This numerical investigation effectively establishes a unique computing exploration for steady magnetohydrodynamic convective streams of tangent hyperbolic nanofluid traveling across a nonlinearly elongating elastic surface with a variable thickness. In addition, the importance of an externally imposed magnetic field of tangent hyperbolic nanofluid is comprehensively analyzed by considering the substantial impact of thermal conductivity and thermal radiation consequences. The governing PDEs (partial differential equations) are transmuted into a nonlinear differential structure of coupled ODEs (ordinary differential equations) using a series of variable similarity transformations. Furthermore, these generated ODEs (ordinary differential equations) are numerically set using a novel revolutionary Runge-Kutta algorithm with a shooting approach constructed in a MATLAB script. In this regard, extensive comparison studies are carried out to validate the acquired numerical results. The interactions between the associated profiles and the relevant parameters are rationally explored and shown using graphs and tabular forms. The velocity distribution declined with improving Weissengberg number $We$ and power-law index m, while the reverse performance can be observed for temperature. As enhancement in Brownian motion, Thermophoretic and radiation parameters significantly rise in temperature distribution. The use of many different technological and industrial systems, including nano-bioconvective systems, nano-droplet evaporation, nano-ink jet printing, and microbial fuel cells, would benefit this research study.

The main objective of this investigation to examine the momentum and thermal transportation of rotating dusty micropolar fluid flux with suspension of conducting dust particles across the stretched sheet. The novelty of the flow model is the exploration of the significance of boosting the volume concentration of dust particles in fluid dynamics. The governing PDEs of the problem for both phase models are transmuted into nonlinear coupled non-dimensional ODEs by utilizing suitable similarity modifications. The bvp4c technique was utilized in MATLAB script to acquire a graphical representation of the experimental results. This study illustrates the analysis of repercussions of pertinent parameters on non-Newtonian fluid and the dusty phase of fluid. By improving the volume concentration of dust particles and rotating parameters, the axial velocity for both phases depreciates, whereas temperature and transverse velocity for both phases have the opposite behavior. The micro-rotation distribution rises with higher contributions of rotating and material parameters, whereas it decreases against larger inputs of volume concentration of dust particles. The growing strength of the dust volume fraction (${\varphi }_d$) caused the coefficient of skin friction to decrease along the x direction, and the skin friction coefficient is raised along the y direction.

This work aims to offer a mathematical model for two-phase flow that investigates the interaction of Casson nanofluid and dust particles across a stretching surface. MHD Darcy–Forchheimer porous medium and Fourier’s law through Cattaneo–Christove thermal flux are also considered. The governing equations for the two phases model are partial differential equations later transmuted into ordinary ones via similarity transforms. The Runge–Kutta method with the shooting tool is utilized numerically to solve the boundary layer equations computed in MATLAB to obtain numerical results for various pertinent parameters. The numerical outcomes of momentum, temperature, and concentration distribution are visible for both phases. The results of the skin friction, heat transfer coefficients, and the Sherwood number are also visible in the graphs. Furthermore, by comparing the current findings to the existing literature, the validity of the results is confirmed and found to be in good agreement. The fluid velocity is reduced against increasing strength of Casson fluid parameter, enhanced the fluid phase and dust phase fluid temperature. The temperature declines against the growing values of the relaxation time parameter in both phases. Dusty fluids are used in various engineering and manufacturing sectors, including petroleum transportation, car smoke emissions, power plant pipes, and caustic granules in mining.

Bioconvection phenomena for MHD Williamson nanofluid flow over an extending sheet of irregular thickness are investigated theoretically, and non-uniform viscosity and thermal conductivity depending on temperature are taken into account. The magnetic field of uniform strength creates a magnetohydrodynamics effect. The basic formulation of the model developed in partial differential equations which are later transmuted into ordinary differential equations by employing similarity variables. To elucidate the influences of controlling parameters on dependent quantities of physical significance, a computational procedure based on the Runge–Kutta method along shooting technique is coded in MATLAB platform. This is a widely used procedure for the solution of such problems because it is efficient with fifth-order accuracy and cost-effectiveness. The enumeration of the results reveals that Williamson fluid parameter $\lambda$, variable viscosity parameter ${\Lambda }_{\mu }$ and wall thickness parameter $\varsigma$ impart reciprocally decreasing effect on fluid velocity whereas these parameters directly enhance the fluid temperature. The fluid temperature is also improved with Brownian motion parameter $Nb$ and thermophoresis parameter $Nt$. The boosted value of Brownian motion $Nb$ and Lewis number $Le$ reduce the concentration of nanoparticles. The higher inputs of Peclet number $Pe$ and bioconvection Lewis number $Lb$ decline the bioconvection distribution. The velocity of non-Newtonian (Williamson nanofluid) is less than the viscous nanofluid but temperature behaves oppositely.

This study aimed to analyze the momentum and thermal transport of a rotating dusty Maxwell nanofluid flow on a magnetohydrodynamic Darcy–Forchheimer porous medium with conducting dust particles. Nanouids are the most important source of effective heat source, having many applications in scientific and technological processes. The dust nanoparticles with superior thermal characteristics offer a wide range of uses in chemical and mechanical engineering eras and modern technology. In addition, nanofluid Cu-water is used as the heat-carrying fluid. The governing equations for the two phases model are partial differential equations later transmuted into ordinary ones via similarity transforms. An efficient code for the Runge–Kutta technique with a shooting tool is constructed in MATLAB script to obtain numeric results. The study is compared to previously published work and determined to be perfect. It is observed that the rising strength of the rotating and magnetic parameters cause to recede the x- and y-axis velocities in the two phase fluid, but the temperature function exhibits an opposite trend. By improving the diameter of nanoparticles $D_m$, the axial velocity improves while transverse velocity and temperature show the opposite behaviors. Furthermore, it is reported that the inclusion of dust particles or nanoparticles both cause to decline the primary and secondary velocities of fluid, and also dust particles decrease the temperature.