Mathematical modelling of the von Kármán flow in a non-Newtonian fluid induced by a rotating disk

Abstract

The von Kármán flow system, created by the hydrodynamic boundary layer flow around a rotating disc, has numerous applications in the engineering industry, such as gas turbines and jet engines, due to their ability to efficiently transmit power, store energy, and perform various mechanical functions. The comprehensive study of the flow patterns around rotating discs, employing a variety of non-Newtonian fluids, is motivated by their distinct properties of shear-thinning or shear-thickening behaviour that possess the more intriguing potential for heat transfer. Hence, this study considers two boundary layer flow problems in different non-Newtonian fluids, namely Reiner-Rivlin fluid and Casson hybrid nanofluid. The axisymmetric flow of the rotating disk is scrutinised considering the disk’s radial shrinking/stretching and the mathematical modelling of these complex transport phenomena is governed by a set of partial differential equations. Thus, a similarity transformation approach was employed, converting the boundary layer equations into similarity equations represented as ordinary differential equations. Subsequently, the bvp4c solver in MATLAB is utilized to numerically solve the set of non-linear ordinary differential equations describing the boundary value problem. The outcomes encompass local skin friction coefficients, the local Nusselt number, as well as velocity and temperature profiles, which were determined for various values of the governing parameters. In summary, it is observed that a pronounced suction effect, decrease in non-Newtonian fluid parameter, and shrinking/stretching of disk delay the boundary layer separation. Additionally, Reiner-Rivlin fluid flow exhibited duality in solutions while Casson fluid flow posses unique solutions only. The upper and lower branch solutions of Reiner-Rivlin fluid flow exhibit opposing behaviours when the suction parameter undergoes variations, while for the shrinking/stretching parameter and fluid material parameter, both solutions display a similar trend. The reliability and accuracy of dual solutions are determined by performing stability analysis which revealed that the upper branch solution is stable, whereas the lower branch solution is unstable. From a mathematical standpoint, this research has notably enhanced the exploration of unique and dual solutions identification and the justification of their reliability.