We are now in a position to begin to consider the solution of heat transfer and fluid mechanics problems by using the equations of motion, continuity, and thermal energy, plus the boundary conditions that were given in the preceding chapter. Before embarking on this task, it is worthwhile to examine the nature of the mathematical problems that are inherent in these equations. For this purpose, it is sufficient to consider the case of an incompressible Newtonian fluid, in which the equations simplify to the forms (2–20), (2–88) with the last term set equal to zero, and (2–93).

The first thing to note is that this set of equations is highly nonlinear. This can clearly be seen in the term u · grad u in (2–88). However, because the material properties such as ρ, Cp, and k are all functions of the temperature θ, and the latter is a function of the velocity u through the convected derivative on the left-hand side of (2–93), it can be seen that almost every term of (2–88) and (2–93) involves a product of at least two unknowns either explicitly or implicitly. In contrast, all of the classical analytic methods of solving partial differential equations (PDEs) (for example, eigenfunction expansions by means of separation of variables, or Laplace and Fourier transforms) require that the equation(s) be linear. This is because they rely on the construction of general solutions as sums of simpler, fundamental solutions of the DEs.