where u — u(x, t) in some domain and v is a parameter. The occurrence of the first derivative in t and the second in x clearly indicates the equation is parabolic, similar to the heat equation, while the interesting additional feature is the occurrence of the non-linear term u du/dx. The equation thus shows a structure roughly similar to that of the Navier-Stokes equations and has actually appeared in two separate problems in aerodynamics. An equation simply related to (1) appears in the approximate theory of a weak non-stationary shock wave in a real fluid. This is discussed in Ref. 1 (pp. 146-154) where a general solution of (1) is given. The equation is also given in J. Burgers' theory of a model of turbulence (Ref. 2) where he notes the relationship between the model theory and the shock wave. Historically, the equation (1) first appears in a paper by H. Bateman (Ref. 3) in 1915 when he mentioned it as worthy of study and gave a special solution. Eq. (1) is of some mathematical interest in itself and may have applications in the theory of stochastic processes. The aim of this paper is to study the general properties of (1) and relate the various applications. I wish to thank Professor P. A. Lagerstrom and F. K. Chuang for helpful collaboration. 2. Relationship of (1) to Shock Wave Theory. The solutions to Eq. (1) can approximately describe the flow through a shock wave in a viscous fluid. They can be related to the shock wave in several ways. In Ref. 1 an approximation based on the NavierStokes equations for one-dimensional non-stationary flow of a compressible viscous fluid gives

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On a quasi-linear parabolic equation occurring in aerodynamics

This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime. The field's action on the particle is difficult to calculate because of its singular nature: the field diverges at the position of the particle. But it is possible to isolate the field's singular part and show that it exerts no force on the particle. What remains after subtraction is a smooth field that is fully responsible for the self-force. The mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime are developed here from scratch. The review begins with a discussion of the basic theory of bitensors. It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle's word line. It continues with a thorough discussion of Green's functions in curved spacetime. The review presents a detailed derivation of each of the three equations of motion. Because the notion of a point mass is problematic in general relativity, the review concludes with an alternative derivation of the equations of motion that applies to a small body of arbitrary internal structure.

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The Motion of Point Particles in Curved Spacetime

http://numericaltank.sjtu.edu.cn/KeyArticles/LIAO-Notes-2009.pdf

Notes on the homotopy analysis method: Some definitions and theorems

A visualization method is used to obtain the main features of the hydrodynamic field for flow past a circular cylinder moving at a uniform speed in a direction perpendicular to its generating lines in a tank filled with a viscous liquid. Photographs are presented to show the particular fineness of the experimental technique. More especially, the closed wake and the velocity distribution behind the obstacle are investigated; the changes in the geometrical parameters describing the eddies with Reynolds number (5 < Re < 40) and with the ratio λ between the diameters of the cylinder and tank are given. A comparison with existing numerical and experimental results is presented and some remarks are made about the calculation techniques proposed up to the present. The limits of the Reynolds-number range for which the twin vortices exist and adhere stably to the cylinder are determined.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112077000147

Experimental determination of the main features of the viscous flow in the wake of a circular cylinder in uniform translation. Part 2. Unsteady flow

Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale “mechanism.” Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

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Mixed-Mode Oscillations with Multiple Time Scales

Matched Asymptotic Expansions

I. Introduction: General Concepts of Singular Perturbation Theory.- II. Layer-type Problems. Ordinary Differential Equations.- III. Layer-type Problems. Partial Differential Equations.- List of References.

Matched Asymptotic Expansions: Ideas and Techniques

The present study was suggested by several problems and difficulties that had appeared in previous experimental and theoretical investigations of viscosity effects in compressible fluids. The outstanding problem was the extension of the classical (Prandtl) boundary-layer theory to high-speed flow, especially supersonic flow. In the boundary-layer theory the equations of motion are simplified by assuming that viscous effects are confined to a narrow region close by the wall through which changes are rapid compared to those in the direction of the wall. Then the resulting non-linear equations are studied with the aim of obtaining the flow field in this narrow region or boundary layer. The pressure is usually obtained from the potential or no-viscous flow about the body. Several authors have studied a boundary-layer theory which has the same basic assumptions but which allows for compressibility and heat conduction. However, in supersonic flow several phenomena are known which show that the basic assumptions of boundary-layer theory do not apply, at least in certain regions.

Problems in the Theory of Viscous Compressible Fluids

Publisher's PrefaceCh. 1The Navier-Stokes Equations for a Viscous Heat-Conducting Compressible Fluid3Ch. 2Review of Viscous Incompressible Fluids45Ch. 3Introductory Discussion of the Navier-Stokes Equations for a Compressible Fluid150Ch. 4Laminar Boundary Layers in Compressible Fluids191Cited References265

P. A. Lagerstrom

Papers

Matched Asymptotic Expansions

Matched Asymptotic Expansions: Ideas and Techniques

Problems in the Theory of Viscous Compressible Fluids

Laminar Flow Theory

Low Reynolds Number Flow Past a Circular Cylinder