This research investigates the two different hybrid nanofluid flows between two parallel plates placed at two different heights, $y_0$and $y_h$, respectively. Water-based hybrid nanofluids are obtained by using $A{l}_2{O}_3$, $Ti{O}_2$ and $Cu$ as nanoparticles, respectively. The upper-level plate is fixed, while the lower-level plate is stretchable. The fluid rotates along the y-axis. The governing equations of momentum, energy and concentration are transformed into partial differential equations by using similarity transformations. These transformed equations are grasped numerically at MATLAB by using the boundary value problem technique. The influence of different parameters are presented through graphs. The numerical outcomes for rotation, Nusselt, Prandtl, and Schmidt numbers are obtained in the form of tables. The heat transfer rate increases by augmentation in the thermophoresis parameter, while it decays by increasing the Reynolds number. Oxide nanoparticles hybrid nanofluid proved more efficient as compared to mixed nanoparticles hybrid nanofluid. This research suggests using oxide nanoparticles for good heat transfer.

This article explores the application of the reduced differential transform method (RDTM) for the computational solutions of two fractional-order cancer tumor models in the Caputo sense: the model based on cancer chemotherapeutic effects which explain the relation between chemotherapeutic drugs, tumor cells, normal cells, and immune cells using a fractional partial differential equations, and the model that describes the different cases of killing rate K of cancer cells (the killing percentage of cancer cells $K$ (I) is dependent on the number of cells, (II) is a function of time only, and (III) is a function of space only). The solutions are presented using Mathematica software as a convergent power series with elegantly computed terms using the suggested technique. The proposed method gives new series form results for various values of gamma. To clarify the complexity of the models, we plot the two- and three-dimensional and contour graphics of the obtained solutions at varied values of fractional-order gamma and the selected system parameters. The solutions are analyzed with fractional and reduced differential transform methods to obtain an idea of invariance regarding the computed solution of the designed mathematical model. The obtained results demonstrate the efficiency and preciseness of the proposed method to achieve a better understanding of chemotherapy effects. It is observed that chemotherapy drugs boost immunity against the specific cancer by decreasing the number of tumor cells, and the killing rate $K$ of cancerous cells depend on the cells concentration.

This article is an attempt to explore the heat transfer features of the steady three-dimensional rotating flow of magneto-hydrodynamic hybrid nanofluids under the effect of nonlinear radiation over the bi-directional stretching surface. For this purpose, two different nano-particles, namely silver (Ag) and molybdenum di-sulfide (MoS₂), were selected. Three different conventional base fluids were utilized to form desired hybrid nanofluids such as water (H₂O), engine oil (EO), and ethylene glycol (EG). We obtained steady three-dimensional highly nonlinear partial differential equations. These highly nonlinear partial differential equations cannot be solved analytically, so these equations were handled in MATLAB with the BVP-4C technique with convergence tolerance at 10⁻⁶. The graph depicts the effect of the magnetization effect, thermal radiation, and stretching ratio on rotating hybrid nanofluids. Additionally, the impact of thermal radiation on the heat coefficient of three different hybrid nanofluids is being investigated. The augmentation in magnetization decreases the primary velocity, whereas the increment in radiation enhances the primary velocity. The stretching ratio and the presence of higher magnetic forces increase the temperature profile. The concentration profile was enhanced with an increment in the magnetic field, stretching, and rotation ratio. The maximum Nusselt number was achieved for the Ag-MoS₂/EO hybrid nanofluid. It was concluded that augmentation in nonlinear radiation enhances the heat transfer coefficient for the examined cases (I) and (II) of the hybrid nanofluids. The Nusselt number doubled for both the examined cases under nonlinear radiation. Moreover, it was discovered that Ag-MoS₂/water produced the best heat transfer results under nonlinear radiation. Therefore, the study recommends more frequent exploration of hybrid nanofluids (Ag-MoS₂/water) when employing nonlinear radiation to analyze the heat transfer coefficient.

This article is focused on investigating the convective magneto-hydrodynamic single-phase flow for comparative analysis of two different base fluids above an exponentially stretchable porous surface under the effect of the chemical reaction. The Buongiorno fluid model is incorporated to observe the Thermophoresis and Brownian diffusion in this study. Boussinesq approximation for temperature and concentration are accounted for flow to be naturally convective. In this study, water and ethanol are assumed for comparative analysis. Additionally, to achieve the outcomes of the designed three-dimensional flow boundary value, problem technique is employed to simulate the problem in MATLAB. Increase in the magnetic field, thermophoresis diffusion, temperature exponent, and Prandtl number expand thermal boundary, whereas contraction is observed with an increase in porosity. Shear stress rates in respective directions have decreased with an increase in the stretching ratio of the surface. Moreover, through comparison, reasonably enhanced Nusselt number is observed for water under influence of study parameters while the Nusselt number abruptly decreases for ethanol. High mass coefficients are observed for both examined fluids.

Entropy is a thermodynamic function in physics that measures the randomness and disorder of molecules in a particular system or process based on the diversity of configurations that molecules might take. Distance-based entropy is used to address a wide range of problems in the domains of mathematics, biology, chemical graph theory, organic and inorganic chemistry, and other disciplines. We explain the basic applications of distance-based entropy to chemical phenomena. These applications include signal processing, structural studies on crystals, molecular ensembles, and quantifying the chemical and electrical structures of molecules. In this study, we examine the characterisation of polyphenylenes and boron ($B_{12}$) using a line of symmetry. Our ability to quickly ascertain the valences of each atom, and the total number of atom bonds is made possible by the symmetrical chemical structures of polyphenylenes and boron $B_{12}$. By constructing these structures with degree-based indices, namely the K Banhatti indices, $ReZ{G}_1$-index, $ReZ{G}_2$-index, and $ReZ{G}_3$-index, we are able to determine their respective entropies.

This study is aimed to explore the magneto-hydrodynamic Carreau fluid flow over a stretching/shrinking surface with a convectively heated boundary. Temperature-dependent variable thermophysical properties are utilized to formulate the problem. The flow governing equations are obtained with boundary layer approximation and constitutive relation of the Carreau fluid. The shooting method is utilized to obtain graphical and numeric outcomes. Additionally, initial guesses are generated with the help of Newton’s method. The effect of Weissenberg number, Magnetization, stretching ratio, Prandtl number, suction/blowing parameter, and Lewis number is obtained on velocity, temperature and species continuity profile and analyzed. Shear stress rates and Nusselt number outcomes under body forces influences are present in tabulated data and discussed. It is observed that in absence of magnetization force, B = 0 and strong mass suction $5\le S\le 7.5$ effect high rates of Nusselt number is obtained. It is concluded that under the influence of power law index and non-linearity parameter maximum heat transfer and reduced shear stress rates are obtained.