In 2012, Rivaie et al. introduced RMIL conjugate gradient (CG) method which is globally convergent under the exact line search. Later, Dai (2016) pointed out abnormality in the convergence result and thus, imposed certain restricted RMIL CG parameter as a remedy. In this paper, we suggest an efficient RMIL spectral CG method. The remarkable feature of this method is that, the convergence result is free from additional condition usually imposed on RMIL. Subsequently, the search direction is sufficiently descent independent of any line search technique. Thus, numerical experiments on some set of benchmark problems indicate that the method is promising and efficient. Furthermore, the efficiency of the proposed method is demonstrated on applications arising from arm robotic model and image restoration problems.
In this numerical investigation, completely developed laminar convective heat transfer characteristics of a Williamson hybrid ferronanofluid over a cylindrical surface are reported. This new model in 2D is engaged to examine the effects of the magnetic field, thermal radiation factor, volume fraction of ultrafine particles, and Weissenberg number with the help of the Keller box method. The numerical calculations are implemented at a magnetic parameter range of 0.4 to 0.8, volume fraction range of 0.0 to 0.1, and a Weissenberg number range of 0.1 to 0.8. The numerical outcomes concluded that the velocity increases when the thermal radiation parameter and the volume fraction of a nanoparticle are increased, but inverse impacts are obtained for the magnetic parameter and the Weissenberg number. The rate of energy transport increases with increasing thermal radiation and volume fraction, while it declines with increasing the magnetic parameter and Weissenberg number. The drag force shows a positive relationship with the thermal radiation parameter and has an opposite relationship with the Weissenberg number and the magnetic parameter. Furthermore, even when the magnetic field, thermal radiation, volume fraction, and Weissenberg number are all present, the heat transfer rate of Williamson hybrid ferronanofluid is greater than that of mono Williamson ferronanofluid.