Spontaneous Breakdown of the Definiteness in Some Convective Heat Transfer Problems

Abstract

In this study, it is shown that above a critical value of a governing parameter, the solutions of some convective heat transfer problems can undergo a bifurcation into a continuum of a non-denumerable infinity of solutions. Thus, the corresponding Nusselt number becomes indeterminate. The origin of this anomalous bifurcation resides in the stability change of the asymptotic state θ(∞) from that of an unstable to that of a stable equilibrium point of the system. As a consequence, the boundary condition θ(∞) = 0 becomes automatically satisfied and thus ineffective in determining the integration constants. Accordingly, the well-posed problem changes spontaneously into an ill-posed one. This remarkable phenomenon will be discussed in detail in the case of an unsteady forced and mixed convection heat transfer problem encountered in an article published recently in Transport in Porous Media. Subsequently, the mentioned loss of definiteness will be explained intuitively with the aid of a simple point-mechanical analogy.