Fluid flow and heat transfer of nanofluids have gained a lot of attention due to their wide application in industry. In this context, the appropriate solution to such phenomena is the study of this exciting and challenging field by the research community. This paper presents an extension of a well-known collocation method (CM) to investigate the accurate solutions to unsteady flow and heat transfer among two parallel plates. First, a mathematical model is developed for the discussed phenomena, then this model is converted into a non-dimensional form using viable similarity variables. In order to inspect the accurate solutions of the accomplished set of nonlinear ordinary differential equations, a collocation method is proposed and applied successfully. Various simulations are performed to analyze the behavior of non-dimensional velocity, temperature, and concentration profiles alongside the deviation of physical parameters present in the model, and then plotted graphically. It is important to mention that the velocity is enhanced due to the higher impact of the parameter Ha. The parameter Nt caused an efficient enhancement in the temperature distribution while the parameters Nt provided a drop in the temperature that actually affected the rate of heat transmission. Dual behavior of concentration is noted for parameter b, while it can be noted that mixed increasing behavior is available for the concentration against Le. The behavior of skin friction, the Nusselt number, and the Sherwood number were also investigated in addition to the physical parameters. It was observed that the Nusselt number increases with the enhancement of the effects of the magnetic field parameter and the Prandtl number. A comparative study shows that the proposed scheme is very effective and reliable in investigating the solutions of the discussed phenomena and can be extended to find the solutions to more nonlinear physical problems with complex geometry.
An improved variational inequality strategy for dealing with variational inequality in a Hilbert space is proposed in this article as an alternative; if Hilbert space is used as the domain of interest, the original extra-gradient method is proposed for resolving variational inequality. This improved variational inequality strategy can be used as a substitute for the original extra-gradient method in some situations. Mann’s mean value method, coupled with the widely used sub-gradient extra-gradient strategy, makes it possible to update all of the previous iterations in a single step, thus saving time and effort. All of this is made feasible via the use of Mann’s mean value technique in conjunction with the convex hull of all prior iterations of the algorithm. It is guaranteed that the mean value iteration will result in an acceptable resolution of a variational inequality issue as long as one or more of the criteria for the averaging matrix are fulfilled. Numerous experiments were performed in order to demonstrate the correctness of the theoretical conclusion obtained.
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