Counter-current (vessel–vessel) heat transfer has been postulated as one of the most important heat transfer mechanisms in living systems. Surprisingly, however, the accurate quantification of the vessel–vessel, and vessel–tissue, heat transfer rates has never been performed in the most general and important case of a finite, unheated/heated tissue domain with noninsulated boundary conditions. To quantify these heat transfer rates, an exact analytical expression for the temperature field is derived by solving the 2-D Poisson equation with uniform Dirichlet boundary conditions. The new results obtained using this solution are as follows: first, the vessel–vessel heat transfer rate can be a large fraction of the total heat transfer rate of each vessel, thus quantitatively demonstrating the need to accurately model the vessel–vessel heat transfer for vessels imbedded in tissues. Second, the vessel–vessel heat transfer rate is shown to be independent of the source term; while the heat transfer rates from the vessels to the tissue show a significant dependence on the source term. Third, while many previous studies have assumed that (1) the total heat transfer rate from vessels to tissue is zero, and/or (2) the heat transfer rates from paired vessels (of different sizes and at different temperatures) to tissue are equal to each other the current analysis shows that neither of these conditions is met. The analytical solution approach used to solve this two vessels problem is general and can be extended for the case of “N” arbitrarily located vessels.
Exponential stability of density-velocity systems with boundary conditions and source term for the H2 norm
