Forced convection problems and some related Fredholm integral equations of the first kind

Abstract

In this thesis a variety of integral equations and partial differential equations which describe two-dimensional and three-dimensional axisymmetric forced convection problems are studied. The correspondence between the integral equation formulation and partial differential equation formulation of such boundary value problems is investigated and this correspondence is used to develop a new technique for the solution of hitherto intractable Fredholm integral equations of the first kind on a finite interval. Basic properties of the physical models are also established by studying fundamental solutions of the partial differential equations. The problems are developed by considering the transport of heat in an inviscid, incompressible, heat conducting fluid whose thermal constants are assumed invariant in temperature, space and time. The transport mechanisms are those of convection in a prescribed velocity field and diffusion, and the temperature field is given by the solution of the partial differential equation (and, less directly, by the integral equation) which is a heat conservation equation and characterises the balance between the diffusion and convection processes. While such problems are highly idealised they have more practical applications in several special cases in which the partial differential equation takes the form of a linear approximation of the Navier - Stokes equations. Circulation is transported by diffusion and convection in a viscous fluid and in certain circumstances the temperature model can be regarded as a model of a viscous fluid. The integral equations also arise in an elastic half-space problem which is described in the first chapter.The two-dimensional problem has been the subject of-many previous investigations. New contributions are made in the domain of the integral equation whose solution has been given explicitly for the first time, and in the dependence of the solutions on the Peclet number, a non-dimensional parameter which characterises the ratio of the diffusive flux to the convective flux, where certain physically motivated results have been confirmed by a more strict mathematical approach. Similar, but considerably more extensive, investigations are made of a family of three-dimensional axisymmetric problems. In these problems, the first order radial derivative term in the partial differential equation has a singular coefficient whose role in the equation is interpreted as that due to a radial component in the forced convection field (‘radial' means 'radial direction in a cylindrical polar coordinate system') The crucial problem here is the realisation of a full understanding of the relationship between the partial differential equation and the integral equation. This is achieved by rewriting the partial differential equation in what is essentially its adjoint form and interpreting the new partial differential equation as a conservation equation which also expresses the balance of diffusion and convection of a quantity called <<-heat in a prescribed convection field. Physically motivated arguments are used to suggest the relationship between the original partial differential equation and the integral equation formulation which is then proved using rigorous mathematical arguments. A Peclet number is defined in this problem and the dependence of the solutions on this parameter is analysed. Representation and uniqueness theorems are also given in classical cases and a discussion of the extension of these results into generalised function spaces is included.