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Here the Cattaneo–Christov double diffusion model explores the mixed convective flow of third-grade nanoliquid on a stretchable surface with Riga device diverging from the traditional Fourier and Fick’s law. The model incorporates entropy optimization and Soret–Dufour effects, offering a unique perspective on heat and mass transfer phenomena. By employing relevant transformations, the complex partial differential equations are converted into more manageable ordinary differential systems. An optimal analysis method is then applied to solve the resulting nonlinear differential system, shedding light on the intricate interplay of various physical variable. Through the utilization of plots, the study delves into the impact of these physical variable, providing insights into the behavior of the system under different conditions. This comprehensive approach not only enhances our understanding of the underlying mechanisms governing the convective flow of nano-liquids, but also highlights the significance of considering nonclassical models in thermal and mass transport studies. The key finding of this study is that fluid velocity enhances for material parameters due to low viscosity. Temperature and nanoparticle concentration enhance for higher values of Dufour and Soret numbers, respectively. For higher estimations of Reynold number, entropy of the system decreases.

The prime aim and essence of this study are to present a closed-form solution of the unidirectional velocity field for thin film flow generated by a third-grade liquid moving over a fixed or movable inclined plane in the presence of partial slip boundary condition. Consideration of partial slip makes this problem different from other published research works. Here lies the novelty of this study. The nondimensional, unidirectional velocity profiles rely on the material parameter of third-grade fluid, slip parameter and initial velocity of the movable plane. Study discloses that the fluid’s velocity decelerates with raising the material property of grade-3 fluid and it accelerates with the slip parameter and initial velocity of the inclined plane. The outcomes of this study are deliberated physically at length.

Numerous industrial and technical dynamic applications of non-Newtonian liquid flow research in thermal and process engineering are increasing every day, owing to their multifunctional relevance. The viscometric flow, heat and mass transfer of an incompressible third-grade liquid model across an exponentially inclined plate according to all of these potential implications are studied in this paper. The inclusion of elements such as mixed convection, heat sink/source, activation energy, thermal heat flux and chemical reaction improves the flow model’s novelty. Using the appropriate similarity transformations, the existing governing expressions are transmuted into nonlinear ordinary differential equations (ODEs). The resulting nonlinear ODEs are numerically solved via the Runge–Kutta (RK) technique in conjunction with a shooting strategy. The dimensionless parameters are graphically illustrated and discussed for the involved profiles. The viscosity of the liquid drops for increased fluid variable, which enhances the velocity field. The velocity profile is declined continuously throughout the boundary layer with increasing buoyancy ratio parameter. The thermal profile inclines for growing values of the radiation and heat source/sink parameters.

In many fields, there are various applications of non-Newtonian fluids. Various complicated fluids (polymer melts, clay coatings and oil) belong to the category of non-Newtonian fluids. The third-grade fluid is one of the most important non-Newtonian fluid models. This paper has the primary object of heat transfer mechanism and boundary layer third-grade fluid flow under the effects of thermal radiation. The time-dependent two-dimensional flow is considered to flow above a permeable stretchable vertical Riga plate. For numerical solutions, the setup of ordinary differential equations (ODEs) is acquired by converting nonlinear governing equations through relevant similarity transformations. The nonlinear setup of ODEs is numerically solved with the aid of a suitable software such as MATLAB via its bvp4c technology. Graphs are sketched to discuss the various flow parameters’ significance for the expression of velocity and temperature fields. Tabulated values of surface drag force and heat transfer rate corresponding to the numerous pertinent parameters are described. The current analysis of the concerned flow mechanism concludes that the fluid parameters descend the temperature distribution but amplify the profile of the fluid velocity. The radiation parameter escalates the temperature field.

This study is important for the fields of pharmaceutical nano-drug suspension, biomedical engineering, pressure surges and food processing systems. The slip condition is necessary for polishing internal cavities and artificial heart valves in a variety of manufactured objects, micro- or nano-channels, and applications. Low Reynolds number (Re$\to 0)$ and long wavelength ($\delta \ll 1$) considerations are used in the formulation of the mathematical model at low non-Newtonian parameter values, nonlinear boundary conditions and the governing nonlinear equation are analytically solved using the perturbation method. The graphs of frictional force, pressure rise, velocity, pressure gradient, and streamline graphs are done using Wolfram MATHEMATICA software. In this paper, we compared the results of the total slip condition with those of the first-order slip condition and the absence of any slip effects. It has been noticed that increasing the suction and injection parameters leads to a decrease the pressure rate with complete slip effects, partial slip effects and no slip effects. We show that an increase in the third grade fluid parameter $\mathrm{\Gamma }$ increases the magnitude of axial velocity. From a physical perspective, it shows the shear thinning characteristic, which causes a decrease in viscosity and an increase in fluid velocity. Frictional force behaves differently when compared to pressure rate. In other words, the pressure gradient acts as an obstacle to the peristalsis-driven flow. The objective of the study is to find the impact of the peristaltic flow phenomena and the impact of peristaltic on third-grade non-Newtonian fluid where the suction and injection are prevailing which is similar to the thing in biomechanical devices, like blood vessels, etc. there is a change of oxygen and carbon dioxide from the tissue layer to the fluid within the blood vessel.

The steady MHD boundary-layer axis-symmetric flow of a third-grade fluid passing through an exponentially expanded cylinder in the vicinity of a magnetic field is investigated in this study. The problem is mathematically modeled. Suitable similarity transformations are carried out to convert the partial differential equations into nonlinear ordinary differential equations. The Runge–Kutta fourth-order shooting technique is used to solve the transmuted system of nonlinear ODEs. Graphical representations of numerical findings are used to examine the effects of various physical factors on the velocity and temperature profiles. The influence of fluid variables on the velocity curve, such as third-grade parameters, second-grade parameters, and Reynolds number, is illustrated and explored. The skin-friction coefficient expression is computed and given. The widths of the velocity and momentum boundary layers are revealed to be increasing functions of the curvature parameter. It is found that the third-grade fluid has a higher velocity profile than Newtonian and second-grade fluids. Also, the stretched cylinders cause a more progressive shift in heat and mass pattern for flow than flat plates do.

This paper presents a stochastic heuristic approach to solve numerically nonlinear differential equation (NLDE) governing the thin film flow of a third-grade fluid (TFF-TGF) on a moving belt. Moreover, the impact on velocity profile due to fluid attribute is also investigated. The estimate solution of the given NLDE is achieved by using the linear combination of Bernstein polynomials with unknown constants. A fitness function is deduced to convert the given NLDE along with its boundary conditions into an optimization problem. Genetic algorithm (GA) is employed to optimize the values of unknown constants. The proposed approach provided results in good agreement with numerical values taken by Runge–Kutta and more accurate than two popular classical methods including Adomian Decomposition Method (ADM) and Optimal Homotopy Asymptotic Method (OHAM). The error is minimized 10${}^{-1}$ times to 10${}^{-2}$ times.

The heat and mass transfer in an unsteady boundary layer flow of an incompressible, laminar, natural convective third-grade fluid is studied. The flow is taken over a semi-infinite vertical porous plate with the temperature-dependent fluid properties by taking into account the effect of viscous dissipation and variable suction. The partial differential equations governing the problem are reduced into the nonlinear, coupled, nondimensional ordinary differential equations with the help of suitable similarity transformations. The Galerkin Finite Element Method is implemented to solve this acquired system. The effects of various significant parameters such as Grashof number, Prandtl number, Eckert number, Solutal Grashof number and Schmidt number on dimensionless velocity, temperature and concentration profiles are presented graphically. The Nusselt number is found to be depressed whereas the Sherwood number is observed to be enhanced with the increasing values of Grashof number and Prandtl number. The study has important applications in chemical process industries such as filtration in food industry, production of drinking water and recovering salts from solutions.