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The power electrical devices are being continuously trimmed down and must disperse a greater amount of heat flux, overheating has become a significant issue. Ferrofluids, also known as magnetic nanofluids, consist of solid magnetic nanoparticles that exhibit superior thermal conductivity compared to their underlying fluids, which can be either aqueous or oil based. This facilitates increased rates of convective heat transfer, but more importantly, it allows for external flow modification using a magnetic field. This work is unique in that it analyses hybridized ferrofluid flow over a shrinking/stretching surface with radiative magnetization and significant magnetic force. The governing equations were solved using the bvp4c function in MATLAB, after being transformed into ordinary differential equations (ODEs) by using similarity transformations. The system acquires a bimodal solution, and stability analysis validates the physical and dynamic stability of the first solution. The local skin friction coefficient, the local Nusselt number, the temperature and velocity profiles, and the substantial influence of governing effects are all studied and graphically displayed. The key findings are the values of Nux1Rex are decreasing for both second and first solutions as values of φCoFe2O4 are increasing. An increasing trend for the values of RexCf is observed for both second and first solutions as values of φCoFe2O4 increases. The results compareded with the results reported in literature and found a significant agreement.

Suspending particles of tiny solid in a fluid used to transport energy can enhance its thermal conductivity and heat transport properties. Our main goal of this examination is to study the radiative unsteady two-dimensional (2D) flow on a continuously diminishing, horizontal sheet. with suction for the hybrid water-based nanofluid and an aligned field of magnetic, including the combined suction, magnetic, and velocity slip conditions effect. The Tiwari & Das model of nanofluid equations is used, which takes into consideration the solid volume percentage. Equations of similarity are derived by employing the transformations of similarity, and the associated equations have been simplified numerically by employing the bvp4c method in MATLAB software for a variety of values of the nanoparticle volume fraction, the unsteadiness, and the wall mass suction in water. It is discovered that, within the given the unsteadiness parameter range, two solutions exist. Moreover, it is found that the fluid velocity slows down in 1st solution as volume fraction of copper nanoparticles rises but speeds up in the second solution at first before slowing down again. Using a temporal stability analysis, it is found that only one of the dual branches is stable over the long run, while the other is unstable.

Here in, heat transfer characteristics of a magnetized Casson sodium alginate-alumina/copper hybrid nanofluid flowing through the permeable moving plane are investigated considering the effects of thermal radiation and Joule heating with Tiwari and Das model. The primary objective of the present study is to investigate the influence of the velocity ratio parameter on the solid volume fraction and the Casson parameter. In addition, the velocity and temperature profiles used in the current study illustrate the effects of the velocity ratio parameter R, solid volume fraction, suction effect, magnetic field, Joule heating, and thermal radiation. To apply the three-stage Lobatto IIIa with boundary value problems (bvp4c) approach to implement the appropriate similarity variables to transform the partial differential equations (PDEs) into a system of nonlinear ordinary differential equations (ODEs). It has been determined that there are two branch solutions for the velocity ratio parameter against using the Casson parameter and copper volume fraction values. Non-unique branch at R Rci. As the Eckert number and radiation grew, the rate of heat transmission surged. The increase of Casson hybrid nanofluid parameter values caused to drop the skin friction in both upper and lower branch.

Abstract Following to improved thermal impact of hybrid nanomaterials, wide range applications of such materials is observed in the thermal engineering, extrusion systems, solar energy, power generation, heat transfer devices etc. The hybrid nanofluid is a modified form of nanofluid which is beneficial for improving energy transfer efficiency. In current analysis, the solid nanoparticles aluminium ( $$\phi_{{{\text{Al}}_{2} {\text{O}}_{3} }}$$ ϕ Al 2 O 3 ) and copper ( $$\phi_{{{\text{Cu}}}}$$ ϕ Cu ) have been mixed with water to produce a new hybrid nanofluid. The investigation of a steady two-dimensional mixed convection boundary layer flow of the resultant hybrid nanofluid on a vertical exponential shrunk surface in the existence of porous, magnetic, thermal radiation, velocity, and thermal slip conditions is carried out. Exponential similarity variables are adopted to transform the nonlinear partial differential equation into a system of ordinary differential equations which has been then solved by employing the shooting method in Maple software. The obtained numerical results such as coefficient of skin friction $$f^{\prime \prime } \left( 0 \right)$$ f ″ 0 , heat transfer rate $$- \theta^{\prime } \left( 0 \right)$$ - θ ′ 0 , velocity $$f^{\prime } \left( \eta \right)$$ f ′ η and temperature $$\left( {\theta \left( \eta \right)} \right)$$ θ η distributions are presented in the form of different graphs. The results revealed that duality exists in solution when the suction parameter $$S \ge S_{ci}$$ S ≥ S ci in assisting flow case. Due to non-uniqueness of solutions, a temporal stability analysis needs to be performed and the result indicates that the upper branch is stable and realizable compared to the lower branch.

Nanoparticles have presented various hurdles to the scientific community during the past decade. The nanoparticles dispersed in diverse base fluids can alter the properties of fluid flow and heat transmission. In the current examination, a mathematical model for the 2D magnetohydrodynamic (MHD) Darcy–Forchheimer nanofluid flow across an exponentially contracting sheet is presented. In this mathematical model, the effects of viscous dissipation, joule heating, first-order velocity, and thermal slip conditions are also examined. Using similarity transformations, a system of partial differential equations (PDEs) is converted into a set of ordinary differential equations (ODEs). The problem is quantitatively solved using the three-step Lobatto-three formula. This research studied the effects of the dimensionlessness, magnetic field, ratio of rates, porosity, Eckert number, Prandtl number, and coefficient of inertia characteristics on fluid flow. Multiple solutions were observed. In the first solution, the increased magnetic field, porosity parameter, slip effect, and volume percentage of the copper parameters reduce the velocity field along the η-direction. In the second solution, the magnetic field, porosity parameter, slip effect, and volume percentage of the copper parameters increase the η-direction velocity field. For engineering purposes, the graphs show the impacts of factors on the Nusselt number and skin friction. Finally, the stability analysis was performed to determine which solution was the more stable of the two.

A modified 3-point Adams block method of order six (3ABM6)to solve neutral delay differential equations (NDDEs) using variable step size strategy is developed. The approximate solution of the retarded and the neutral terms of the neutral delay differential equations at the grid points is obtained using the Newton divided difference interpolation technique. The proposed method will approximate the solution in each step using the three-points concurrently. To determine the performance of the proposed method, the maximum errors (MAXERR) and total number of function calls (FNC) will be compared with the method of 2-point order six predictor-corrector. The numerical results show that the 3ABM6 reduces the number of function calls and better accuracy in term of MAXERR.

A modified 3-point Adams block method of order six (3ABM6)to solve neutral delay differential equations (NDDEs) using variable step size strategy is developed. The approximate solution of the retarded and the neutral terms of the neutral delay differential equations at the grid points is obtained using the Newton divided difference interpolation technique. The proposed method will approximate the solution in each step using the three-points concurrently. To determine the performance of the proposed method, the maximum errors (MAXERR) and total number of function calls (FNC) will be compared with the method of 2-point order six predictor-corrector. The numerical results show that the 3ABM6 reduces the number of function calls and better accuracy in term of MAXERR.

A modified 3-point Adams block method of order six (3ABM6)to solve neutral delay differential equations (NDDEs) using variable step size strategy is developed. The approximate solution of the retarded and the neutral terms of the neutral delay differential equations at the grid points is obtained using the Newton divided difference interpolation technique. The proposed method will approximate the solution in each step using the three-points concurrently. To determine the performance of the proposed method, the maximum errors (MAXERR) and total number of function calls (FNC) will be compared with the method of 2-point order six predictor-corrector. The numerical results show that the 3ABM6 reduces the number of function calls and better accuracy in term of MAXERR.