Some Aspects of the Asymptotic Dynamics of Solutions of the Homogeneous Navier–Stokes Equations in General Domains

In the first part of the paper we study decays of solutions of the Navier–Stokes equations on short time intervals. We show, for example, that if w is a global strong nonzero solution of homogeneous Navier–Stokes equations in a sufficiently smooth (unbounded) domain Ω ⊆ R³ and β ∈[1/2, 1) , then there exist C₀ > 1 and δ₀ ∈ (0, 1) such that

Second-order nonlinear differential equations with infinite set of periodic solutions

For the differential equation u″ = f(t, u, u′), where the function f: R × R ² → R is periodic in the first variable and f (t, x, 0) ≡ 0, sufficient conditions for the existence of a continuum of nonconstant periodic solutions are found. Published in Neliniini Kolyvannya, Vol. 11, No. 4, pp. 495–500, October–December, 2008.     

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.     

Asymptotic Stability of Riemann Solutions for a Class of Multidimensional Systems of Conservation Laws with Viscosity

We prove the asymptotic stability of two-state nonplanar Riemann solutions for a class of multidimensional hyperbolic systems of conservation laws when the initial data are perturbed and viscosity is added. The class considered here is those systems whose flux functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. In particular, we obtain the uniqueness of the self-similar L^∞ entropy solution of the two-state nonplanar Riemann problem. The asymptotic stability to which the main result refers is in the sense of the convergence as t→∞ in L_loc¹ of the space of directions ξ = x/t. That is, the solution u(t, x) of the perturbed problem satisfies u(t, tξ)→R(ξ) as t→∞, in L_loc¹(ℝⁿ), where R(ξ) is the self-similar entropy solution of the corresponding two-state nonplanar Riemann problem.     

Asymptotic Behavior of Global Classical Solutions to the Mixed Initial-Boundary Value Problem for Quasilinear Hyperbolic Systems with Small BV Data

In this paper, we investigate the asymptotic behavior of global classical solutions to the mixed initial-boundary value problem with small BV data for linearly degenerate quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space {(t,x)|t≥0,x≥0}. Based on the existence result on the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of C ¹ traveling wave solutions, provided that the C ¹ norm of the initial and boundary data is bounded and the BV norm of the initial and boundary data is sufficiently small. Applications to quasilinear hyperbolic systems arising in physics and mechanics, particularly to the system describing the motion of the relativistic string in the Minkowski space-time R ¹⁺ⁿ , are also given.     

One Equation Model for Turbulent Channel Flow with Second Order Viscoelastic Corrections

A modified second order viscoelastic constitutive equation is used to derive a k–l type turbulence closure to qualitatively assess the effects of elastic stresses on fully-developed channel flow. Specifically, the second order correction to the Newtonian constitutive equation gives rise to a new term in the momentum equation involving the time-averaged elastic shear stress and in the turbulent kinetic energy transport equation quantifying the interaction between the fluctuating elastic stress and rate of strain tensors, denoted by P _w , for which a closure is developed and tested. This closure is based on arguments of isotropic turbulence and equilibrium in boundary layer flows and a priori P _w could be either positive or negative. When P _w is positive, it acts to reduce the production of turbulent kinetic energy and the turbulence model predictions qualitatively agree with direct numerical simulation (DNS) results obtained for more realistic viscoelastic fluid models with memory which exhibit drag reduction. In contrast, P _w  < 0 leads to a drag increase and numerical breakdown of the model occurs at very low values of the Deborah number, which signifies the ratio of elastic to viscous stresses. Limitations of the turbulence model primarily stem from the inadequacy of the k–l formulation rather than from the closure for P _w . An alternative closure for P _w , mimicking the viscoelastic stress work predicted by DNS using the Finitely Extensible Nonlinear Elastic-Peterlin fluid model, which is mostly characterized by P _w  > 0 but has also a small region of negative P _w in the buffer layer, was also successfully tested. This second model for P _w leads to predictions of drag reduction, in spite of the enhancement of turbulence production very close to the wall, but the equilibrium conditions in the inertial sub-layer were not strictly maintained.     

The Laplace transform method for Burgers' equation

The Laplace transform method (LTM) is introduced to solve Burgers' equation. Because of the nonlinear term in Burgers' equation, one cannot directly apply the LTM. Increment linearization technique is introduced to deal with the situation. This is a key idea in this paper. The increment linearization technique is the following: In time level t, we divide the solution u(x, t) into two parts: u(x, t^k) and w(x, t), t^k⩽t⩽t^k+1, and obtain a time‐dependent linear partial differential equation (PDE) for w(x, t). For this PDE, the LTM is applied to eliminate time dependency. The subsequent boundary value problem is solved by rational collocation method on transformed Chebyshev points. To face the well‐known computational challenge represented by the numerical inversion of the Laplace transform, Talbot's method is applied, consisting of numerically integrating the Bromwich integral on a special contour by means of trapezoidal or midpoint rules. Numerical experiments illustrate that the present method is effective and competitive. Copyright © 2009 John Wiley & Sons, Ltd.     

Static Solutions of the Vlasov-Einstein System

Static solutions of the SO(3)-symmetric Vlasov-Einstein system are studied via a variational approach. For the constitutive relation of the Emden-Fowler type φ(E,F)≡E ^{σ+ 1} F ^k we prove the existence of such solutions of sufficiently small mass-energy, provided 0<σ < k+3/2. These solutions are local minimizers of the energy-Casimir functional, subjected to a variational barrier. Accepted July 16, 2000?Published online January 22, 2001     

Exact analytic solutions for free convection boundary layers on a heated vertical plate with lateral mass flux embedded in a saturated porous medium

 The problem of the self-similar boundary flow of a “Darcy-Boussinesq fluid” on a vertical plate with temperature distribution T _w(x) = T _∞+A·x ^λ and lateral mass flux v _w(x) = a·x ^(λ−1)/2, embedded in a saturated porous medium is revisited. For the parameter values λ = 1,−1/3 and −1/2 exact analytic solutions are written down and the characteristics of the corresponding boundary layers are discussed as functions of the suction/ injection parameter in detail. The results are compared with the numerical findings of previous authors. Received on 8 March 1999     

Three views of viscoelasticity for Cox–Merz materials

A slight rearrangement of the classical Cox and Merz rule suggests that the shear stress value of steady shear flow, , and complex modulus value of small amplitude oscillatory shear, G^∗(ω) = (G^′2 + G^″2)^1/2, are equivalent in many respects. Small changes of material structure, which express themselves most sensitively in the steady shear stress, τ, show equally pronounced in linear viscoelastic data when plotting these with G^∗ as one of the variables. An example is given to demonstrate this phenomenon: viscosity data that cover about three decades in frequency get stretched out over about nine decades in G^∗ while maintaining steep gradients in a transition region. This suggests a more effective way of exploiting the Cox–Merz rule when it is valid and exploring reasons for lack of validity when it is not. The τ −G^∗ equivalence could also further the understanding of the steady shear normal stress function as proposed by Laun.     

Singular perturbation of boundary value problem for a vector fourth order nonlinear differential equation

We study the vector boundary value problem with boundary perturbations: ε~2y~((4))=f(x,y,y″,ε, μ) ( μ<χ<1-μ) y(χ,ε,μ)l_(χ-μ)= A_1(ε,μ), y(χ,ε,μ)l_(χ-1-μ)=B_1(ε,μ) y″(χ,ε,μ)l_(χ-μ)=A_2(ε,μ),y″(χ,ε,μ)l_(χ-1-μ)=B_2(ε,μ)where yf, A_j and B_j (j=1,2) are n-dimensional vector functions and ε,μ are two small positive parameters. This vector boundary value problem does not appear to have been studied, although the scalar boundary value problem has been treated. Under appropriate assumptions, using the method of differential inequalities we find a solution of the vector boundary value problem and obtain the uniformly valid asymptotic expansions.     

Wave Packets, Signaling and Resonances in a Homogeneous Waveguide

We solve the initial-boundary-value linear stability problem for small localised disturbances in a homogeneous elastic waveguide formally by applying a combined Laplace – Fourier transform. An asymptotic evaluation of the solution, expressed as an inverse Laplace – Fourier integral, is carried out by means of the mathematical formalism of absolute and convective instabilities. Wave packets, triggered by perturbations localised in space and finite in time, as well as responses to sources localised in space, with the time dependence satisfying e^−iωt + O(e^−ɛt ), for t → ∞, where Im ω₀ = 0 and ω > 0 , that is, the signaling problem, are treated. For this purpose, we analyse the dispersion relation of the problem analytically, and by solving numerically the eigenvalue stability problem. It is shown that due to double roots in a wavenumber k of the dispersion relation function D(k, ω), for real frequencies ω, that satisfy a collision criterion, wave packets with an algebraic temporal decay and signaling with an algebraic temporal growth, that is, temporal resonances, are present in a neutrally stable homogeneous waveguide. Moreover, for any admissible combination of the physical parameters, a homogeneous waveguide possesses a countable set of temporally resonant frequencies. Consequences of these results for modelling in seismology are discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Estimating the viscoelastic moduli of complex fluids using the generalized Stokes–Einstein equation

We obtain the linear viscoelastic shear moduli of complex fluids from the time-dependent mean square displacement, <Δr ²(t)>, of thermally-driven colloidal spheres suspended in the fluid using a generalized Stokes–Einstein (GSE) equation. Different representations of the GSE equation can be used to obtain the viscoelastic spectrum, G˜(s), in the Laplace frequency domain, the complex shear modulus, G ^*(ω), in the Fourier frequency domain, and the stress relaxation modulus, G _r (t), in the time domain. Because trapezoid integration (s domain) or the Fast Fourier Transform (ω domain) of <Δr ²(t)> known only over a finite temporal interval can lead to errors which result in unphysical behavior of the moduli near the frequency extremes, we estimate the transforms algebraically by describing <Δr ²(t)> as a local power law. If the logarithmic slope of <Δr ²(t)> can be accurately determined, these estimates generally perform well at the frequency extremes. Received: 8 September 2000/Accepted: 9 March 2000     

Planar elongation of soft polymeric networks

A new test fixture for the filament stretch rheometer (FSR) has been developed to measure planar elongation of soft polymeric networks with application towards pressure-sensitive adhesives (PSAs). The concept of this new geometry is to elongate a tube-like sample by keeping the perimeter constant. To validate this new technique, soft polymeric networks of poly(propylene oxide) (PPO) were investigated during deformation. Particle tracking and video recording were used to detect to what extent the imposed strain rate and the sample perimeter remained constant. It was observed that, by using an appropriate choice of initial sample height, perimeter, and thickness, the planar stretch ratio will follow l(t) = h(t)/h₀ = exp([(e)\dot] t)\lambda(t) = h(t)/h_0= \exp({\dot{\varepsilon}} t), with h(t) being the height at time t and [(e)\dot]{\dot{\varepsilon}} the imposed constant strain rate. The perimeter would decrease by a few percent only, which is found to be negligible. The ideal planar extension in this new fixture was confirmed by finite element simulations. Analysis of the stress difference, σ _zz  − σ _xx , showed a network response similar to that of the classical neo-Hookean model. As the Deborah number was increased, the stress difference deviated more from the classical prediction due to the dynamic structures in the material. A modified Lodge model using characteristic parameters from linear viscoelastic measurements gave very good stress predictions at all Deborah numbers used in the quasi-linear regime.     

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).     

Supersonic leeside flow topology on delta wings revisited

The leeside vortex structures on delta wings with sharp leading edges were studied for supersonic flow at the Institute of Theoretical and Applied Mechanics of the Russian Academy of Sciences in Novosibirsk. The experiments were carried out with three wings with sweep angles of χ=68°, 73°, and 78° and parabolic profiles in the 0.6 × 0.6 m² test section of the blow-down wind tunnel T-313 of the institute. The test conditions were varied from Mach numbers M=2 to 4, unit Reynolds numbers from Re _l=26 × 10⁶ to 56 × 10⁶ m⁻¹, and angles of attack from α=0° to 22°. The results of the investigations revealed that for certain flow conditions shocks are formed above, below, and between the primary vortices. The experimental data were accurate enough to detect the onset of secondary and tertiary separation as well as other boundaries. The various flow regimes discussed in the literature were extended in several cases. The major findings are reported. Received: 6 September 1999/Accepted: 24 January 2000     

A unified approach to closed-form solutions of moving heat-source problems

A closed-form model for the computation of temperature distribution in an infinitely extended isotropic body with a time-dependent moving-heat sources is discussed. The temperature solutions are presented for the sources of the forms: (i) ₀₁(t)=₀ exp(−λt), (ii) ₀₂(t) =₀(t/t ^*)exp(−λt), and ₀₃(t)=₀[1+a cos(ωt)], where λ and ω are real parameters and t ^* characterizes the limiting time. The reduced (or dimensionless) temperature solutions are presented in terms of the generalized representation of an incomplete gamma function Γ(α,x;b) and its decomposition C _Γ and S _Γ. The solutions are presented for moving, -point, -line, and -plane heat sources. It is also demonstrated that the present analysis covers the classical temperature solutions of a constant strength source under quasi-steady state situations. Received on 13 June 1997     

Lagrangian Time-Scales in Homogeneous Non-Gaussian Turbulence

Lagrangian time-scales in homogeneous non-Gaussian turbulence were studied using a one-dimensional Lagrangian Stochastic Model. The existence of two time-scales τ _L and T _L , one typical of the inertial subrange and the other which is an integral property, is outlined. Variations of the ratio T _L /τ _L in the plane skewness-flatness (S, F) are shown and a connection with the statistical constraint F ≥S ² + 1 is evidenced. The Lagrangian autocorrelation function ρ(t) of particle velocity was computed for some values of (S, F). It is shown that for small times, say t < T _L , the influence of non-Gaussianity is negligible and ρ(t) presents the same behaviour as in the Gaussian case regardless of variations in (S, F).As the time increases, departures from Gaussianity are observed and autocorrelation turns out to be always larger than in the Gaussiancase. This is supported by some considerations in terms of information entropy, which is shown to decrease with increasing departures from Gaussianity. Spectral analysis of Lagrangian velocity shows that non-Gaussianity is relevant only to large scales of the stochastic process and that the expected inertial subrange decay ω⁻² is attained by spectra of all simulations, except for one case in which the model probability density function is bimodal, due to the vicinity to the statistical limit. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Decay Rate and Higher Approximation of Mild Solutions to the Navier�CStokes Equations

In this paper, we consider v(t) = u(t) − e ^tΔ u ₀, where u(t) is the mild solution of the Navier–Stokes equations with the initial data u₀ ? L²(\mathbb Rⁿ)?Lⁿ(\mathbb Rⁿ){u_0\in L^2({\mathbb R}^n)\cap L^n({\mathbb R}^n)} . We shall show that the L ² norm of D ^β v(t) decays like t^{-\frac |b|-1 2-\frac n4}{t^{-\frac {|\beta|-1} {2}-\frac n4}} for |β| ≥ 0. Moreover, we will find the asymptotic profile u ₁(t) such that the L ² norm of D ^β (v(t) − u ₁(t)) decays faster for 3 ≤ n ≤ 5 and |β| ≥ 0. Besides, higher-order asymptotics of v(t) are deduced under some assumptions.     

Homogenization of Non-linear Scalar Conservation Laws

We study the limit as ε → 0 of the entropy solutions of the equation . We prove that the sequence u ^ε two-scale converges toward a function u(t, x, y), and u is the unique solution of a limit evolution problem. The remarkable point is that the limit problem is not a scalar conservation law, but rather a kinetic equation in which the macroscopic and microscopic variables are mixed. We also prove a strong convergence result in .