Boundary Integral Equation Solution of Non-linear Plane Potential Problems

This paper presents the boundary integral equation (BIE) formulation,and numerical solution procedure for two-dimensional problemsgoverned by Laplac's equation and subject to non-linear boundaryconditions. The introduction of non-linear terms constitutesa fundamental extension of the BIE method, as previous applicationshave been restricted entirely to linear problems. Furthermore,non-linearities necessitate the use of iterative solution techniqueswhich present the conceptual disadvantage that a solution isnot guaranteed. However, such difficulties were not encounteredwith the Newton—Raphson method employed in this study.The various features of the BIE technique are illustrated bythe application to a physical problem which is of significancein heat exchanger design.     

The Boundary-Element Method for an Improperly Posed Problem

This paper investigates the numerical solution of an inverseLaplace problem which is improperly posed. Three different mathematicalmodels, using direct, least-squares, and minimal-energy methodsare presented for four test problems. The boundary-element methodis used, and it is found that the minimal-energy method alwaysgives a good stable approximation to the solution, whereas thedirect and least-squares methods do not.     

Flow Past an Impulsively Started Semi-infinite Flat Plate in the Presence of a Magnetic Field

The solution of the motion of a viscous electrically conductingfluid past a semi-infinite flat plate, which is started impulsivelyfrom rest with a constant velocity parallel to itself, in thepresence of an applied magnetic field which is parallel to theplate at infinity, is investigated. The governing partial differentialequations can be written in similarity form and reduced to aset of coupled, non-linear, singular parabolic partial differentialequations. Numerical methods have been developed previouslywhich deal with uncoupled equations. The analytical work forlarge time indicates that the steady state solution is approachedexponentially when the equations uncouple but a simple extensionof this analysis is not possible when the equations are coupled.     

Plane Sub-Critical Flow Past a Lifting Aerofoil

In this paper a numerical technique is used in the solutionof the equations governing the two-dimensional flow past a liftingaerofoil. The method as originated by Sells is modified by theuse of an extra conformal mapping such that more mesh pointsoccur in the region where the flow varies most. Thus more accurateresults are obtained without having to use substantially morecomputer storage or time. Results illustrating the method are presented for the 10% RAE101 section, at zero and non-zero incidence.     

Spacewise coefficient identification in steady and unsteady one-dimensional diffusion problems

In this paper the determination of the spacewise dependent materialproperty coefficients and the function solution in both steadyand unsteady diffusion problems are analysed. For a one-dimensionalquasi-heterogeneous material with square-root harmonic conductivityit is shown that a single measurement of the conductivity andthe flux on the boundary is sufficient to determine uniquelythe unknown physical property and the function solution.     

Free Convection from a Vertical Plate in a Porous Medium Subjected to a Porous Medium Subjected to a Sudden Change in Surface Heat Flux

An analysis is made of the transient free convection from a vertical flat plate which is embedded in a fluid-saturated porous medium. It is assumed that for time a steady state temperature or velocity has been obtained in the boundary-layer which occurs due to a uniform flux dissipation rate . Then at time the heat flux on the plate is suddenly changed to and maintained at this value for 0$$ " align="middle" border="0"> . An analytical solution has been obtained for the temperature/velocity field for small times in which the transport effects are confined within an inner layer adjacent to the plate. These effects cause a new steady boundary layer. A numerical solution of the full boundary-layer equations is then obtained for the whole transient from to the steady state, firstly by means of a step-by-step method and then by a matching technique. The transition between the two distinct solution methods is always observed to occur very near to the turning point of the plate surface temperature, a time at which the fluid temperature is close to its steady state profile. The solution obtained using the step-by-step method shows excellent agreement with the small time analytical solution. Results are presented to illustrate the occurrence of transients from both small and large increases and decreases in the levels of existing energy inputs.     

The Magnetogasdynamic Boundary Layer on a Thermally Insulated Flat Plate

This paper examines the effect of compressibility on the flowin the boundary layer on a semi-infinite, thermally insulatedflat plate placed at zero incidence to a uniform stream of electricallyconducting gas, with an aligned magnetic field at large distancesfrom the plate. The present discussion is limited to small values of the conductivityparameter = 4µv, and the Prandtl number is taken to beunity. The latter assumption permits a simplification of theanalysis, and the former allows the dependence of the flow onthe parameters ß = µH²/4U² and M = U/cto beadequately illustrated without excessive computation. A seriessolution valid for small values of the conductivity parameterand for Mach numbers not too large is derived. Values of ß = 0.3 and 0.5, = 0.01 and 0.1 are consideredand for those values the skin friction decreases with increasingMach number, similar to the case when ß = 0. The analysissuggests that for larger values of ß the skin frictionmight even increase with the Mach number initially. This iscertainly the case with the tangential component of the magneticfield, which for ß = 0.5 exhibits a maximum at approximatelyunit Mach number. The reason for this behaviour lies in thefact that, in view of the temperature changes taking place inthe flow, the electrical conductivity and thereby the localvalue of can change by more than an order of magnitude. Thishas the effect of giving results which are akin to those forarbitrary large in incompressible flow even though the valueof based on the main stream gas properties remains low.