Partial Differential Equations in Mechanics 2

An important common theme in the developments presented in connection with Laplace’s equation, the diffusion equation and the wave equation is that they are all of the second-order and represent the fundamental equations which govern elliptic, parabolic and hyperbolic partial differential equations, respectively. A further general observation in previous expositions is that as the phenomena that are being modelled becomes either more complex or encompasses more complicated fundamental processes, the partial differential equations which describe such phenomena are expected to acquire a higher order. This was evident in the description of advection-diffusion phenomena governing the transport of chemicals in porous media. In the presence of only advective phenomena the transport process can be described by a first-order partial differential equation; when diffusive processes are taken into consideration, the transport process can be described by a second-order partial differential equation. The biharmonic equation is one such partial differential equation which arises as a result of modelling more complex phenomena encountered in problems in science and engineering. The term biharmonic is indicative of the fact that the function describing the processes satisfies Laplace’s equation twice explicitly. The exact first usage of the biharmonic equation is not entirely clear since every harmonic function which satisfies Laplace’s equation is also a biharmonic function.

Poisson’s equation is the inhomogeneous equivalent of Laplace’s equation. It is encountered in the modelling of a variety of problems in mechanics and physics, ranging from the study of fluid flows in porous media to the theory of gravitation, In Chapter 5 dealing with Laplace’s equation, we have briefly encountered Poisson’s equation in connection with the development of the Green’s function for Laplace’s equation. In this Chapter, however, the emphasis is on the examination of specific types of problems where the governing partial differential equation corresponds to the inhomogeneous form of Laplace’s equation. Furthermore, the function which contributes to the inhomogeneous form results from the incorporation of specific aspects of the mechanics and physics of a problem. The origins of Poisson’s equation is generally associated with the study of problems in field theory, such as electrostatics, mechanics and physics of conducting media and in the description of gravitational potential in regions with mass density. In these instances, the variation of the field characterized by either an electric potential or a gravitational potential in the presence of a spatial distribution (of either an electronic charge density or a mass density exterior to a point), gives rise to Poisson’s equation. The range of applicability of Poisson’s equation also extends to the study of steady fluid flow through porous media which are internally subjected to either fluid influx or withdrawal.