Two examples are aircraft modeling and weather prediction models. In the case of an aircraft, the lift forces are determined by the velocity with three components, the pressure and in many cases the temperatures. So, there are five quantities which vary with 3D space. Such problems require very complicated models, and the need for faster languages, algorithms and computing hardware is essential to give realistic numerical simulations. Consider an electric transformer that is used on a power line.
Heat is generated by the electrical current flowing through the wire inside the transformer.
In order to cool the transformer, fins which are not long or very thin in any direction are attached to the transformer. Thus, there will be significant temperature distributions in each of the three directions, and consequently, there will be heat diffusion in all three directions. The problem is to determine the steady state heat distribution in the 3D fin so that one can determine the fin's cooling effectiveness. 3D Fin on an Electrical Transformer 3.8.3. The heat flowing through these six surfaces will be given by the Fourier heat law applied to each of the three directions.