Non-self-similar analysis of fluid flow and heat transfer in a narrow conical gap with a rotating cone: a third-order asymptotic expansion method

Purpose This paper aims to derive a third-order approximate analytical solution using the asymptotic expansion method for the non-self-similar problem of laminar fluid flow and heat transfer in a narrow conical gap between a stationary disk and a rotating cone at small conicity angles (up to 4°). Design/methodology/approach The mathematical model incorporates a third-order series expansion in terms of a small parameter Reynolds number Re for both the three velocity components and the static pressure. Approximate analytical solutions for temperature profiles and Nusselt numbers are developed for moderate Prandtl numbers Pr = 10 in the regime prior to the formation of a separate thermal boundary layer on the stationary disk. Findings The analysis determines how non-self-similar effects influence the profiles of velocity, static pressure and shear stress, as well as the tangent of the flow swirl angle at the stationary disk. Specific conditions are identified under which the derivatives of the radial and tangential velocity components (and the shear stresses) at the disk vanish. Furthermore, results show that the flow swirl angle at the disk is fully consistent with the self-similar solution at low Reynolds numbers. Research limitations/implications The thermal solutions depend on a previously derived similarity parameter and are strictly valid within a limited range of the parameter Sv < 0.8. Practical implications The obtained analytical solutions provide a valuable tool for analyzing and verifying experimental data on fluid flow and heat transfer in gases, water and aqueous solutions. Originality/value For the first time, a third-order approximate analytical solution for this specific non-self-similar conical flow configuration is presented, uniquely expanding the static pressure fields alongside velocity fields within the asymptotic framework.