Subharmonic non-linear acoustic resonances in open tubes—Part I,theory

Summary An open tube is considered, driven at one-half of the fundamental frequency. The interesting fact appears that acoustic theory breaks down in the inviscid case. At a sufficient intensity of a resonant overtone, the distortion effects become too strong to use an expansion scheme.

---

Zusammenfassung Betrachtet wird ein offenes Rohr, das bei der halben Resonanzfrequenz angetrieben wird. Es ergibt sich die bemerkenswerte Tatsache, dass die akustische Theorie im reibungsfreien Fall ihre Gültigkeit verliert. Bei genügend grosser Amplitude eines resonanten Obertones werden die Dispersionseffekte so stark, dass die Entwicklung der Lösungen nach kleinen Machzahlen ihre Gültigkeit verliert.

     

Subharmonic non-linear acoustic resonances in closed tubes

Summary A closed tube is considered driven at one-half of the fundamental frequency. The interesting fact appears that acoustic theory breaks down in the inviscid case. At a sufficient intensity of a first resonant overtone, the theory of Chester applies.

---

Zusammenfassung Betrachtet wird ein geschlossenes Rohr, das bei ber halben Resonanz-frequenz angetrieben wird. Es ergibt sich die bemerkenswerte Tatsache, dass die akustische Theorie im reibungsfreien Fall ihre Gültigkeit verliert. Bei genügend grosser Amplitude eines ersten Obertones ergibt sich die Theorie von Chester.

     

Localisation of the discharge tubes Part II

Proceedings - Mathematical Sciences - The phenomenon of the localisation of the discharge in ordinary discharge tubes has been studied quantitatively as a function of the gas in the discharge tube...     

The numerical solution of a periodic parabolic problem subject to a nonlinear boundary condition. II

Approximating the differential equation by the standard fully implicit finite difference approximation reduces the problem to that of solving a set of non-linear algebraic equations. In the first part of this paper it was shown that Newton's method provided a satisfactory technique for the solution of this set of equations. In this paper the implementation of Newton's method is discussed, and it is shown that a device due toBickley andMacNamee can be used very successfully. A more accurate difference formula is also considered and a summary of numerical results presented.     

Smoothing by mollifiers. Part II: nonlinear optimization

This article complements the paper (Jongen, Stein, Smoothing by mollifers part I: semi-infinite optimization J Glob Optim doi:), where we showed that a compact feasible set of a standard semi-infinite optimization problem can be approximated arbitrarily well by a level set of a single smooth function with certain regularity properties. In the special case of nonlinear programming this function is constructed as the mollification of the finite min-function which describes the feasible set. In the present article we treat the correspondences between Karush–Kuhn–Tucker points of the original and the smoothed problem, and between their associated Morse indices.      

Numerical investigation of quenching for a nonlinear diffusion equation with a singular Neumann boundary condition

For a nonlinear diffusion equation with a singular Neumann boundary condition, we devise a difference scheme which represents faithfully the properties of the original continuous boundary value problem. We use non‐uniform mesh in order to adequately represent the spatial behavior of the quenching solution near the boundary. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 429–440, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10013     

Thermally driven acoustic oscillations. Part II: Stability limit for helium

Summary The stability limits for thermally driven acoustic oscillations in a helium-filled tube are given. Calculations are partly based on a step-by-step integration of the differential equation for the acoustic pressure, but most results refer to a model using a discontinuous temperature distribution; in the latter case a transcendental stability-frequency equation has been numerically solved. Results are presented in the form of critical temperature ratios versus a parameter representing the ratio between the tube radius and the extent of the viscous flow region. Several asymptotic limits of the stability curves are analytically treated. Finally, limitations of the theory are discussed which occur when the solid tube wall specific heats become vanishingly small at the cold end.

---

Zusammenfassung Die Stabilitätsgrenze von thermisch getriebenen akustischen Schwingungen von Helium in einem Rohr wird angegeben. Rechnungen beruhen zum Teil auf der schrittweisen Integration der Differentialgleichung für den akustischen Druck, doch wurden die meisten Resultate für ein Modell mit einem Temperatursprung erhalten; in diesem Falle wurde eine transzendente Stabilitäts-Frequenz-Gleichung numerisch gelöst. Resultate geben ein kritisches Temperaturverhältnis über einen Parameter, welches das Verhältnis von Rohr-Radius zur Größe der Reibungsschicht der Strömung gibt. Verschiedene Asymptoten der Stabilitätskurve werden analytisch behandelt. Schliesslich werden die Begrenzungen der Theorie diskutiert, die durch die verschwindend kleine spezifische Wärme der Rohrwand am kalten Ende bedingt sind.

     

Numerical investigation of blood flow. Part II: In capillaries

In order to understand the normal and pathologic behavior of the human vascular system, detailed knowledge of blood flow and the response of blood vessels is required. In fact the ability to predict the flow hydrodynamics at any site in the vessels can lead to a better understanding of the behavior of blood flow. Simulation can play an important role in understanding the hemodynamic forces. The objective of the present attempt was to simulate the behavior of blood flow in microvessels using computational fluid dynamics (CFD). Numerical analysis is performed using a commercially available CFD package Fluent 6.2 which is based on the finite volume method. A continuum approach is proposed in which fluid structure interaction has been taken into account. Based on limitations imposed by computational resources, a more simplified model based on volume of fluid (VOF) approach is suggested to simulate movements of RBCs in capillaries and also to predict RBCs’ deformation. Three-dimensional incompressible laminar flow fields are obtained by solving continuity and Navier–Stokes equations computationally. It was found that multiphase CFD simulations may give further insight into the dynamic characteristics of blood flow under complex flow conditions.     

High-order three-point schemes for boundary value problems. II: nonlinear problems

A highly accurate method for solving nonlinear boundary value problems of the convection-diffusion type is presented. It is shown that the rate of convergence of the present method is very rapid, superior to that of the classical central difference methods. Its main feature is that the converged solution can be achieved within a reasonable amount of iterations for any small value of the mesh diffusion coefficient. All these features are supported by comparisons that are presented for several example problems.     

Existence of nonlinear boundary layer to the Boltzmann equation with reverse reflection boundary condition

Many physical models have boundaries. When the Boltzmann equation is used to study a physical problem with boundary, there usually exists a layer of width of the order of the Knudsen number along the boundary. Hence, the research on the boundary layer problem is important both in mathematics and physics. Based on the previous work, in this paper, we consider the existence of boundary layer solution to the Boltzmann equation for hard sphere model with positive Mach number. The boundary condition is imposed on incoming particles of reverse reflection type, and the solution is assumed to approach to a global Maxwellian in the far field. Similar to the problem with Dirichlet boundary condition studied in [S. Ukai, T. Yang, S.H. Yu, Nonlinear boundary layers of the Boltzmann equation: I. Existence, Comm. Math. Phys. 3 (2003) 373-393], the existence of a solution is shown to depend on the Mach number of the far field Maxwellian. Moreover, there is an implicit solvability condition on the boundary data. According to the solvability condition, the co-dimension of the boundary data related to the number of the positive characteristic speeds is obtained.     

Analysis of resonances in the spin-orbit problem in Celestial Mechanics: Higher order resonances and some numerical experiments (Part II)

In this work we study the stability of the non-synchronous resonances in the spin-orbit problem in Celestial Mechanics. Using a mathematical model obtained making some restrictions on the physical problem, we trap the 32 resonance between invariant surfaces for realistic values of the physical and orbital parameters in the cases of the Moon and other three satellites of Saturn. We also provide some remarks on higher order resonances and on the Mercury-Sun system.Moreover we apply some numerical techniques in order to determine the break-down threshold of the invariant tori. In the last section we give some remarks on the robustness of the golden mean torus.     

Global solutions of the heat equation with a nonlinear boundary condition

We consider the heat equation with a nonlinear boundary condition, $$(P) \left\{\begin{array}{ll} \partial_t u = \Delta u, & x \in \Omega, \quad t > 0, \\ \partial_\nu u=u^p, & x \in \partial \Omega,\quad t > 0,\\ u (x,0) = \phi (x),& x\in\Omega, \end{array}\right.$$ where ${\Omega = \{x = (x^{\prime},x_N) \in {\bf R}^{N} : x_N > 0\}, N \ge 2, \partial_t = \partial{/}\partial t , \partial_\nu = -\partial{/}\partial x_{N}}$ , p > 1 + 1/N, and (N ? 2)p < N. In this paper we give a complete classification of the large time behaviors of the nonnegative global solutions of (P).     

非线性抛物方程在非线性边界条件下解的爆破时间的下界

研究了非线性抛物方程在非线性边界条件下的解的爆破问题,通过构造一个能量表达式,运用微分不等式的方法,得到该能量表达式所满足的微分不等式,然后通过积分得到当爆破发生时解在非线性边界条件下的爆破时间的下界.     

Superconvergence analysis of the finite element method for nonlinear hyperbolic equations with nonlinear boundary condition

This paper discusses the semidiscrete finite element method for nonlinear hyperbolic equations with nonlinear boundary condition. The superclose property is derived through interpolation instead of the nonlinear H^1 projection and integral identity technique. Meanwhile, the global superconvergence is obtained based on the interpolated postprocessing techniques.     

Solvability of quasilinear elliptic equations with nonlinear boundary conditions. II

It is shown that solvability of the second order quasilinear elliptic equation Qu = 0 in Ω with first order nonlinear boundary condition Nu = 0 on ?Ω can be inferred from appropriate estimates on solutions of the problem $\text{Qu = ?}$ in Ω, Nu = 0 on ?Ω as ? varies over a suitable function class. This result improves previous work of the author, where estimates were required on solutions of $\text{Qu = ?}$ in Ω, Nu = ? on ?Ω as (?, ?) varies over some function space. The value of this improvement is demonstrated by some examples.     

Critical exponents of the non-Newtonian polytropic filtration equation with nonlinear boundary condition

This work is concerned with the critical exponent of the non-Newtonian polytropic filtration equation with nonlinear boundary conditions. We obtain the critical global existence exponent and critical Fujita exponent by constructing various self-similar supersolutions and subsolutions.     

On positive solutions of the heat equation satisfying a nonlinear boundary condition

We consider a boundary value problem for the heat equation in the exterior of a bounded domain of space variables. On the boundary of the domain, we pose a nonlinear boundary condition. We find sharp nonlinearity exponents for which there exists no global solution.     

Asymptotic regularity and attractors of the reaction-diffusion equation with nonlinear boundary condition

We consider the dynamical behavior of the reaction-diffusion equation with nonlinear boundary condition for both autonomous and non-autonomous cases. For the autonomous case, under the assumption that the internal nonlinear term f is dissipative and the boundary nonlinear term g is non-dissipative, the asymptotic regularity of solutions is proved. For the non-autonomous case, we obtain the existence of a compact uniform attractor in H¹(Ω) with dissipative internal and boundary nonlinearities.