It is still not quite apparent how suspended nanoparticles improve heat transmission. Multiple investigations have demonstrated that the aggregation of nanoparticles is a critical step in improving the thermal conductivity of nanofluids. However, the thermal conductivity of the nanofluid would be greatly affected by the fractal dimension of the nanoparticle aggregation. The purpose of this research is to learn how nanoparticle aggregation, joule heating, and a heat source affect the behavior of an ethylene glycol-based nanofluid as it flows over a permeable, heated, stretched vertical Riga plate and through a porous medium. Numerical solutions to the present mathematical model were obtained using Mathematica's Runge-Kutta (RK-IV) with shooting technique. In the stagnation point flow next to a permeable, heated, extending Riga plate, heat transfer processes and interrupted flow phenomena are defined and illustrated by diagrams in the proposed mixed convection, joule heating, and suction variables along a boundary surface. Data visualizations showed how different variables affected temperature and velocity distributions, skin friction coefficient, and the local Nusselt number. The rates of heat transmission and skin friction increased when the values of the suction parameters were raised. The temperature profile and the Nusselt number both rose because of the heat source setting. The increase in skin friction caused by changing the nanoparticle volume fraction from $\phi =0.0$ to $\phi =0.01$ for the without aggregation model was about 7.2% for the case of opposing flow area $\left(\lambda =-1.0\right)$ and 7.5% for the case of aiding flow region $\left(\lambda =1.0\right)$. With the aggregation model, the heat transfer rate decreases by approximately 3.6% for cases with opposing flow regions $\left(\lambda =-1.0\right)$ and 3.7% for cases with assisting flow regions $\left(\lambda =1.0\right)$, depending on the nanoparticle volume fraction and ranging from $\phi =0.0$ to $\phi =0.01$, respectively. Recent findings were validated by comparing them to previously published findings for the same setting. There was substantial agreement between the two sets finding.

We introduce a topology on the set of vertices of a directed graph and we call the topological space as pathless directed topological space. We study relation between the relative topologies and pathless directed topological spaces of E-generated subdirected graphs. Then, we study connectedness, isomorphic and homeomorphic properties in digraphs and pathless directed topological spaces. Moreover, we apply our results to blood circulation process in human heart and disprove Shokry and Aly [M. Shokry and R. E. Aly, Topological properties on graph vs medical application in human heart, Int. J. Appl. Math., 15 (2013), 1103-1109], Nada et al. [S. Nada, A. E. F. El Atik and M. Atef, New types of topological structures via graphs, Math. Method. Appl. Sci., 41 (2018), 5801-5810] and Nawar et al. [A. S. Nawar and A. E. F. A. El-Atik, A model of a human heart via graph nano topological spaces, Int. J. Biomath., 12 (2019), p.1950006]. We show that pathless directed topology is accurately describing the circulation of blood in the heart of human body.

The topic of convex and nonconvex mapping has many applications in engineering and applied mathematics. The Aumann and fuzzy Aumann integrals are the most significant interval and fuzzy operators that allow the classical theory of integrals to be generalized. This paper considers the well-known fuzzy Hermite–Hadamard (HH) type and associated inequalities. With the help of fuzzy Aumann integrals and the newly introduced fuzzy number valued up and down convexity ($U\mathcal{D}$-convexity), we increase this mileage even further. Additionally, with the help of definitions of lower $U\mathcal{D}$-concave (lower $U\mathcal{D}$-concave) and upper $U\mathcal{D}$-convex (concave) fuzzy number valued mappings ($\mathcal{F}\mathcal{N}\mathcal{V}\mathcal{M}$s), we have gathered a sizable collection of both well-known and new extraordinary cases that act as applications of the main conclusions. We also offer a few examples of fuzzy number valued $U\mathcal{D}$-convexity to further demonstrate the validity of the fuzzy inclusion relations presented in this study.

Here we give the notion of L2−directed topological spaces of directed graphs, and some results about this notion such as Alexandroff property. Next, we study the form of L2−directed topological space on E-generated subdirected graphs and their relation with the relative topologies. The relations between some fundamental properties in topological spaces with their corresponding properties in graphs such as the isomorphically and connectedness are introduced.