This study explores the heat and mass transfer of Casson nanofluid flow containing gyrotactic microorganisms past a swirling cylinder. Fluid flow is generated owing to the torsional movement of the cylinder. An analysis is performed in the presence of gyrotactic microorganisms. The effects of chemical reaction, magnetohydrodynamics, heat generation/absorption, and zero mass flux condition are also considered. The Cattaneo-Christov heat flux model is initiated instead of conventional Fourier heat flux. Apposite transformations are betrothed to attain the coupled system of equations. The numerical solution is developed from the novel mathematical model via bvp4c function utilizing MATLAB software. Numerous graphs and tables are established to portray the inspiration of embroiled parameters on the flow distributions. To corroborate the presented results; a comparison to an already done published paper is also made. An excellent synchronization between the two results is obtained thus endorsing the presented model. Also, form the graphical structures and numerically erected tables, it is professed that concentration of the fluid is lessened owing to an upsurge in values of Reynolds number and Brownian motion parameter. Furthermore, diminishing density of microorganism is perceived for mounting estimates of bioconvection Péclet number.

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Zadeh’s fuzzy sets are very useful tool to handle imprecision and uncertainty, but they are unable to characterize the negative characteristics in a certain problem. This problem was solved by Zhang et al. as they introduced the concept of bipolar fuzzy sets. Thus, fuzzy set generalizes the classical set and bipolar fuzzy set generalize the fuzzy set. These theories are based on the set of real numbers. On the other hand, the set of complex numbers is the generalization of the set of real numbers so, complex fuzzy sets generalize the fuzzy sets, with wide range of values to handle the imprecision and uncertainty. So, in this article, we study complex bipolar fuzzy sets which is the generalization of bipolar fuzzy set and complex fuzzy set with wide range of values by adding positive membership function and negative membership function to unit circle in the complex plane, where one can handle vagueness in a much better way as compared to bipolar fuzzy sets. Thus this paper leads us towards a new direction of research, which has many applications in different directions. We develop the notions of union, intersection, complement, Cartesian product and De-Morgan’s Laws of complex bipolar fuzzy sets. Furthermore, we develop the complex bipolar fuzzy relations, fundamental operations on complex bipolar fuzzy matrices and some operators of complex bipolar fuzzy matrices. We also discuss the distance measure on complex bipolar fuzzy sets and complex bipolar fuzzy aggregation operators. Finally, we apply the developed approach to a numerical problem with the algorithm.

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