The calculus of variations is utilised to study the behaviour of a rippled graphene sheet supported on a metal substrate. We propose a model that is underpinned by two key parameters, the bending rigidity of graphene γ, and the van der Waals interaction strength ξ. Three cases are considered, each of which addresses a specific configuration of a rippled graphene sheet located on a flat substrate. The transitional case assumes that both the graphene sheet length and substrate length are constrained. The substrate constrained case assumes only the substrate has a constrained length. Finally, the graphene constrained case assumes only the length of the graphene sheet is constrained. Numerical results are presented for each case, and the interpretation of these results demonstrates a continuous relationship between the total energy per unit length and the substrate length, that incorporates all three configurations. The present model is in excellent agreement with earlier results of molecular dynamics (MD) simulations in predicting the profiles of graphene ripples.

A mathematical model is developed to study the folding behaviour of multi–layer graphene sheets supported on a substrate. The conformation of the fold is determined from variational considerations based on two energies, namely the graphene elastic energy and the van der Waals (vdW) interaction energy between graphene layers and the substrate. The model is nondimensionalized and variational calculus techniques are then employed to determine the conformation of the fold. The Lennard–Jones potential is used to determine the vdW interaction energy as well as the graphene–substrate and graphene–graphene spacing distances. The folding conformation is investigated under three different approximations of the total line curvature. Our findings show good agreement with experimental measurements of multi–layer graphene folds from the literature.

We present a novel analytical prediction for the effective bending rigidity γ _eff of multi–layer graphene sheets. Our approach involves using a variational model to determine the folding conformation of multi–layer graphene sheets where the curvature of each graphene layer is taken into account. The Lennard–Jones potential is used to determine the van der Waals interaction energy per unit area and the spacing distance between graphene layers. The mid–line of the folded multi–layer graphene is described by a solution derived in previous work for folded single– and multi–layer graphene. Several curves are obtained for the single–layer solution using different values of the bending rigidity γ, and compared to the mid–line of the folded multi–layer graphene. The total area between these curves and the mid–line is calculated, and the value of γ _eff is determined by the single–layer curve for which this area is minimized. While there is some disagreement in the literature regarding the relationship between the bending rigidity and the number of layers, our analysis reveals that the bending rigidity of multi–layer graphene follows an approximate square–power relationship with the number of layers N, where N < 7. This trend is in line with theoretical and experimental studies reported in the literature.

We introduce a Bézier form of Schurer‐type modification of the quantum λ ‐Bernstein operators, extending the classical Schurer operators through the Bézier basis with shape parameter −1 ≤ λ ≤ 1. By applying Korovkin’s theorem, we obtain both global and local approximation results. Furthermore, the convergence behavior is analyzed in terms of the standard and second‐order moduli of continuity, together with estimates involving Lipschitz classes and Peetre’s K ‐functional.

Nanofluids find extensive applications in enhancing the thermodynamic efficiency of thermal systems across various domains of engineering and scientific disciplines. This study aims to explore the complex relationship between the varying thermal conductivity and viscosity impact in nanofluid dynamics. The main objective of this study is to examine the three-dimensional stagnation flow of Casson nanofluid across a stretching and spinning disk, influenced by a magnetic source. The Navier–Stokes model for flow systems includes Brownian diffusion and thermophoresis. By using scaling variables, the complex system of partial differential equations is simplified into a set of coupled high degree nonlinear ordinary differential equations with convective boundary conditions. The homotopy technique is applied for analytic solutions. The optimization analysis is conducted on heat transfer rate and surface drag force coefficient using response parameters. The influence of the different parameters for the flow problem has been discussed and is shown through graphs. The finding of our study is that the velocity profiles increase in both radial directions as well as in the azimuthal direction by varying the rotation parameter strength, while the temperature gradient profile dwindles. Finally, the precision of the presented model is reaffirmed by means of a graphical juxtaposition with published data under a specific limiting scenario.