Heat Flux Vector Potential in Convective Heat Transfer

The heat function concept introduced by Kimura and Bejan (1983, “The Heatline Visualization of Convective Heat Transfer,” ASME J. Heat Transfer, 105(4), pp. 916–919) for two-dimensional (2D) heat transfer is being extended in this note to three dimensions. It is shown that a heat flux vector potential exists and can be used in three-dimensional (3D) heat convection problems. It is further shown that this heat flux vector potential degenerates to the heat function introduced by Kimura and Bejan (1983, “The Heatline Visualization of Convective Heat Transfer,” ASME J. Heat Transfer, 105(4), pp. 916–919) when the heat convection is two-dimensional.

Asymptotics for the spectral heat function and bounds for integrals of Dirichlet eigenfunctions

We introduce the spectral heat function H associated with the Dirichlet-Laplace–Beltrami operator ΔM on a compact smooth Riemannian manifold M with a non-empty smooth boundary. We obtain two-term asymptotics for H without assuming any billiard conditions on M. As a corollary, we obtain estimates for the integral of the k′th Dirichlet eigenfunction of ΔM over M for k→∞.