Significance of Shape Factor in Heat Transfer Performance of Molybdenum-Disulfide Nanofluid in Multiple Flow Situations; A Comparative Fractional Study

In this modern era, nanofluids are considered one of the advanced kinds of heat transferring fluids due to their enhanced thermal features. The present study is conducted to investigate that how the suspension of molybdenum-disulfide (MoS2) nanoparticles boosts the thermal performance of a Casson-type fluid. Sodium alginate (NaAlg) based nanofluid is contained inside a vertical channel of width d and it exhibits a flow due to the movement of the left wall. The walls are nested in a permeable medium, and a uniform magnetic field and radiation flux are also involved in determining flow patterns and thermal behavior of the nanofluid. Depending on velocity boundary conditions, the flow phenomenon is examined for three different situations. To evaluate the influence of shape factor, MoS2 nanoparticles of blade, cylinder, platelet, and brick shapes are considered. The mathematical modeling is performed in the form of non-integer order operators, and a double fractional analysis is carried out by separately solving Caputo-Fabrizio and Atangana-Baleanu operators based fractional models. The system of coupled PDEs is converted to ODEs by operating the Laplace transformation, and Zakian’s algorithm is applied to approximate the Laplace inversion numerically. The solutions of flow and energy equations are presented in terms of graphical illustrations and tables to discuss important physical aspects of the observed problem. Moreover, a detailed inspection on shear stress and Nusselt number is carried out to get a deep insight into skin friction and heat transfer mechanisms. It is analyzed that the suspension of MoS2 nanoparticles leads to ameliorating the heat transfer rate up to 9.5%. To serve the purpose of achieving maximum heat transfer rate and reduced skin friction, the Atangana-Baleanu operator based fractional model is more effective. Furthermore, it is perceived that velocity and energy functions of the nanofluid exhibit significant variations because of the different shapes of nanoparticles.     

A New Ferrocene‐Containing Polyamide Prepared from an Improved Synthesis of 1,1′‐Ferrocene Dicarbonyl Chloride and Ferrocene‐Based Diamine

1,1′‐Ferrocene dicarbonyl chloride was prepared by an improved and efficient conversion method from 1,1′‐ferrocene dicarboxylic acid and reacted by esterification with p‐nitrophenol, followed by reduction, to form a ferrocene‐based diamine, 1,1′‐ferrocene bis (p‐amino phenylate). The diamine was characterized by elemental analysis, ¹H NMR, and Fourier transform infrared (FTIR) spectroscopy and subsequently condensed with 1,1′‐ferrocene dicarbonyl chloride to form a novel main chain ferrocene‐containing polyamide, poly{imino ferrocene bis (p‐amino phenylate) ferrocenyl}. Its polymeric nature was confirmed by its physical properties, elemental analysis, FTIR spectroscopy, differential scanning calorimetry, and thermogravimetric studies.     

Synthesis and physico-analytical studies of some novel ferrocenyl Schiff base derivatives

A series of ferrocenyl Schiff base derivatives was synthesized by condensation reactions of 1,1′-ferrocenedicarboxaldehyde and aromatic amines containing long chain alkyl groups as free ends which were characterized by their physical properties, elemental, FTIR, ¹H NMR, ¹³C NMR spectral and thermal analysis. The thermal behaviour of the synthesized compounds was studied by differential scanning calorimetry (DSC) which revealed that these compounds may exhibit mesomorphic properties. The DSC results of aromatic amines and ferrocenyl Schiff bases were compared to study the effects of structure, i.e. rigid core and terminal chain length, on the phase transition behaviour.     

Fourier transform and distributional representation of the generalized gamma function with some applications

We present a Fourier transform representation of the generalized gamma functions, which leads to a distributional representation for them as a series of Dirac-delta functions. Applications of these representations are shown in evaluation of the integrals of products of the generalized gamma function with other functions. The results for Euler’s gamma function are deduced as special cases. The relation of the generalized gamma function with the Macdonald function is exploited to deduce new identities for it.