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Nanofluids are a very productive etymology of intensifying the process of heat and mass transport systems linked with the industrial and thermal engineering systems. Nanomaterials have effective thermal properties and various applications in our daily life like in heat transfer, electronic cooling systems, energy production and biomedicine and also in the food industry. Keeping the entire motivating potential ramifications of nanoparticles in mind, this work is visualized in the mathematical model developed to show the heat and mass transport behavior of swimming motile organisms in the existence of the magnetic field, heat conduction source, thermal radiation, chemical processes and viscous dissipation. The flow of mass and heat transport under consideration is governed by nonlinear partial differential equations (PDEs) transformed into ordinary differential equations (ODEs) by implementing an eminent method called similarity transform and then numerical results obtained through MATLAB inbuilt package 'bvp4c'. Numerical solution is visualized through the comparison of Casson fluid results with Newtonian fluid. The impact of numerous nondimensional parameters of temperature, heat transfer, velocity and concentration profiles involved in governing equations is debated and visualized graphically. Furthermore, the effects of parameters and local Nusselt number, motile organism's number, Biot number, Sherwood number, thermal radiation and microorganism concentration are elaborated through graphical representation. From these results, we clearly see that the velocity profile shows a decrement by raising the values of Buoyancy ratio Nr and Bioconvection Rayleigh number Nc, thermal profile depicted propagation by incrementing the values of Biot and radiation variables, concentration profile decreases by incrementing Lewis parameter Le and microorganisms profile revealed an increase and decrease by the presence of magnetic M and bioconvection Lewis variable Lb, respectively. [ABSTRACT FROM AUTHOR]

Abstract: We extend results about critically -colorable graphs to choosability and paintability (list colorability and on-line list colorability). Using a strong version of Brooks’ Theorem, we generalize Gallai’s Theorem about the structure of the low-degree subgraph of critically -colorable graphs, and introduce a more adequate lowest-degree subgraph. We prove lower bounds for the edge density of critical graphs, and generalize Heawood’s Map-Coloring Theorem about graphs on higher surfaces to paintability. We also show that on a fixed given surface, there are only finitely many critically -paintable/-choosable/-colorable graphs, if . In this situation, we can determine in polynomial time -paintability, -choosability and -colorability, by giving a polynomial time coloring strategy for “Mrs. Correct”. Our generalizations of -choosability theorems also concern the treatment of non-constant list sizes (non-constant ). Finally, we use a Ramsey-type lemma to deduce all 2-paintable, 2-choosable, critically 3-paintable and critically 3-choosable graphs, with respect to vertex deletion and to edge deletion. [Copyright &y& Elsevier]

Topological indices are numerical numbers assigned to the graph/structure and are useful to predict certain physical/chemical properties. In this paper, we give explicit expressions of novel Banhatti indices, namely, first K Banhatti index B 1 G , second K Banhatti index B 2 G , first K hyper-Banhatti index HB 1 G , second K hyper-Banhatti index HB 2 G , and K Banhatti harmonic index H b G for hyaluronic acid curcumin and hydroxychloroquine. The multiplicative version of these indices is also computed for these structures. [ABSTRACT FROM AUTHOR]