A rapid and effective way of working out the optimum parameter of convergence control in the homotopy analysis method (HAM) is introduced in this paper. As compared with the already known ways of evaluating the convergence control parameter in HAM either through the classical h-level curves with h being the convergence control parameter or from the classical squared residual formula as adopted in the HAM society, an elegant way of calculating the convergence control parameter yielding the same optimum values is offered. In most cases, the new method is shown to perform quicker and better against the residual error method when integrations are much harder to evaluate or even by numerical means. Examples originating from real life applications selected from the literature demonstrate the validity and usefulness of the introduced technique.

The object of this study is the development of the fluid flow and resultant heat transfer caused by a rotating disk moving vertically upward or downward during an unsteady flow motion. The problem is formulated such that the similarity equations governing the physical phenomenon eventually reduced to those reported in the traditional viscous pumping study of von Karman for a vertically motionless but still rotating disk. The non-rotating disk with the upward or downward motion leads to the formation of a two-dimensional flow over the disk. Otherwise, the rotation and vertical action of the disk sets up a three-dimensional flow over the surface. It is observed that the upward and downward motion of the disk exerts an effect similar to that of the injection/suction through the wall, albeit with observable differences. Moreover, the viscous pumping is found to be a jet-like radial velocity as the disk moves upward fast. Although the downward movement of the disk suppresses the velocity field, a growth in the boundary layer thickness is anticipated, contrary to the traditional wall suction. The temperature field is shown to be highly dependent on the form of the wall temperature, which is maintained at a time-varying function. Moreover, the impact of the vertical wall movement is observed to be overwhelmed by high disk rotations.

An exact solution for the steady state Navier-Stokes equations in cylindrical coordinates is presented in this work. It serves to investigate the fluid flow and heat transfer occurring between two stretchable disks rotating co-axially at constant distance apart. The governing equations of motion and energy are first transformed into a set of nonlinear differential equation system by the use of von Karman similarity transformations, which are later solved numerically. The small Reynolds number case allows us to extract closed-form solutions for the physical phenomenon. The effects of the same or opposite direction rotation, as well as the stretching parameter and the Reynolds number, are discussed on the flow and heat characteristics. The main physical implication of the results is that stretching action of a disk surface alters considerably the classical flow behavior occurring between two disks and the physically interesting quantities like the torque and heat transfer are elucidated in the presence of a new physical mechanism; that is the surface stretching in the current research.