The free jets as boundary-layer flows induced by continuous stretching surfaces

This paper proves that the free laminar jets of the classical hydrodynamics may be identified with certain boundary-layer flows induced by continuous surfaces immersed in quiescent incompressible fluids and stretched with well-defined velocities. In this sense: (i) Schlichting's round jet of momentum flow coincides with the axisymmetric flow induced by a thin continuous wire issuing from a small orifice at x=0 and stretching along the x-axis with velocity U _w(x) = 3J˙/(8πρνx), and (ii) the Schlichting–Bickley plane jet of momentum flow J˙ coincides with the boundary-layer flow induced by an impermeable plane wall issuing from a long slit (of length l) and stretching with velocity U _w(x)= [{3J˙ ²/(32νρ² l ² x)}]^1/3.     

Homogenization of a degenerate parabolic problem in a highly heteregeneous mediumHomogénéisation d'un problème parabolique dégénéré dans un milieu fortement hétérogène

We consider the homogenization of a time-dependent heat transfer problem in a highly heteregeneous periodic medium made of two connected components having finite heat capacities c_α(x) and heat conductivities a_α(x), α=1,2, of order one, separated by a third material with thickness of order ε the size of the basic periodicity cell, but with conductivity λa₃(x) where a₃=O(1) and λ tends to zero with ε. Assuming only that c_i(x)?0 a.e., such that the problem can degenerate (parabolic-elliptic), we identify the homogenized problem following the values of δ=lim_ε→0ε²/λ. To cite this article: M. Mabrouk, A. Boughammoura, C. R. Mecanique 331 (2003).     

Quadrature formulas of maximum algebraic accuracy for cauchy type integrals with complex exponents of the Jacobi weight function

The present paper presents a Gauss type quadrature formula for a Cauchy type integral whose density is the product of a Hölder function by the weight function (1 ? x) ^α (1 + x) ^β (Re α, Reβ > ?1) of orthogonal Jacobi polynomials. It is shown that at the roots of the function of the second kind corresponding to the Jacobi polynomial P _n ^(α,β) (x), the quadrature formula with n nodes gives the exact value of a Cauchy type integral for an arbitrary polynomial of order k ≤ 2n. This formula was tested when solving several contact and mixed problems of the theory of elasticity.     

Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons

We consider the dynamics of N boson systems interacting through a pair potential N ^?1 V _a (x _i ?x _j ) where V _a (x)=a ^?3 V(x/a). We denote the solution to the N-particle Schrödinger equation by Ψ _{N, t} . Recall that the Gross-Pitaevskii (GP) equation is a nonlinear Schrödinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if u _t solves the GP equation, then the family of k-particle density matrices solves the GP hierarchy. Under the assumption that a=N ^?? for 0N→∞ the limit points of the k-particle density matrices of Ψ _{N, t} are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by ∫V(x)dx. The uniqueness of the solutions of this hierarchy remains an open question.     

Determination of source parameter in parabolic equations

The authors consider the problem of finding u=u(x, t) and p=p(t) which satisfy u = Lu + p(t) + F(x, t, u, _x, p(t)) in Q _T=Ω×(0, T], u(x, 0)=ø(x), x∈Ω, u(x, t)=g(x, t) on ?Ω×(0, T] and either ∫_G(t) Φ(x,t)u(x,t)dx = E(t), 0 ? t ? T or u(x₀, t)=E(t), 0≤t≤T, where Ω?R ⁿ is a bounded domain with smooth boundary ?Ω, x ₀∈Ω, L is a linear elliptic operator, G(t)?Ω, and F, ø, g, and E are known functions. For each of the two problems stated above, we demonstrate the existence, unicity and continuous dependence upon the data. Some considerations on the numerical solution for these two inverse problems are presented with examples.     

On the fundamental linear fractional order differential equation

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

Preparation and characterization of LiAl_xCo_yNi_1-x-yO₂ particles

Lithium-aluminum-cobalt-nickel oxide LiAlxCoyNi1xyO2 particles, generally used as cathode of lithium battery, were prepared by chemical coprecipitation from an aqueous solution of LiOH, Al(NO3)3, Co(NO3)2 and Ni(NO3)2 with NH4OH. XRD, SEM and FTIR were used to examine the effect of nickel content on the product. FTIR patterns showed that increase in nickel content decreased the absorption strength of the peak of spinel structure of the product, attributed to the occupation by nickel in the aluminum sites. Particle size and electrical properties of the lithium-aluminum-cobalt-nickel oxide (abbreviated as LACNO) particles were also determined.     

Surface responses induced by point load or uniform traction moving steadily on an anisotropic half-plane

Surface responses induced by point load or uniform traction moving steadily with subsonic speed on an anisotropic half-plane boundary are investigated. It is found that the effects of the material constant on surface displacements are through matrices L^?1(v) and S(v)L^?1(v), while those on surface stress components are through matrices Ω(v) and Γ(v). Explicit expressions for the elements of these four matrices are expressed in terms of elastic stiffness for general anisotropic materials. The special cases of monoclinic materials with symmetry plane at x₁ = 0, x₂ = 0 and x₃ = 0, and the case for orthotropic materials are all deduced. Results for isotropic material may be recovered from present results. For monoclinic materials with a plane of symmetry at x₃ = 0, two of the elements of matrix Ω(v) are found to be independent of subsonic speed.     

Some new classes of inverse coefficient problems in non-linear mechanics and computational material science

Three classes of inverse coefficient problems arising in engineering mechanics and computational material science are considered. Mathematical models of all considered problems are proposed within the J₂-deformation theory of plasticity. The first class is related to the determination of unknown elastoplastic properties of a beam from a limited number of torsional experiments. The inverse problem here consists of identifying the unknown coefficient g(ξ²) (plasticity function) in the non-linear differential equation of torsional creep −(g(|∇u|²)u_x1)_x1−(g(|∇u|²)u_x2)_x2=2?, x∈Ω⊂R², from the torque (or torsional rigidity) T(?), given experimentally. The second class of inverse problems is related to the identification of elastoplastic properties of a 3D body from spherical indentation tests. In this case one needs to determine unknown Lame coefficients in the system of PDEs of non-linear elasticity, from the measured spherical indentation loading curve P=P(α), obtained during the quasi-static indentation test. In the third model an inverse problem of identifying the unknown coefficient g(ξ²(u)) in the non-linear bending equation is analyzed. The boundary measured data here is assumed to be the deflections w_i[τ_k]?w(λ_i;τ_k), measured during the quasi-static bending process, given by the parameter τ_k, , at some points , of a plate. An existence of weak solutions of all direct problems are derived in appropriate Sobolev spaces, by using monotone potential operator theory. Then monotone iteration schemes for all the linearized direct problems are proposed. Strong convergence of solutions of the linearized problems, as well as rates of convergence is proved. Based on obtained continuity property of the direct problem solution with respect to coefficients, and compactness of the set of admissible coefficients, an existence of quasi-solutions of all considered inverse problems is proved. Some numerical results, useful from the points of view of engineering mechanics and computational material science, are demonstrated.     

Steady Flow of a Navier-Stokes Fluid Around a Rotating Obstacle

Let ? be a body immersed in a Navier-Stokes liquid ? that fills the whole space. Assume that ? rotates with prescribed constant angular velocity ω. We show that if the magnitude of ω is not “too large”, there exists one and only one corresponding steady motion of ? such that the velocity field v(x) and its gradient grad?v(x) decay like |x|^?1 and |x|^?2, respectively. Moreover, the pressure field p(x) and its gradient grad?p(x) decay like |x|^?2 and |x|^?3, respectively. These solutions are “physically reasonable” in the sense of Finn. In particular, they are unique and satisfy the energy equation. This result is relevant to several applications, including sedimentation of heavy particles in a viscous liquid.     

Linearized Stability for Semiflows Generated by a Class of Neutral Equations,with Applications to State-Dependent Delays

We prove a principle of linearized stability for semiflows generated by neutral functional differential equations of the form x′(t) = g(? x _t , x _t ). The state space is a closed subset in a manifold of C ²-functions. Applications include equations with state-dependent delay, as for example x′(t) = a x′(t + d(x(t))) + f (x(t + r(x(t)))) with \({a\in\mathbb{R}, d:\mathbb{R}\to(-h,0), f:\mathbb{R}\to\mathbb{R}, r:\mathbb{R}\to[-h,0]}\).     

Carleman Estimate for Elliptic Operators with Coefficients with Jumps at an Interface in Arbitrary Dimension and Application to the Null Controllability of Linear Parabolic Equations

In a bounded domain of R ⁿ⁺¹, n ≧ 2, we consider a second-order elliptic operator, ${A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}In a bounded domain of R ⁿ⁺¹, n ≧ 2, we consider a second-order elliptic operator, A=-?_x0² - ?_x ·(c(x) ?_x){A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}, where the (scalar) coefficient c(x) is piecewise smooth yet discontinuous across a smooth interface S. We prove a local Carleman estimate for A in the neighborhood of any point of the interface. The “observation” region can be chosen independently of the sign of the jump of the coefficient c at the considered point. The derivation of this estimate relies on the separation of the problem into three microlocal regions and the Calderón projector technique. Following the method of Lebeau and Robbiano (Comm Partial Differ Equ 20:335–356, 1995) we then prove the null controllability for the linear parabolic initial problem with Dirichlet boundary conditions associated with the operator ?_t - ?_x ·(c(x) ?_x){{\partial_t - \nabla_x \cdot (c(x) \nabla_x)}} .     

Period bounds for generalized Rayleigh equation

Simple upper and lower bounds are obtained for the least period T of any non-constant periodic solution x(t) of the differential equation x″ − F(x') + g(x) = 0.     

Universal Attractors¶for the Navier-Stokes Equations¶of Compressible and Heat-Conductive Fluid¶in Bounded Annular Domains in ℝn

This paper is concerned with the dynamics for the Navier-Stokes equations for a polytropic viscous heat-conductive ideal gas in bounded annular domains Ω _n in ? ⁿ (n= 2, 3). One of the important features of this problem is that the metric spaces H ⁽¹⁾ and H ⁽²⁾ we work with are two incomplete metric spaces, as can be seen from the constraints θ >0 and u> 0, withθ and u being absolute temperature and specific volume respectively. For any constants δ₁, δ₂, δ₃, δ₄, δ₅ satisfying certain conditions, two sequences of closed subspaces H ^{( i )} _δ?H ^{( i )} (i= 1,2) are found, and the existence of two (maximal) universal attractors in H ⁽¹⁾ _δ and H ⁽²⁾ _δ is proved.     

Similarity solution of mixed convection over a horizontal moving plate

Investigation to the mixed convective heat and mass transfer over a horizontal plate has been carried out. By applying transformation group theory to analysis of the governing equations of continuity, momentum, energy and diffusion, we show the existence of similarity solution for the problem provided that the temperature and concentration at the wall are proportional to x ^4/(7-5n) and that the moving speed of the plate is proportional to x ^(3-n)/(7-5n), and further obtain a similarity representation of the problem. The similarity equations have been solved numerically by a fourth-order Runge–Kutta scheme. The numerical results obtained for Pr=0.72 and various values of the parameters Sc, K ₁, K ₂ and K ₃ reveals the influence of the parameters on the flow, heat and mass transfer behavior.     

Operator method for solving the plane elasticity problem for a strip with periodic cuts

In the present paper, we use the conformal mapping z/c = ζ?2a sin ζ (a, c?const, ζ = u + iv) of the strip {|v| ≤ v ₀, |u| < ∞} onto the domain D, which is a strip with symmetric periodic cuts. For the domain D, in the orthogonal system of isometric coordinates u, v, we solve the plane elasticity problem. We seek the biharmonic function in the form F = C ψ ₀ + S ψ*₀ + x(C ψ ₁ ? S ψ ₂) + y(C ψ ₂ + S ψ ₁), where C(v) and S(v) are the operator functions described in [1] and ψ ₀(u), …, ψ ₂(u) are the desired functions. The boundary conditions for the function F posed for v = ±v ₀ are equivalent to two operator equations for ψ ₁(u) and ψ ₂(u) and to two ordinary differential equations of first order for ψ ₀(u) and ψ*₀(u) [2]. By finding the functions ψ _j (u) in the form of trigonometric series with indeterminate coefficients and by solving the operator equations, we obtain infinite systems of linear equations for the unknown coefficients. We present an efficient method for solving these systems, which is based on studying stable recursive relations. In the present paper, we give an example of analysis of a specific strip (a = 1/4, v ₀ = 1) loaded on the boundary v = v ₀ by a normal load of intensity p. We find the particular solutions corresponding to the extension of the strip by the longitudinal force X ^∞ and to the transverse and pure bending of the strip due to the transverse force Y ^∞ and the constant moment M ^∞, respectively. We also present the graphs of normal and tangential stresses in the transverse cross-section x = 0 and study the stress concentration effect near the cut bottom.     

Entire Solutions with Merging Fronts to Reaction–Diffusion Equations

We deal with a reaction–diffusion equation u _t = u _xx + f(u) which has two stable constant equilibria, u = 0, 1 and a monotone increasing traveling front solution u = φ(x + ct) (c > 0) connecting those equilibria. Suppose that u = a (0 < a < 1) is an unstable equilibrium and that the equation allows monotone increasing traveling front solutions u = ψ₁(x + c ₁ t) (c ₁ < 0) and ψ₂(x + c ₂ t) (c ₂ > 0) connecting u = 0 with u = a and u = a with u = 1, respectively. We call by an entire solution a classical solution which is defined for all . We prove that there exists an entire solution such that for t≈ − ∞ it behaves as two fronts ψ₁(x + c ₁ t) and ψ₂(x + c ₂ t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c > − c ₁, we show the existence of an entire solution which behaves as ψ₁( − x + c ₁ t) in and φ(x + ct) in for t≈ − ∞.     

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.     

Study of the aerodynamic characteristics of cone-sphere models in a hypersonic flow

The aerodynamic characteristics of cone-sphere models are studied at Mach numbers M = 6, 8.4, and 12 to 13 over a wide Reynolds number range. Models of a braking device (sphere) were connected with a load (frustum of a cone) by means of shrouds. The dependences of the aerodynamic coefficients C _x and C _y on the angle of attack α were obtained for different relative dimensions of the load and the braking device, shroud lengths, and Mach and Reynolds numbers. The effect of the above-mentioned parameters on the aerodynamic characteristics of the models is analyzed. The C _x (Re_D) dependences of load-parachutemodels in a symmetric flow are determined over the wide Mach and Reynolds number ranges 6 ≤ M ≤ 13 and 3 · 10³ ≤ Re_D ≤ 3 · 10⁶.     

Stationary problem of third-grade fluids in two and three dimensions: existence and uniqueness

This paper is devoted to the stationary problem of third-grade fluids in two and three dimensions. In two dimensions, we show existence of solutions and uniqueness, for a boundary of class C^2,1 and small data, by generalizing the method used by J.M. Bernard for the stationary problem of second-grade fluids (we deal with a polynomial of four degrees instead of two degrees). Contrary to the case of two dimensions, the resolution of the problem of third-grade fluids in three dimensions requires the physical condition |α₁+α₂|<(24νβ)^1/2. From this condition, we derive two “pseudo ellipticities” for the operator ν|A(u)|²+(α₁+α₂)tr(A(u)³)+β|A(u)|⁴, where A(u) is a 3-order symmetric matrix such that tr(A(u))=0. Thus, with, in addition, a sharp estimate of the scalar product (|A(u)|²A(u)-|A(v)|²A(v),A(u)-A(v)), we are able to prove existence of solutions and uniqueness, for a boundary of class C^2,1 and small data, in three dimensions.

Résumé

Cet article est consacré au problème stationnaire des fluides de grade trois en dimension deux et trois. En dimension deux, nous montrons l’existence de solutions et l’unicité, pour une frontière de classe C^2,1 et une donnée petite, en généralisant la méthode utilisée par J.M. Bernard pour le problème stationnaire des fluides de grade deux (nous avons affaire à un polynôme de degré quatre au lieu de deux). Contrairement au cas de la dimension deux, la résolution du problème des fluides de grade trois en dimension trois requière la condition physique |α₁+α₂|<(24νβ)^1/2. De cette condition, nous déduisons deux “pseudo matrice” pour l’opérateur ν|A(u)|²+(α₁+α₂)tr(A(u)³)+β|A(u)|⁴, où A(u) est une matice symétrique d’ordre 3 à trace nulle. De là, avec, en plus, une fine estimation du produit scalaire (|A(u)|²A(u)-|A(v)|²A(v),A(u)-A(v)), nous sommes capables de prouver l’existence de solutions et l’unicité, pour une frontière de classe C^2,1 et une donnée petite, en dimension trois.