Analysis of von Kármán’s swirling flow on a rotating disc in Bingham fluids

Abstract

In this article, the flow above a rotating disc, which was first studied by von Kármán for a Newtonian fluid, has been investigated for a Bingham fluid in three complementary but separate ways: by computational fluid dynamics (CFD), by a semi-analytical approach based on a new transformation law, and by another semi-analytical approach based on von Kármán’s transformation. The full equations, which consist of a set of partial differential equations, are solved by CFD simulations. The semi-analytical approach, in which a set of ordinary differential equations is solved, is developed here by simplifying the full equations invoking several assumptions. It is shown that the new transformation law performs better and reduces to von Kármán’s transformation as a limiting case. The present paper provides a closed-form expression for predicting the non-dimensional moment coefficient which works well in comparison with values obtained by the full CFD simulations. Detailed variations of tangential, axial, and radial components of the velocity field as a function of Reynolds number (Re) and Bingham number (Bn) have been determined. Many subtle flow physics and fluid dynamic issues are explored and critically explained for the first time in this paper. It is shown how two opposing forces, viz., the viscous and the inertial forces, determine certain important characteristics of the axial-profiles of non-dimensional radial velocity (e.g., the decrease of maxima, the shift of maxima, and the crossing over). It has been found that, at any Re, the maximum value of the magnitude of non-dimensional axial velocity decreases with an increase in Bn, thereby decreasing the net radial outflow. A comparison between the streamline patterns in Newtonian and Bingham fluids shows that, for a Bingham fluid, a streamline close to the disc-surface makes a higher number of complete turns around the axis of rotation. The differences between the self-similarity in a Newtonian fluid flow and the non-similarity in a Bingham fluid flow are expounded with the help of a few compelling visual representations. Some major differences and similarities between the flow of a Newtonian fluid above a rotating disc and that of a Bingham fluid, deduced in the present investigation, are brought together in a single table for ready reference. Two limiting cases, viz. Bn → 0 and Re → ∞, are considered. The present results show that the Bingham fluid solution progressively approaches the von Kármán’s solution for a Newtonian fluid as the Bingham number is progressively reduced to zero (Bn → 0). It is also established here that, for finite values of Bn, the Bingham fluid solution progressively approaches the von Kármán’s solution for a Newtonian fluid as the non-dimensional radius and Reynolds number increase. The higher the value of Bn, the higher is the required value of Re at which convergence with the solution for Newtonian fluid occurs.