Analytical solutions for heat transfer on fractal and pre-fractal domains

Fractals can be used to represent intricate self-similar geometries, but their application to the representation of physical systems is beset with difficulties which stem from an inability to define traditionally derived-physical quantities such as stress, pressure, strain, heat etc. This paper describes a method for the determination of analytical heat-transfer solutions on pre-fractal and fractal domains. The approach requires the construction of maps from pre-fractal domains to the continuum, which facilitate the application of traditional continuum solution methods. Solutions on fractal domains are achievable with this approach, and are defined to be the limit solution of analytical solutions obtained on the pre-fractals approximating the fractal of interest. This approach avoids many of the complications and technical difficulties arising from the use of measure theory and fractional derivatives, but also infers that the governing heat transfer equations are valid on all pre-fractals. The fractals considered are necessarily deterministic and relatively simple in form to demonstrate the solution methodology. The solutions presented are limited to one and two-dimensional domains and, in 1-D, are applied to an idealised composite material consisting of relatively small particles of infinitely low thermal conductivity embedded in a relatively large matrix of infinitely high thermal conductivity. The fractal composite system is thus not truly representative of a realistic physical system, but the methods presented do serve to demonstrate how analytical solutions can be attained on dust-like fractal domains. It is demonstrated that a measurable temperature is possible on a fractal structure along with finite measures of heat flux and energy. Transient and steady state thermal solutions are presented. The solutions on a selection of the pre-fractals are compared against finite element predictions to reinforce the validity of the approach.