Flow and heat transfer in annuli owing to inner shrinking and outer stationary cylinder

Abstract

The scope of the present work is to investigate the motion of the fluid flow and heat exchange within two concentric cylinders. It is assumed that the inner cylinder is moving (shrunk) and the outer cylinder is stationary. It derives the physical phenomenon, which can find industrial practical applications. The gap between the cylinders, curvature of cylinders, and Prandtl number are three physical parameters controlling the hydro-thermal flow motion. Asymptotic solutions are available when the gap size is moderately small. These and other solutions based on the numerical simulations clearly indicate that there are critical gap sizes beyond which either no solutions or dual solutions are possible. The common salient characteristic of solutions is that the axial fluid dragged into the cylinder as a result of surface shrinkage is thrown radially out through the outer cylinder. The non-existence of solutions after identifying critical gap sizes explains why the shrinking sheet and shrinking cylinder problems with an unbounded flow zone could not have been treated in the literature so far. This is the prime reason that with infinite gap size, other controlling mechanisms like surface transpiration, unsteadiness, velocity slip or magnetic field need to be included in the mathematical model to get a solvable problem under the surface shrinking process often studied among researchers. One of the solutions called branch I solution, has rational profiles with small pertinent physical quantities like skin friction, vertical/radial thrown velocity, and heat transfer rate. However, from a stability analysis, it can be shown that the branch II solution, possesses comparatively larger such quantities, making them questionable to appear in real physics with unstable eigenvalues.