In this paper, the heat transfer properties in the three-dimensional (3D) magnetized with the Darcy-Forchheimer flow over a shrinking surface of the $$Cu + Al_{2} O_{3} /$$C u + A l 2 O 3 / water hybrid nanofluid with radiation effect were studied. Valid linear similarity variables convert the partial differential equations (PDEs) into the ordinary differential equations (ODEs). With the help of the shootlib function in the Maple software, the generalized model in the form of ODEs is numerically solved by the shooting method. Shooting method can produce non-unique solutions when correct initial assumptions are suggested. The findings are found to have two solutions, thereby contributing to the introduction of a stability analysis that validates the attainability of first solution. Stability analysis is performed by employing if bvp4c method in MATLAB software. The results show limitless values of dual solutions at many calculated parameters allowing the turning points and essential values to not exist. Results reveal that the presence of dual solutions relies on the values of the porosity, coefficient of inertia, magnetic, and suction parameters for the specific values of the other applied parameters. Moreover, it has been noted that dual solutions exist in the ranges of $$F_{s} \le F_{sc}$$F s ≤ F sc , $$M \ge M_{C}$$M ≥ M C , $$S \ge S_{C} ,$$S ≥ S C , and $$K_{C} \le K$$K C ≤ K whereas no solution exists in the ranges of $$F_{s} > F_{sc}$$F s > F sc , $$M < M_{c}$$M < M c , $$S < S_{c}$$S < S c , and $$K_{C} > K$$K C > K . Further, a reduction in the rate of heat transfer is noticed with a rise in the parameter of the copper solid volume fraction.

The modern world moves towards the art of nanotechnology which is impossible without the analysis of thermal performance and thermophysical featuring of nanofluids. Therefore, a case study for Darcy–Forchheimer Flow (DFF) (γ-Al₂O₃/H₂O)nf over a permeable Stretching Rotating Inclined Disk (SRID) under the impacts of thermal radiation and viscus dissipation is organized. The nanofluid is synthesized by novel γ-aluminum nanomaterial and pure water. Then, the problem is formulated properly via similarity equations by inducing empirical correlations of (γ-Al₂O₃/H₂O)nf with their thermophysical attributes. A numerical algorithm is successfully implemented for mathematical analysis and furnished the results for DFF of (γ-Al₂O₃/H₂O)nf. It is inspected that the F_r opposes the motion and the fluid moves promptly by increasing the strength of stretching parameter. The temperature of (γ-Al₂O₃/H₂O)nf enhances due to higher dissipation and fraction factor favors the thermophysical attributes of (γ-Al₂O₃/H₂O)nf. Therefore, the nanofluid has high thermal performance rate and would be better for industrial and engineering purposes.

We deal in this paper with a diffusive SIR epidemic model described by reaction–diffusion equations involving a fractional derivative. The existence and uniqueness of the solution are shown, next to the boundedness of the solution. Further, it has been shown that the global behavior of the solution is governed by the value of $$R_0$$R 0 , which is known in epidemiology by the basic reproduction number. Indeed, using the Lyapunov direct method it has been proved that the disease will extinct for $$ R_0 <1 $$R 0 < 1 for any value of the diffusion constants. For $$R_0>1$$R 0 > 1 , the disease will persist and the unique positive equilibrium is globally stable. Some numerical illustrations have been used to confirm our theoretical results.

This research presents a review of an analytical simulation of heat and mass transmission features of steady, non-Newtonian Casson fluid motion across a permeable medium through a stretching surface. The effects of heat production and thermal emission are put into discussion. Mathematically, the governing model is manipulated by a series of nonlinear partial equations, which are then modified into ordinary differential equations with the assistance of appropriate conversion. Analytical results for such equations are then achieved by invoking the notable technique of the homotopy analysis method (HAM), and its solution sounds good while achieving the convergence guaranteed in the convergence table. Some achievements have been made. The consequence of raising the value of the Casson parameter is comprehended to be putting down the velocity field while increasing the temperature field. Also, the concentration field falls with an increase in the Schmidt number, while it rises with an enhancement in the Soret number. The electric parameter due to Lorentz’s force is capable of accelerating the temperature of the fluid but downsizing the velocity.

The analysis of nanofluid dynamics in a bounded domain attained much attention of the researchers, engineers, and industrialists. These fluids became much popular in the researcher’s community due to their broad uses regarding the heat transfer in various industries and fluid flowing in engine and in aerodynamics as well. Therefore, the analysis of Cu‐kerosene oil and Cu‐water is organized between two Riga plates with the novel effects of thermal radiations and surface convection. The problem reduced in the form of dimensionless system and then solved by employing variational iteration and variation of parameter methods. For the sake of validity, the results checked with numerical scheme and found to be excellent. Further, it is examined that the nanofluids move slowly by strengthen Cu fraction factor. The temperature of Cu‐kerosene oil and Cu‐water significantly rises due to inducing thermal radiations and surface convection. The behaviour of shear stresses is in reverse proportion with the primitive parameters, and local Nusselt number increases due to varying thermal radiations, Biot number, and fraction factor, respectively.

This study is an attempt to investigate the three-dimensional flow of engine oil-based nanofluid under the impact of rotation and partial slip phenomenon over a stretchable surface. Partial slip condition is incorporated into the system at boundaries and the arising quotient of angular speed to stretching constant is regarded as rotation impression on freely moving three-dimensional flow of copper oxide-engine oil and aluminum oxide-engine oil nanofluid. The designed problem has been tackled numerically with the boundary value problem technique, this numerical mechanism requires one to transform the flow governing highly nonlinear PDEs into first-order differential equations along with associated boundary conditions. This procedure is carried out by employing a suitable similarity transformation. The investigation determines the impact of slip and rotation on primary and secondary velocity profiles, shear stress rates, and heat transfer coefficient. Minimum skin friction and Maximum Nusselt number has been observed for nanofluid under partial slip parameter as compared to nanofluid. Reduced skin coefficients were achieved for both nanofluids. Moreover, the outcomes of the study have been related with already available literature and were found in good agreement.