Heat transfer unidirectional flows

Conductive heat transfer in all solid components, radiative heat exchange between all diffusive surfaces in the unidirectional solidification furnace, and the Navier-Stokes equations for the melt flow in the crucible are coupled. 

Chapter 3 Unidirectional and One-Dimensional Flow and Heat Transfer Problems... 

For all of the flows that can be classified as unidirectional, the analysis of Section A shows that the governing equations reduce to solving a single heat equation for the magnitude of the scalar velocity component in the flow direction. For heat transfer applications, mathematically analogous problems involve heat transport by pure conduction. As noted earlier, there are excellent comprehensive books devoted exclusively to the solution of this class of problems.4 Here, we consider a related problem, which is chosen because it addresses the physically important coupling of heat transfer in the presence of phase change and also because it is another ID problem that exhibits a self-similar solution. 

In this section, we instead consider two well-known examples of heat transfer in the fully developed, laminar, and unidirectional flow of a Newtonian fluid in a straight circular tube. We begin with a problem in which there is a prescribed heat flux into the fluid at the walls of the tube, so that there is a steady-state temperature distribution in the tube. At the end of the section, we consider the transient evolution of the temperature distribution beginning with an initially sharp temperature jump within the fluid at a fixed position (say z = 0), which illustrates an important phenomenon that is known as Taylor dispersion. 

---