This research investigates the effects of slip and thermo-diffusion on magnetohydrodynamic fluid flow and heat transfer through a porous medium, using a fractional second-grade fluid model. A semi-analytical solution is obtained through Laplace transform techniques, revealing detailed insights into concentration, temperature, and velocity distributions. The study sheds light on the significant impact of key parameters, including Prandtl and Schmidt numbers, Grashof numbers, and the second-grade parameter, on the intricate relationships between thermo-diffusion, slip conditions, and fluid flow in porous media.
The chemical vapour deposition method is widely used to synthesise high quality graphene with a large surface area. However, the cooling process leads to the formations of ripples and wrinkles in the graphene structure. When a self-adhered wrinkle achieves the maximum height, it then folds onto the surface and leads to a collapsed wrinkle. The presence of such deformations often affects the properties of graphene. In this article, we describe a novel mathematical model to understand the formation and geometry of these wrinkles. The stability of these wrinkles is examined based on variational derivations for the energy of each structure. The model provides detailed explanations for the geometry of these wrinkles which would help in tuning their properties. References J. Aljedani, M. J. Chen, and B. J. Cox. Variational model for collapsed graphene wrinkles. Appl. Phys. A 127.11, 886 (2021), pp. 1–13. doi: 10.1007/s00339-021-05000-y A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau. Superior thermal conductivity of single-layer graphene. Nano Lett. 8.3 (2008), pp. 902–907. doi: 10.1021/nl0731872 S. Chen, Q. Li, Q. Zhang, Y. Qu, H. Ji, R. S. Ruoff, and W. Cai. Thermal conductivity measurements of suspended graphene with and without wrinkles by micro-Raman mapping. Nanotech. 23.36, 365701 (2012). doi: 10.1088/0957-4484/23/36/365701 on p. C85). B. J. Cox, T. Dyer, and N. Thamwattana. A variational model for conformation of graphene wrinkles formed on a shrinking solid metal substrate. Mat. Res. Express 7.8, 085001 (2020). doi: 10.1088/2053-1591/abaa8f A. K. Geim. Graphene: Status and prospects. Science 324.5934 (2009), pp. 1530–1534. doi: 10.1126/science.1158877 on p. C85). K. Kostarelos and K. S. Novoselov. Graphene devices for life. Nature Nanotech. 9 (2014), pp. 744–745. doi: 10.1038/nnano.2014.224 F. Long, P. Yasaei, R. Sanoj, W. Yao, P. Král, A. Salehi-Khojin, and R. Shahbazian-Yassar. Characteristic work function variations of graphene line defects. ACS Appl. Mat. Inter. 8.28 (2016), pp. 18360–18366. doi: 10.1021/acsami.6b04853 R. Muñoz and C. Gómez-Aleixandre. Review of CVD synthesis of graphene. Chem. Vapor Dep. 19.10–12 (2013), pp. 297–322. doi: 10.1002/cvde.201300051 L. Spanu, S. Sorella, and G. Galli. Nature and strength of interlayer binding in graphite. Phys. Rev. Lett. 103.19, 196401 (2009). doi: 10.1103/PhysRevLett.103.196401 T. Verhagen, B. Pacakova, M. Bousa, U. Hübner, M. Kalbac, J. Vejpravova, and O. Frank. Superlattice in collapsed graphene wrinkles. Sci. Rep. 9.1, 9972 (2019). doi: 10.1038/s41598-019-46372-9 C. Wang, Y. Liu, L. Li, and H. Tan. Anisotropic thermal conductivity of graphene wrinkles. Nanoscale 6.11 (2014), pp. 5703–5707. doi: 10.1039/C4NR00423J W. Wang, S. Yang, and A. Wang. Observation of the unexpected morphology of graphene wrinkle on copper substrate. Sci. Rep. 7.1 (2017), pp. 1–6. doi: 10.1038/s41598-017-08159-8 Y. Wang, R. Yang, Z. Shi, L. Zhang, D. Shi, E. Wang, and G. Zhang. Super-elastic graphene ripples for flexible strain sensors. ACS Nano 5.5 (2011), pp. 3645–3650. doi: 10.1021/nn103523t Y. Wei, B. Wang, J. Wu, R. Yang, and M. L. Dunn. Bending rigidity and Gaussian bending stiffness of single-layered graphene. Nano Lett. 13.1 (2013), pp. 26–30. doi: 10.1021/nl303168w Z. Xu and M. J. Buehler. Interface structure and mechanics between graphene and metal substrates: A first-principles study. J. Phys.: Cond. Mat. 22.48, 485301 (2010). doi: 10.1088/0953-8984/22/48/485301 Y. Zhang, N. Wei, J. Zhao, Y. Gong, and T. Rabczuk. Quasi-analytical solution for the stable system of the multi-layer folded graphene wrinkles. J. Appl. Phys. 114.6, 063511 (2013). doi: 10.1063/1.4817768 W. Zhu, T. Low, V. Perebeinos, A. A. Bol, Y. Zhu, H. Yan, J. Tersoff, and P. Avouris. Structure and electronic transport in graphene wrinkles. Nano Lett. 12.7 (2012), pp. 3431–3436. doi: 10.1021/nl300563h
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.
In this paper, the method of upper and lower solutions is employed to obtain uniqueness of solutions for a boundary value problem at resonance. The shift method is applied to show the existence of solutions. A monotone iteration scheme is developed and sequences of approximate solutions are constructed that converge monotonically to the unique solution of the boundary value problem at resonance. Two examples are provided in which explicit upper and lower solutions are exhibited.
Abstract 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.
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
Abstract 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.
<p>We have proposed a $ q $-analogue $ c(\mathcal{F}(q)) $ and $ c_0(\mathcal{F}(q)) $ of Fibonacci sequence spaces, where $\mathcal{F}(q) = (f^q_{km})$ denotes a $ q $-Fibonacci matrix defined in the following manner:</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ f^q_{km} = \begin{cases} q^{m+1} \frac{f_{m+1}(q)}{f_{k+3}(q) - 1}, & \text{if } 0 \leq m \leq k, \\ 0, & \text{if } m > k, \end{cases} $\end{document} </tex-math></disp-formula></p><p>for all $ k, m \in \mathbb{Z}^+_0 $, where $(f_k(q))$ denotes a sequence of $ q $-Fibonacci numbers. We developed a Schauder basis and determined several important duals ($ \alpha $-, $ \beta $-, $ \gamma $-) of the aforesaid constructed spaces $ c(\mathcal{F}(q)) $ and $ c_0(\mathcal{F}(q)) $. Additionally, we examined certain characterization results for the matrix class $(\mathfrak{U}, \mathfrak{V})$, where $\mathfrak{U} \in \{c(\mathcal{F}(q)), c_0(\mathcal{F}(q))\}$ and $\mathfrak{V} \in \{\ell_{\infty}, c, c_0, \ell_1\}$. Essential conditions for the compactness of the matrix operators on the space $ c_0(\mathcal{F}(q)) $ via the Hausdorff measure of noncompactness (Hmnc) were presented.</p>
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