Direct integration of the energy equation at fluid flow in narrow conical gaps with one surface rotating

Purpose This study aims to obtain for the first time a direct analytical solution to the problem of laminar fluid heat transfer in a conical gap between a disk and a cone (one surface rotates and the other is fixed) with small cone angles up to 4°. Design/methodology/approach Approximate analytical solution was obtained via a direct integration of the energy equation in combination with the simplified Navier–Stokes equations for small conicity angles. Findings Simplified Navier–Stokes equations for small conicity angles were strictly obtained for the first time in the known literature. A new solutions for a thermal boundary layer thickness and for the temperature profile on a rotating surface. Practical implications The obtained solutions are important for analyzing experimental data on heat transfer in gases, water, aqueous solutions and mineral oils, as well as convective diffusion in electrochemistry. Originality/value The novelty of this article is (a) an analytical solution obtained by directly integrating the energy equation; (b) a new concept for a thermal boundary layer on a rotating surface, with a thickness equal to 75% of that by the IAEMEE for high Pr numbers; (c) improved accuracy of prediction of the dimensionless temperature profiles in comparison with the self-similar solution, which do not degenerate to null at the outer boundary of the thermal boundary layer and (d) a rigorous mathematical justification of a simplified system of Navier–Stokes equations for narrow conical gaps.