Self-similar solutions of the magnetohydrodynamic boundary layer system for a dilatable fluid

Summary A rigorous mathematical analysis is given for a magnetohydrodynamic boundary layer problem, which arises in the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting dilatable fluid along an isolated surface in the presence of an exterior magnetic field orthogonal to the flow. In the self-similar case, the problem is transformed into a third-order nonlinear ordinary differential equation with certain boundary conditions, which is proved to be equivalent to a singular initial value problem for an integro-differential equation of first order. With the aid of the singular initial value problem, the uniqueness, existence, and nonexistence results for generalized normal solutions are established.     

反对称正交铺层剪切圆柱壳在外压下的屈曲和后屈曲

给出了反对称正交铺层剪切圆柱壳广义大挠度Donnell 型方程, 并运用位移型摄动技术构造出该圆柱壳在均匀外压作用下的后屈曲渐近级数解。考虑到边界效应对中短圆柱壳的影响及边值问题摄动解的一致性, 详细研究了该圆柱壳端部边界层方程和奇异摄动解, 以便与中部正则摄动解相匹配。文中同时给出一些典型例子并讨论了横向剪切变形、Batdo rf 数、弹性模量比和初始几何缺陷对圆柱壳屈曲与后屈曲性态的影响。比较显示, 横向剪切变形对圆柱壳的屈曲与后屈曲有重要影响。      

Blow-up rates for degenerate parabolic equations coupled via equation and boundary

In this article, we consider a degenerate parabolic system coupled with an equation and a boundary condition. We get the sufficient condition and necessary conditions of the solutions blowing up in finite time. For blow-up solutions, we establish the precise blow-up estimates.     

On extra boundary condition in the stagnation point flow of a second grade fluid

The two dimensional stagnation point flow of a second grade fluid is considered. The flow is governed by a boundary value problem in which the order of differential equations is one more than the number of available boundary conditions. It is shown that without augmenting the boundary conditions at infinity it is possible to obtain a numerical solution of the problem for all values of K, where K is the dimensionless viscoelastic fluid parameter. The numerical results using the algorithm foreshadow an asymptotic behavior for large K. The asymptotic solution is derived up to terms of O(K⁻¹). Perturbation solutions are also obtained up to the terms of O(K²). Finally an approximate solution is developed, based on stretching of the independent variable and minimizing the residual of the differential equation in the least square sense. All these solutions are compared with the exact numerical solution and the appropriate conclusions are drawn.     

Interface waves along an anisotropic imperfect interface between anisotropic solids

First and second order asymptotic boundary conditions are introduced to model a thin anisotropic layer between two generally anisotropic solids. Such boundary conditions can be used to describe wave interaction with a solid-solid imperfect anisotropic interface. The wave solutions for the second order boundary conditions satisfy energy balance and give zero scattering from a homogeneous substrate/layer/substrate system. They couple the in-plane and out-of-plane stresses and displacements on the interface even for isotropic substrates. Interface imperfections are modeled by an interfacial multiphase orthotropic layer with effective elastic properties. This model determines the transfer matrix which includes interfacial stiffness and inertial and coupling terms. The present results are a generalization of previous work valid for either an isotropic viscoelastic layer or an orthotropic layer with a plane of symmetry coinciding with the wave incident plane. The problem of localization of interface waves is considered. It is shown that the conditions for the existence of such interface waves are less restrictive than those for Stoneley waves. The results are illustrated by calculation of the interface wave velocity as a function of normalized layer thickness and angle of propagation. The applicability of the asymptotic boundary conditions is analyzed by comparison with an exact solution for an interfacial anisotropic layer. It is shown that the asymptotic boundary conditions are applicable not only for small thickness-to-wavelength ratios, but for much broader frequency ranges than one might expect. The existence of symmetric and SH-type interface waves is also discussed.     

The asymptotic boundary layer on a circular cylinder in axisymmetric micropolar fluid flow

An asymptotic boundary layer analysis is presented using the theory of micropolar fluids due to Eringen. The laminar boundary layer induced on the outside of a long, slender cylinder due to the flow of an incompressible micropolar fluid parallel to the axis of the cylinder is investigated. For reasons of both analytical and practical interests the boundary layer characteristics far down stream from the leading edge are analyzed on the basis of their asymptotic nature. Asymptotic series solutions for the velocity and micro-rotation fields are obtained. An expression for the new micropolar boundary layer thickness is derived. Central to the present investigation is the result that while calculating the skin friction one should take into account the total surface stress effects, not only due to the usual shear stresses but also due to the couple stresses. As a result, it is shown that the micropolar theory does predict a reduction in skin friction as is observed in experiments thus confirming Eringen's well known conjecture.     

Boundary layer on a permeable wall with suction of gas

The differential model of turbulence, supplemented with transport equation for turbulent heat flux, is used to perform a numerical investigation of the boundary layer on a permeable wall with suction of gas. It is demonstrated that the protraction of transition from laminar to turbulent mode of flow and the laminarization of the initial turbulent boundary layer occur under conditions of suction of gas. This is evidenced both by the behavior of integral and local characteristics of flow and heat transfer and by the degeneracy of turbulence when the suction of laminar turbulent layer becomes asymptotic. The critical values of the suction parameter are determined.     

A NUMERICAL STUDY OF THE FLOW OF GRANULAR MATERIALS BETWEEN TWO VERTICAL FLAT PLATES WHICH ARE AT DIFFERENT TEMPERATURES

The present study considers an assembly of spherical particles, densely packed, between two vertical flat plates which are at different temperatures. A continuum model is used and the flow due to such a temperature difference is investigated. For a fully developed flow of these materials, the governing equations reduce to a system of coupled, non-linear ordinary differential equations. The equations are integrated numerically. In so doing, the equations are discretized using the conventional central finite difference approximation technique and the unknown conditions are assumed to reduce the problem to an initial value problem. After the trial solutions the known boundary conditions at the edge of the integration interval are compared with the correspondent values provided by the trial solutions. If the solutions at this point do not agree with the known boundary conditions the Newton-Raphson method is used to correct the initial guesses and the iteration is repeated. This procedure continues until the solutions converge to the given values. The technique is found to be powerful for this type of application.     

A NUMERICAL STUDY OF THE FLOW OF GRANULAR MATERIALS BETWEEN TWO VERTICAL FLAT PLATES WHICH ARE AT DIFFERENT TEMPERATURES

ABSTRACT

The present study considers an assembly of spherical particles, densely packed, between two vertical flat plates which are at different temperatures. A continuum model is used and the flow due to such a temperature difference is investigated. For a fully developed flow of these materials, the governing equations reduce to a system of coupled, non-linear ordinary differential equations. The equations are integrated numerically. In so doing, the equations are discretized using the conventional central finite difference approximation technique and the unknown conditions are assumed to reduce the problem to an initial value problem. After the trial solutions the known boundary conditions at the edge of the integration interval are compared with the correspondent values provided by the trial solutions. If the solutions at this point do not agree with the known boundary conditions the Newton-Raphson method is used to correct the initial guesses and the iteration is repeated. This procedure continues until the solutions converge to the given values. The technique is found to be powerful for this type of application.     

Long wave motion in layered elastic media

Long wave motion in a geometrically symmetric 3-layer laminated elastic structure is investigated. The associated dispersion relation is established for three different boundary value problems. For all three cases, numerical solutions are presented and a long wave asymptotic analysis carried out, in each case the cut-off frequencies being shown to satisfy transcendental equations. Long wave approximations are employed to determine the asymptotic orders of the displacement components in the various long wave regimes. The asymptotic structures in a single layer plate associated with bending, extension, thickness stretch resonance and thickness shear resonance are well-known. It is shown that these structures are preserved within the multi-layer problem. This work provides the theoretical framework to generalise the above mentioned theories.     

Wave breaking and shock waves for a periodic shallow water equation

This paper is devoted to the study of a recently derived periodic shallow water equation. We discuss in detail the blow-up scenario of strong solutions and present several conditions on the initial profile, which ensure the occurrence of wave breaking. We also present a family of global weak solutions, which may be viewed as global periodic shock waves to the equation under discussion.     

Blow-up and control of marginally separated boundary layers

Interactive solutions for steady two-dimensional laminar marginally separated boundary layers are known to exist up to a critical value Gamma(c) of the controlling parameter (e.g. the angle of attack of a slender airfoil) Gamma only. Here, we investigate three-dimensional unsteady perturbations of such boundary layers, assuming that the basic flow is almost critical, i.e. in the limit Gamma(c)-Gamma-->0. It is then shown that the interactive equations governing such perturbations simplify significantly, allowing, among others, a systematic study of the blow-up phenomenon observed in earlier investigations and the optimization of devices used in boundary-layer control.     

Intermediate asymptotics for Richards' equation in a finite layer

Perturbation methods are applied to study an initial-boundary-value problem for Richards' equation, describing vertical infiltration of water into a finite layer of soil. This problem for the degenerate diffusion equation with convection and Dirichlet/Robin boundary conditions exhibits several different regimes of behavior. Boundary-layer analysis and short-time asymptotics are used to describe the structure of similarity solutions, traveling waves, and other solution states and the transitions connecting these different intermediate asymptotic regimes.     

Stabilization of a system modeling temperature and porosity fields in a Kelvin�CVoigt-type mixture

In this paper, we investigate the asymptotic behavior of solutions to the initial boundary value problem for the interaction between the temperature field and the porosity fields in a homogeneous and isotropic mixture from the linear theory of porous Kelvin?CVoigt materials. Our main result is to establish conditions which insure the analyticity and the exponential stability of the corresponding semigroup. We show that under certain conditions for the coefficients we obtain a lack of exponential stability. A numerical scheme is given.     

Flow with convective acceleration through a porous medium

Summary The flow streaming into a porous and permeable medium with arbitrary but smooth wall surface is considered on the basis of the Euler equation (in the pure fluid region) and a generalized Darcy's law in which the convective acceleration is taken into account. The asymptotic behavior of the flow for small permeability of the medium is investigated. It is shown that the flow in the porous medium is irrotational except in the boundary layer next to the surface. The velocity distribution in the boundary layer is given in a universal form. Proper boundary conditions connecting the potential flow in the pure fluid region and the potential flow in the porous medium are obtained when the boundary layer is neglected.On leave from Department of Aeronautical Engineering, Kyoto University, Kyoto, Japan.     

Arbitrary axisymmetric steady streaming: flow,force and propulsion

A well-developed method to induce mixing on microscopic scales is to exploit flows generated by steady streaming. Steady streaming is a classical fluid dynamics phenomenon whereby a time-periodic forcing in the bulk or along a boundary is enhanced by inertia to induce a non-zero net flow. Building on classical work for simple geometrical forcing and motivated by the complex-shaped oscillations of elastic capsules and bubbles, we develop the mathematical framework to quantify the steady streaming of a spherical body with arbitrary axisymmetric time-periodic boundary conditions. We compute the flow asymptotically for small-amplitude oscillations of the boundary in the limit where the viscous penetration length scale is much smaller than the body. In that case, the flow has a boundary layer structure, and the fluid motion is solved by asymptotic matching. Our results, presented in the case of no-slip boundary conditions and extended to include the motion of vibrating free surfaces, recover classical work as particular cases. We illustrate the flow structure given by our solution and propose one application of our results for small-scale force generation and synthetic locomotion.     

A note on the Blasius and Sakiadis flow of a non-Newtonian power-law fluid in a constant transverse magnetic field

The Blasius and Sakiadis flows of a non-Newtonian power-law conducting fluid under the effect of a constant transverse magnetic field is considered. The boundary layer equations are transformed into a non-dimensional form and a new dimensionless magnetic parameter is introduced. The transformed boundary layer equations are solved with a finite difference method. Both Blasius and Sakiadis flows reach an asymptotic state and become one-dimensional with the increase of the coordinate in the streamwise direction. The flow behavior in the intermediate region is studied and exact analytical solutions have been found for the asymptotic state. The characteristics of physical and engineering interest are discussed in the paper in detail.     

Boundary integral methods in low frequency acoustics

Abstract

This expository paper is concerned with the direct integral formulations for boundary value problems of the Helmholtz equation. We discuss unique solvability for the corresponding boundary integral equations and its relations to the interior eigenvalue problems of the Laplacian. Based on the integral representations, we study the asymptotic behaviors of the solutions to the boundary value problems when the wave number tends to zero. We arrive at the asymptotic expansions for the solutions, and show that in all the cases, the leading terms in the expansions are always the corresponding potentials for the Laplacian. Our integral equation procedures developed here are general enough and can be adapted for treating similar low frequency scattering problems.     

Blow-up for doubly degenerate parabolic system with nonlocal sources

This article deals with the blow-up properties of the solution to a doubly degenerate parabolic system with nonlocal sources and inner absorptions, subject to homogeneous Dirichlet boundary conditions. We first establish the local existence and uniqueness of its classical solutions. Then we show that the critical exponent is determined by the interaction among all the six nonlinear exponents from all three kinds of the nonlinearities. Finally, we give the precise blow-up estimates and the uniform blow-up profiles.