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1.
We prove short-time well-posedness of the Cauchy problem for incompressible strongly elliptic hyperelastic materials. Our method consists in:
- Reformulating the classical equations in order to solve for the pressure gradient (The pressure is the Lagrange multiplier corresponding to the constraint of incompressibility.) This formulation uses both spatial and material variables.
- Solving the reformulated equations by using techniques which are common for symmetric hyperbolic systems. These are:
- Using energy estimates to bound the growth of various Sobolev norms of solutions.
- Finding the solution as the limit of a sequence of solutions of linearized problems.
2.
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 nvn) 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),ut(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. 相似文献
3.
For aqueousn-heptanol solutions and in a nearly two-dimensional flow, two strikingfeatures have been detected:
- a shift of the minimum of the surface tension
- a discrepancy between the observed Marangoni flow velocities and the expected ones from static surface tension values.
4.
The paper is concerned with the asymptotic behavior as t → ∞ of solutions u(x, t) of the equation ut—uxx—∞;(u)=O, x∈(—∞, ∞) , in the case ∞(0)=∞(1)=0, ∞′(0)<0, ∞′(1)<0. Commonly, a travelling front solution u=U(x-ct), U(-∞)=0, U(∞)=1, exists. The following types of global stability results for fronts and various combinations of them will be given.
- Let u(x, 0)=u 0(x) satisfy 0≦u 0≦1. Let \(a\_ = \mathop {\lim \sup u0}\limits_{x \to - \infty } {\text{(}}x{\text{), }}\mathop {\lim \inf u0}\limits_{x \to \infty } {\text{(}}x{\text{)}}\) . Then u approaches a translate of U uniformly in x and exponentially in time, if a? is not too far from 0, and a+ not too far from 1.
- Suppose \(\int\limits_{\text{0}}^{\text{1}} {f{\text{(}}u{\text{)}}du} > {\text{0}}\) . If a ? and a + are not too far from 0, but u0 exceeds a certain threshold level for a sufficiently large x-interval, then u approaches a pair of diverging travelling fronts.
- Under certain circumstances, u approaches a “stacked” combination of wave fronts, with differing ranges.
5.
C. A. Stuart 《Archive for Rational Mechanics and Analysis》1993,125(2):145-200
We discuss the propagation of electromagnetic waves through a stratified dielectric. The ability of such a device to support guided waves depends upon the way in which the refractive index varies across the layers. In the present discussion, we show how nonlinear effects and appropriate stratification can be used to obtain any one of the following behaviours:
- guidance occurs only at low power.
- guidance occurs only at high power.
- guidance occurs at all powers.
- there is no guidance.
6.
K. N. Katyal 《Flow, Turbulence and Combustion》1971,23(1):281-294
The flow of a dipolar fluid between two parallel plates with and without heat transfer is studied. The following cases are discussed:
- Isothermal flow due to the relative motion of the plates,
- Isothermal flow due to a constant pressure gradient with the plates at rest,
- Nonisothermal flow with linearly varying plate temperatures.
7.
The problem of axial variation of stress concentrations at the periphery and normal to the axis of a circular tunnel is solved by means of the three-dimensional photoelasticity technique, under the following conditions:
- The center lines of two horizontal tunnels of equal diameter (2r) are separated by a distanceK and include an angle α.
- K and α assume values of 0, 3r, 7/2r, 4r and 30 deg, 60 deg, 90 deg, respectively.
- The tunnels are located in a uniform, uniaxial stress field normal to the axes of the tunnels.
8.
It is shown that the complex poles z of the scattering matrix satisfy the inequality: Im z≧a+b log ¦z¦, b>0, in three instances of classical scattering in three space dimensions described by the wave equation ut t?c2Δu+qu=0.
- c and q smooth with c=1 and q=0 for ¦x¦>p, all rays going to infinity, and the energy form positive definite.
- c=1 and q=0 outside of a convex body on which u=0.
- c=1, q bounded and measurable, q=0 for ¦x¦>p, and the energy form not necessarily positive definite.
9.
ATaylor-series correction to the maximum inplane shear stress was studied as a means of extending the data zone in photoelastic determination of stress-intensity factors beyond the singular region of a two-degree-of-freedom analysis. Convergence properties were obtained by comparing with several complete two-dimensional solutions. Experiments were performed on two kinds of three-dimensional problems, plates containing surface flows in both bending and extension. Results were analyzed by both a two-degree-of-freedom and aTaylor-series correction method (TSCM). Results were compared to theories of F. W. Smith and A. S. Kobayashi and R. C. Shah. It was concluded that:
- The TSCM program converges rapidly to accurateK I values and will accommodate the scatter inherent in experimental data if the series is properly truncated.
- The TSCM program is essentially equivalent to the two-parameter representation when only the crack-surface effects dominate.
- When effects other than crack surfaces are important, TSCM requires more terms but still predictsK I with reasonable accuracy.
10.
A numerical procedure on the basis of the Marker and Cell-method [1] was developed in order to solve the conservation equations for mass, momentum and energy for the case of bubble growth on a heating surface. This procedure was used to calculate steam bubble growth on a horizontal stainless steel heating surface under saturated pool boiling conditions at a system pressure of 1 bar and different superheatings. The essential results obtained are: -Good agreement was found between calculations and experiments concerning bubble growth rates, bubble shape and temperature field in the liquid surrounding the bubble. -During its growth the bubble penetrates the temperature boundary layer formed in the liquid on the heating surface, simultaneously liquid is displaced aside. -The microlayer evaporation fraction of the total bubble growth increases with growth time from 20 % to 50%. 相似文献
11.
Prof. Dr. A. P. Szumowski Prof. Dr. F. Obermeier Prof. Dr. G. E. A. Meier 《Experiments in fluids》1995,18(3):145-152
Experimental investigations of Laval nozzle flow show for relatively low supply to exit pressure ratios, which correspond to shock wave positions close to the nozzle throat, three different, oscillatory instabilities.
- Shock pattern oscillations where the root of a λ-like shock front remains nearly in constant position, but where the proportion between the normal part and the oblique part of the shock changes periodically.
- Shock wave and separation bubble oscillations where the motion of the shock wave is accompanied by displacements of the separation bubble.
- Flow rate oscillations where the shock waves leave periodically through the nozzle throat in upstream direction.
12.
Dr.-Ing. H. M. Sofrata 《Heat and Mass Transfer》1980,14(2):119-129
The aim of this paper is to develop a fast and simple accurate graphical method for designing the required cooling coil for an air conditioning system in which both the sensible and latent heat are transferred. The method has the following advantages:
- Direct solution utilizing only Psychrometer charts.
- Solution of combined heat and mass transfer problems occuring at pressures other than that of atmosphere.
- Direct determination of boundary temperature at which dehumidification begins for that type of problem where a portion of the surface is in a dry condition.
- Less effort required than trial and error method in determination of air condition leaving a counterflow coil of a given area.
13.
Mahmood H. Rana 《Experimental Mechanics》1969,9(12):538-549
An important parameter involved in the viscoelastic deformation of structural materials is the coefficient of “solid” viscosity. Determination of this parameter is necessary, if it is to be used in structural design. This paper deals with pertinent analytical considerations concerning solid viscosity and describes the procedures followed in the determination of parameters for structural and true viscosity of a Queenston limestone. The following three techniques were used:
- Relaxation technique
- Uniaxial compressive loading
- Cantilever-beam loading
14.
A series of stress-freezing photoelastic experiments were performed with multiple replications upon edge-cracked strips for three types of “cracks” in current use:
- Rectangular slots 0.152 mm wide,
- 1.59-mm-wide slots terminating in a 30-deg vee notch of approximately 0.025-mm root radius, and
- Natural cracks (approximately 0.0025-mm root radius).
15.
It is shown that the well-known variational principles for the ideal compressible fluid model in Eulerian coordinates have the following deficiencies:
- They are not related to the corresponding variational principles in Lagrangian coordinates;
- The variation procedure in these variational problems does not lead to the equations of motion themselves in the Euler form; rather it leads to relations which correspond to definite classes of solutions of the Euler equations. Here allowance for the equations of the constraints imposed by the adiabaticity and continuity conditions limits the region of application of these variational principles to only potential flows;
- More general results, involving flows other than potential, are achieved by artificial selection of certain additional constraint conditions imposed on the quantities being varied, and in this case additional clarification is required to ascertain whether any inviscid compressible fluid flow is the extremum of the corresponding variational problem.
16.
H. Janeschitz-Kriegl 《Rheologica Acta》1982,21(4-5):388-393
An exposition is given of results, as obtained with the aid of Doi's sliplink model, being considered as the most simple version of the famous “reptation model”. It turns out that this model which exhibits three distinct phases of relaxation (an extremely fast phase, an equilibration phase and a slow disengagement phase) is capable of explaining several peculiar features of polymer melt rheology:
- The molecular mass dependence of the breadth of the rubber plateau in the storage modulus, of the zero-shear viscosity and of the normal-stress coefficients.
- The molecular mass independence of the equilibrium (shear and tensile) compliances for monodisperse polymers (semi-quantitative prediction).
- The seemingly contradictory sensitivity of these compliances for the breadth of the molecular mass distribution.
- The critical value of the shear stress at which melt fracture occurs in capillary flow.
- An equilibration phase in tensile experiments on unvolcanized rubber.
17.
Joseph Der Hovanesian 《Experimental Mechanics》1974,14(6):233-236
This paper presents a new “hybrid” method whereby the ratio of the isochromatic-fringe visibility/isopachic-fringe visibility may be easily and continuously varied. This simple procedure merely combines a conventional polariscope with a holographic system. A variable beam splitter permits an incoherent superposition of the reconstruction of a doubly exposed hologram with real-time isochromatics, either dark or light field. By varying the ratios of the above two, in the image plane, numerous interesting results may be obtained including:
- Isochromatics only, without errors in position
- Isochromatics-isopachic fringes identical to those obtained through classical interferometry
- Isochromatic-isopachic fringes whereby the amplitude modulation between the two may be minimized
- Continuously variable isopachic/isochromatic-fringe visibility.
18.
《Wave Motion》2018
In this paper we present a strategy for the synthesis of acoustic sources with controllable near fields in free space and constant depth homogeneous ocean environments. We first present the theoretical results at the basis of our discussion and then, to illustrate our findings we focus on the following three particular examples:
- 1.acoustic source approximating a prescribed field pattern in a given bounded sub-region of its near field.
- 2.acoustic source approximating different prescribed field patterns in given disjoint bounded near field sub-regions.
- 3.acoustic source approximating a prescribed back-propagating field in a given bounded near field sub-region while maintaining a very low far field signature.
19.
Jordan Goblet 《Journal of Dynamics and Differential Equations》2012,24(3):483-494
The present note is a continuation of the author??s effort to study the existence of continuously differentiable solutions to the semi-implicit system of differential equations (1) $$f(x^{\prime}(t)) = g(t, x(t))$$ (2) $$\quad x(0) = x_0,$$ where ${\quad\Omega_g \subseteq \mathbb{R} \times\mathbb{R}^n}$ is an open set containing (0, x 0) and ${g:\Omega_g \rightarrow\mathbb{R}^n}$ is a continuous function, ${\quad\Omega_f \subseteq \mathbb{R}^n}$ is an open set and ${f:\Omega_f\rightarrow\mathbb{R}^n}$ is a continuous function. The transformation of (1)?C(2) into a solvable explicit system of differential equations is trivial if f is locally injective around an element ${\gamma\in \Omega_f\cap f^{-1}(g(0,x_0))}$ . In this paper, we study (1)?C(2) when such a translation is not possible because of the inherent multivalued nature of f ?1. 相似文献
20.
We study supersonic flows of an electrically conductive gas in crossed electric and magnetic fields [1] in the presence of shock waves. It is shown that three steady flow regimes can exist, and that these are defined by the electrical conductivity of the gas as a function of temperature and density.
- The normal regime is characterized by a tendency for the shock to move toward the channel entrance on increase of the static pressure at the channel exit. The steady regime of this type exists and is stable.
- The anomalous regime (formally constructed) is characterized by a tendency for the shock to move toward the exit on increase of the static pressure at the channel exit. This regime is unstable and the flow in the MHD-channel may be either entirely supersonic or entirely subsonic.
- The limiting (boundary) regime is intermediate between the normal and anomalous regimes and is characterized by the fact that the stationary position of the shock wave and its amplitude are not uniquely defined. Steady flow in this case is not unique.