The key purpose of the existing article is to discuss the effects of various hybrid nanofluids and a simple nanofluid over the heat transfer and friction drags along a stretched surface. The various kinds of hybrid nanofluids and a simple nanofluid together with the effects of aligned magnetic field, nonlinear radiation and suction have been taken into consideration. These hybrid nanofluids are prepared by suspending a couple of distinct nanoparticles $$Cu$$Cu and $$A{l}_{2}{O}_{3}$$A l 2 O 3 into the base fluids $${H}_{2}O$$H 2 O and $${C}_{2}{H}_{6}{O}_{2}$$C 2 H 6 O 2 . The comparison of various graphical results of skin friction coefficient, rate of heat transfer, velocity and temperature for two different hybrid nanofluids $$Cu{-}A{l}_{2}{O}_{3}$$C u - A l 2 O 3 / $${H}_{2}O$$H 2 O , $$Cu{-}A{l}_{2}{O}_{3}$$C u - A l 2 O 3 / $${H}_{2}O{-}{C}_{2}{H}_{6}{O}_{2}$$H 2 O - C 2 H 6 O 2 and a simple nanofluid $$A{l}_{2}{O}_{3}$$A l 2 O 3 / $${H}_{2}O$$H 2 O is considered. Moreover, the impact of surface stretching, aligned magnetic field and thermal radiation over the velocity, temperature, skin friction coefficient and local Nusselt number are also considered. The outcomes drawn from this modern research is that the hybrid nanofluid $$Cu{-}A{l}_{2}{O}_{3}$$C u - A l 2 O 3 / $${H}_{2}O{-}{C}_{2}{H}_{6}{O}_{2}$$H 2 O - C 2 H 6 O 2 is quite effective in cooling and heating in comparison to the other hybrid nanofluids $$Cu{-}A{l}_{2}{O}_{3}$$C u - A l 2 O 3 / $${H}_{2}O$$H 2 O , $$Cu{-}A{l}_{2}{O}_{3}$$C u - A l 2 O 3 / $${C}_{2}{H}_{6}{O}_{2}$$C 2 H 6 O 2 and a simple nanofluid $$A{l}_{2}{O}_{3}/{H}_{2}O$$A l 2 O 3 / H 2 O . Based on these findings we could say that the suspension of multiple particles in the composition of two or more base fluids provides a better rate of heat transfer and limits the friction drag.

In this paper, an interior penalty virtual element method (IPVEM) is developed for solving the biharmonic problem on polygonal meshes. By modifying the existing $H^2$ -conforming virtual element, an $H^1$ -nonconforming virtual element is obtained with the same degrees of freedom as the usual $H^1$ -conforming virtual element, such that it locally has $H^2$ -regularity on each polygon in meshes. To enforce the $C^1$ continuity of the solution, an interior penalty formulation is adopted. Hence, this new numerical scheme can be regarded as a combination of the virtual element space and discontinuous Galerkin scheme. Compared with the existing methods, this approach has some advantages in reducing the degree of freedom and capability of handling hanging nodes. The well-posedness and optimal convergence of the IPVEM are proven in a mesh-dependent norm. We also derive a sharp estimate of the condition number of the linear system associated with IPVEM. Some numerical results are presented to verify the optimal convergence of the IPVEM and the sharp estimate of the condition number of the discrete problem. Besides, in the numerical test, the IPVEM has a good performance in computational accuracy by contrast with the other VEMs solving the biharmonic problem.

We prove convergence and quasi-optimal complexity of an adaptive finite element algorithm on triangular meshes with standard mesh refinement. Our algorithm is based on an adaptive marking strategy. In each iteration, a simple edge estimator is compared to an oscillation term and the marking of cells for refinement is done according to the dominant contribution only. In addition, we introduce an adaptive stopping criterion for iterative solution which compares an estimator for the iteration error with the estimator for the discretization error.