right),$$ wherex=(x ₁,x ₂,...x _n )∈R _n ,?=(?/?x ₁, ?/?x ₂,...?/?x _n ), and the functionsc(x _n),σ(x _n) satisfy $$c(x_n ) = \left\{ \begin{gathered} c_1 , 0 \leqq x_n< h, \hfill \\ c_2 , x_n \geqq h, \hfill \\ \end{gathered} \right.$$ and $$\rho (x_n ) = \left\{ \begin{gathered} \rho _1 , 0 \leqq x_n< h, \hfill \\ \rho _2 , x_n \geqq h, \hfill \\ \end{gathered} \right.$$ wherec ₁,c ₂,? ₁,? ₂, andh are positive constants. The operator arises in the study of acoustic wave propagation in a layer of water having sound speedc ₁ and density? ₁ which overlays a bottom having sound speedc ₂ and density? ₂. In this paper it is shown that the operatorA, acting on a class of functions u (x) which are defined for x_n≧0 and vanish for x_n=0, defines a selfadjoint operator on the Hilbert space whereR ₊ ⁿ ={x∈R ⁿ :x _n >0} anddx =dx ₁ dx ₂...dx _n denotes Lebesgue measure in R ₊ ⁿ . The spectral family ofA is constructed and the spectrum is shown to be continuous. Moreover an eigenfunction expansion for A is given in terms of a family of improper eigenfunctions. Whenc ₁≧c ₂ each eigenfunction can be interpreted as a plane wave plus a reflected wave. When c₁< c₂, additional eigen-functions arise which can be interpreted as plane waves that are trapped in the layer 0n h by total reflection at the interface x_n=h.     

Finite difference analysis of unsteady natural convection flow along an inclined plate with variable surface temperature and mass diffusion

An analysis is performed to study the transient laminar natural convection flows along an inclined semi-infinite flat plate in which the wall temperatureT _w ^′ and species concentration on the wallC _w ^′ vary as the power of the axial co-ordinate in the formT _w ^′ (x)=T _∞ ^′ +ax ⁿ andC _w ^′ =C _∞ ^′ +bx ^m respectively. The dimensionless governing equations considered here are unsteady, two-dimensional, coupled and non-linear integro-differential equations. A finite difference scheme of Crank-Nicolson type is employed to solve the problem. The velocity, temperature, concentration, skin friction, Nusselt number and Sherwood number are studied in detail for various sets of values of parameters. Correlation equations are also established for Nusselt number and Sherwood number in terms of parameters.     

Bounded solutions of some nonlinear elliptic equations in cylindrical domains

The existence of a (unique) solution of the second-order semilinear elliptic equation $$\sum\limits_{i,j = 0}^n {a_{ij} (x)u_{x_i x_j } + f(\nabla u,u,x) = 0}$$ withx=(x ₀,x ₁,?,x _n )?(s ₀, ∞)× Ω′, for a bounded domainΩ′, together with the additional conditions $$\begin{array}{*{20}c} {u(x) = 0for(x_1 ,x_2 ,...,x_n ) \in \partial \Omega '} \\ {u(x) = \varphi (x_1 ,x_2 ,...,x_n )forx_0 = s_0 } \\ {|u(x)|globallybounded} \\ \end{array}$$ is shown to be a well-posed problem under some sign and growth restrictions off and its partial derivatives. It can be seen as an initial value problem, with initial value?, in the spaceC ₀ ⁰ $(\overline {\Omega '} )$ and satisfying the strong order-preserving property. In the case thata _ij andf do not depend onx ₀ or are periodic inx ₀, it is shown that the corresponding dynamical system has a compact global attractor. Also, conditions onf are given under which all the solutions tend to zero asx ₀ tends to infinity. Proofs are strongly based on maximum and comparison techniques.     

A Concentration Phenomenon for Semilinear Elliptic Equations

For a domain ${\Omega \subset \mathbb{R}^{N}}$ we consider the equation $$-\Delta{u} + V(x)u = Q_n(x)|{u}|^{p-2}u$$ with zero Dirichlet boundary conditions and ${p\in(2, 2^*)}$ . Here ${V \geqq 0}$ and Q _n are bounded functions that are positive in a region contained in ${\Omega}$ and negative outside, and such that the sets {Q _n  > 0} shrink to a point ${x_0 \in \Omega}$ as ${n \to \infty}$ . We show that if u _n is a nontrivial solution corresponding to Q _n , then the sequence (u _n ) concentrates at x ₀ with respect to the H ¹ and certain L ^q -norms. We also show that if the sets {Q _n  > 0} shrink to two points and u _n are ground state solutions, then they concentrate at one of these points.     

Fractional boundary value problems with singularities in space variables

We are concerned with the existence of solutions for the singular fractional boundary value problem $^{c}\kern-1pt D^{\alpha}u = f(t,u)$ , u(0)+u(1)=0, u′(0)=0, where α∈(1,2), f∈C([0,1]×(??{0})) and lim _x→0 f(t,x)=∞ for all t∈[0,1]. Here, $^{c}\kern-1pt D$ is the Caputo fractional derivative. Increasing solutions of the problem vanish at points of (0,1), that is, they “pass through” the singularity of f inside of (0,1). The results are based on combining regularization and sequential techniques with a nonlinear alternative. In limit processes, the Vitali convergence theorem is used.     

The Linearized 2D Inviscid Shallow Water Equations in a Rectangle: Boundary Conditions and Well-Posedness

We consider the linearized 2D inviscid shallow water equations in a rectangle. A set of boundary conditions is proposed which make these equations well-posed. Several different cases occur depending on the relative values of the reference velocities (u ₀, v ₀) and reference height ${\phi_0}$ (sub- or super-critical flow at each part of the boundary).     

Well-Posedness of Nematic Liquid Crystal Flow in {L^{3}_{\rm uloc}(\mathbb{R}^3)}

In this paper, we establish the local well-posedness for the Cauchy problem of a simplified version of hydrodynamic flow of nematic liquid crystals in ${\mathbb{R}^3}$ for any initial data (u ₀, d ₀) having small ${L^{3}_{\rm uloc}}$ -norm of ${(u_{0}, \nabla d_{0})}$ . Here ${L^{3}_{\rm uloc}(\mathbb{R}^3)}$ is the space of uniformly locally L ³-integrable functions. For any initial data (u ₀, d ₀) with small ${\|(u_0, \nabla d_0)\|_{L^{3}(\mathbb{R}^3)}}$ , we show that there exists a unique, global solution to the problem under consideration which is smooth for t > 0 and has monotone deceasing L ³-energy for ${t \geqq 0}$ .