This study discusses the flow of hybrid nanofluid over an infinite disk in a Darcy–Forchheimer permeable medium with variable thermal conductivity and viscosity. The objective of the current theoretical investigation is to identify the thermal energy characteristics of the nanomaterial flow resulting from thermo-solutal Marangoni convection on a disc surface. By including the impacts of activation energy, heat source, thermophoretic particle deposition and microorganisms the proposed mathematical model becomes more novel. The Cattaneo-Christov mass and heat flux law is taken into account when examining the features of mass and heat transmission rather than the traditional Fourier and Fick heat and mass flux law. MoS₂ and Ag nanoparticles are dispersed in the base fluid water to synthesize the hybrid nanofluid. PDEs are transformed to ODEs by using similarity transformations. The RKF-45th order shooting method is used to solve the equations. With the use of appropriate graphs, the effects of a number of non-dimensional parameters on velocity, concentration, microorganism, and temperature fields are addressed. The local Nusselt number, density of motile microorganisms and Sherwood number are calculated numerically and graphically to derive correlations in terms of the relevant key parameters. The findings show that as we increase the Marangoni convection parameter, skin friction, local density of motile microorganisms, Sherwood number, velocity, temperature and microorganisms profiles increase, whereas Nusselt number and concentration profile exhibit an opposite behavior. The fluid velocity is reduced as a result of enhancing the Forchheimer parameter and Darcy parameter.

Ternary hybrid nanofluids (THNFs) are modern fluids introduced to enhance the performance of conventional hybrid nanofluids (HNFs). Their unique properties make them suitable for diverse applications, ranging from heat exchangers to advanced industrial and medical treatments. Due to the practical applications and innovative features of THNFs, this paper aims to analyze the performance of these fluids to improve the efficiency of modern devices. The THNF is formulated by adding nanoparticles of three different kinds of aluminum oxide (Al ₂ O ₃ ), silicon dioxide (SiO ₂ ), and copper (Cu) into water (H ₂ O) based ethylene glycol (C ₂ H ₆ O ₂ ). The momentum equation is formulated considering the influences of Darcy Forchheimer, permeability, and magnetic field. Thermal radiation, intermolecular friction force, and Joule heating effects are accounted in the thermal field equation. Mass concentration relation is acquired considering binary chemical reaction and activation energy (AE). Additionally, the influence of stratifications (thermal and solutal) at the boundary of the cylinder is considered. The physical phenomenon representing partial differential equations is reduced into ordinary ones utilizing the transformations and then solved via Runge–Kutta Fehlberg (RKF‐45) numerical scheme in Mathematics. Influence of involved sundry variables on HNF and THNF velocity, thermal field, mass concentration, surface drag force (skin friction coefficient), mass, and heat transfer rates were examined. The results showed that the velocity fields of THNF and HNF decay through variables Darcy Forchheimer, porosity, and Hartman number. Thermal field of THNF and HNF improves via radiation parameter, Eckert number, and Hartman number. Local heat transfer rate upsurges versus curvature variable and Prandtl number.

The current study scrutinizes heat and mass transfer features of magnetized flow of $${\mathrm{ZnO}-SAE50}$$ZnO-SAE50 nanolubricant over Riga plate in a Darcy Forchheimer medium. The effects of variable viscosity, thermal radiation, variable thermal conductivity, viscous dissipation and uniform heat source/sink are examined in this study. The diffusion model presented by Cattaneo–Christov is incorporated in this study to enclose heat and mass transport phenomenon. Additionally, the mass transfer rate is inspected subjected to the effects of variable solutal diffusivity and higher order chemical reaction. Heat and mass transfer phenomena have significant applications in the disciplines of science and technology that can be seen everywhere in nature. This simultaneous transportation phenomenon indicates a variety of applications in manufacturing processes, aerodynamics, cooling systems, environmental sciences, oceanography, food industries, biological disciplines, and energy transport systems etc. The modeled system of PDEs is metamorphosed to nonlinear ODEs with the introduction of appropriate transformations. An eminent bvp4c method in MATLAB has been incorporated to execute the resulting system of ODEs numerically. The outcomes of velocity, temperature and concentration profiles corresponding to various emerging parameters have been exposed graphically. The motion of $${\mathrm{ZnO}-SAE50}$$ZnO-SAE50 nanolubricant tends to enhance significantly with larger modified Hartmann number, whereas converse behavior is reported by increasing porosity parameter and variable viscosity parameter. The greater heat transfer rate is observed for variable thermal conductivity parameter. The rates of heat and mass transfer slow down for thermal and solutal time relaxation parameters respectively. The concentration profile gets enriched by growing the order of the chemical reaction and variable mass diffusivity parameter. It is concluded that by increasing solid volume fraction up to $$1.5\%$$1.5% , the viscosity of the nanolubricant enhances up to $$12\%$$12% which consequently slows down motion of the nanolubricant but increases temperature and concentration profiles.

In recent years, using the idea of analytic and bi-univalent functions, many ideas have been developed by different well-known authors, but the using Gegenbauer polynomials along with certain bi-univalent functions is very rare in the literature. We are essentially motivated by this recent research going on, here in our present investigation, we make use of certain q-derivative operator and Gegenbauer polynomials and define a new subclass of analytic and bi-univalent functions. We then obtain certain coefficient bounds, the Fekete–Szegö inequalities and upper bounds for the second-order Hankel determinant for the defined functions class.

In this paper, heat and mass transportation flow of swimming gyrotactic microorganisms (microbes) and solid nanoparticles under the viscous dissipation effect is investigated. The flow model PDEs are renovated with ordinary ones using suitable boundary layer approximations. The system governing the flow model dimensionless equations as well as boundary conditions is numerically treated with the SOR (successive over relaxation) technique. The flow, heat, and mass transport characteristics are examined against the prime parameters. A comparison is examined to be in a good agreement with the earlier results. It is found here that flow and thermal characteristics of the problem are substantially affected by the porous medium. The outcomes evidently point out that porous medium causes an enhancement in the skin friction and density of the motile microbes. Further, the rate of heat transport is devaluated at the surface of sheet due to viscous dissipation and elevated due to suction.

The present evaluation focused on heat transfer and magnetic flux along the moving non-conducting wedge shape with thermal conductivity and surface heat flux effects. A suitable and well-known similarity transformation for integration is used for the coupled non-linear partial differential equations with stream formulations. The Keller Box technique is utilized to integrate the final non-similar equations numerically. The discretized algebraic equations are displayed graphically and numerically using the MATLAB software. The physical characteristics like temperature, magnetic field, velocity profiles, skin friction, heat transfer and magnetic intensity are investigated with various factors. The influence of relevant characteristics like thermal conductivity $$, Prandtl number Pr, wedge parameter $$, variable viscosity $$, and Biot number Bi is interpreted graphically and numerically. It is noted that velocity graph effects are declined at lower value of Biot number and enhanced value is obtained at the higher value of Biot number. The fluid temperature with highest thermal slip effects is displayed with minimum value of thermal conductivity but minimum value is obtained at large value of thermal conductivity. The maximum heat flux value is obtained at large value of moving parameter.