An asymptotic expansion method vs a self-similar solution for convective heat transfer in rotating cone-disk systems

Abstract

This paper compares an asymptotic expansion method and a self-similar solution for modeling Couette flow and convective heat transfer in a conical gap at small conicity angles up to 4°. The cases of rotation of a cone with a stationary disk and rotation of a disk with a stationary cone are considered. The self-similar system of equations provides the best agreement with experiments compared to the asymptotic expansion method. In any case, both methods are applicable only to conicity taper angles up to 4°, while at large conicity angles, the calculation results become significantly inaccurate. Calculations also showed that, at small conicity angles, convective heat transfer can be modeled using the self-similar energy equation in the boundary-layer approximation without considering radial heat conduction. In this study, analytical solutions were also obtained for limiting cases of a stationary fluid in a gap at small conicity angles without and with allowance for radial heat conduction.

Improved asymptotic expansion method for laminar fluid flow and heat transfer in conical gaps with disks rotating

“…In the present study, the simplified system of equations ( 7)-(9) will be solved using the new improved asymptotic expansion method first developed in the recent paper of Shevchuk (2023c). This solution will be compared with the self-similar solution of system (1)-(4) (Shevchuk, 2004b(Shevchuk, , 2022(Shevchuk, , 2023b(Shevchuk, , 2023c.…”

“…The need to take into account the term with radial thermal conductivity in the heat transfer equation in a self-similar formulation is discussed in detail in recent works of Shevchuk (2022Shevchuk ( , 2023aShevchuk ( , 2023b and Shevchuk (2023c). It was shown there that, strictly speaking, the self-similar formulation of the problem of flow and heat transfer is valid and leads to accurate calculation results only at small conicity angles up to 4°.…”

“…In a recent work by Shevchuk (2022Shevchuk ( , 2023a, it was shown that the implementation of the asymptotic expansion method by Sdougos et al (1984), Buschmann (2002) and Buschmann et al (2005) guarantees accurate results only for very small Reynolds numbers of the cone rotation Re, whereas the inaccuracy of this method increases with the rotational Reynolds number of the cone. The relation for the radial velocity component of Sdougos et al (1984), Buschmann (2002) and Buschmann et al (2005) agrees very well with the self-similar solution, whereas the profiles of the axial and tangential velocity components differ significantly from the self-similar solution, which generates significant errors in the calculation of the tangent of the flow swirl angle and Nusselt numbers on the disk based on this technique.…”

“…In contrast to the classical problem of a rotating cone with a fixed disk, the problem of a rotating disk with a fixed cone has attracted much less attention from researchers. In the pioneering work of Shevchuk (2004b), as well as in subsequent works (Shevchuk, 2009(Shevchuk, , 2011(Shevchuk, , 2015(Shevchuk, , 2022, the problem was solved in a self-similar statement. Turkyilmazoglu (2020) also Improved asymptotic expansion method performed the simulation of fluid flow and convective heat transfer for the case of disk rotation with a fixed cone using the self-similar approach.…”

The flow and heat in the conical region of a rotating cone and an expanding disk

“…The recent publication in Turkyilmazoglu (2020) points to the fact that for small conical gap sections, streamwise heat conduction terms are important in the cone-disk heat transfer problems. An excellent recent work was given in Shevchuk (2022) presenting the Couette flow and convective heat transfer at small conicity angles by means of an asymptotic expansion and simulation methods. The Buongiorno nanofluid model was recently added to the classical problem in Basavarajappaa and Bhatta (2022), where it was proven that nanoparticles with Brownian and thermophoresis effects contribute to lower Nusselt numbers.…”