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This research examines the flow of 2D micropolar fluid across an inclined linear shrinking/stretching surface with suction, convective, slip and thermal radiation impact. The governing partial differential equations (PDEs) are transformed into an ordinary differential equations (ODEs) system using a linear similarity transformation, and the bvp4c technique is then used to solve the ODEs system in MATLAB. Comparisons of both solutions are presented. The effects of regulating flow parameters on dimensionless temperature profiles, angular velocity, velocity, skin friction, wall couple stress and heat transfer are shown graphically. Stable velocity profiles are inversely related to velocity slip and material parameter. For a given suction and stretching/shrinking parameter value, graphs show double solutions. The couple stress and skin friction in the first solution, drop then increase as the material parameter [Formula: see text] increases. Critical points of the stretching/shrinking parameter, [Formula: see text] where both solutions occur, are indicated by the symbols [Formula: see text] (i.e., [Formula: see text], and that there is no solution when [Formula: see text]. A table summarizes the stability study’s results and highlights.

Hybrid nanofluid is a modified form of nanofluid which is beneficial for improving energy transfer efficiency. In current investigation, solid nanoparticles aluminum (ΦAl2O3) and copper (ΦCu) 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 (PDEs) into a system of ordinary differential equations (ODEs) which has been then solved by employing the shooting method in Maple 2018 software. The obtained numerical results such as coefficient of skin friction f''(0), heat transfer rate -θ'(0), velocity f'(η) and temperature (θ(η)) distributions are presented in the form of different graphs. The results revealed that duality exists in solution when the suction parameter S ≥ Sci 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.

This research investigates the flow of stagnation point magnetohydrodynamic (MHD) and heat transfer along the stretched sheet in the existence of radiation and slip effects. With the help of similarity variables, the governing partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs). The BVP4C technique in Matlab function has been used to simplify the governing ODEs. The numerical outcomes for temperature and velocity profiles, coefficient of skin friction and Nusselt Number have been achieved and matched with the findings in literature. The findings are compared to previously reported results. In addition, the impacts of numerous related parameters on the profiles of velocity and temperature are shown, and the results of every related parameter are presented using graphs. The velocity profile decreases as the magnetic force, suction, and permeability parameters rise.

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.

One of the most pressing issues in contemporary applied mathematics is the regulation of energy transfer via the application of external forces. The processes of heat transfer are affected by magnetic force, which has many practical uses in industry, engineering, and medicine. This research explores the magnetohydrodynamics (MHD) three-dimensional stable axisymmetric boundary layer over a permeable moving plate, which consists of water as a base liquid and binary distinct nanoparticles to generate a hybrid nanofluid. In all of these, flow beyond the boundary layer area might be calculated by a small crosswise velocity. As a result of its high thermal conductivity, a pair of distinct kinds of nanoparticles have been considered, namely alumina and copper, which are integrated into the base water. The mathematical model is built within a boundary of specified geometry and then converted into a set of ordinary differential equations (ODEs). Resultant ODEs are solved numerically using the technique of three-stage Lobatto IIIa in bvp4c solver in 2017, MATLAB software. Results revealed that two branches exist in certain ranges of moving parameter. The impacts of an increasing physical parameter on profiles of velocities and temperature with skin friction as well as with heat transfer rate are represented in graphs. Furthermore, as the volume fraction of copper increases, so does the skin friction coefficient in the positive direction of [Formula: see text]. The effect of viscous dissipation on the temperature profile in the [Formula: see text]-direction has the same rising results as observed in the [Formula: see text]-direction. According to the results of the temporal stability analysis, the upper branch is realizable and stable.

PurposeThis study aims to investigate the flow and heat transfer of a hybrid nanofluid through an exponentially stretching/shrinking sheet along with mixed convection and Joule heating. The nanoparticles alumina (Al2O3) and copper (Cu) are suspended into a base fluid (water) to form a new kind of hybrid nanofluid (Al2O3-Cu/water). Also, the effects of constant mixed convection parameter and Joule heating are considered.Design/methodology/approachThe governing partial differential equations are transformed into ordinary differential equations (ODEs) using appropriate similarity transformations. The transformed nonlinear ODEs are solves using the bvp4c solver available in MATLAB software. A comparison of the present results shows a good agreement with the published results.FindingsDual solutions for hybrid nanofluid flow obtained for a specific range of the stretching/shrinking parameter values. The values of the skin friction coefficient increases but the local Nusselt number decreases for the first solution with the increasing of the magnetic parameter. Enhancing copper volume fraction and Eckert number reduces the surface temperature, which intimates the decrement of heat transfer rate for the first and second solutions for the stretching/shrinking sheet. In detail, the first solution results show that when the Eckert number increases as 0.1, 0.4 and 0.7 at λ = 1.5, the temperature variations reduced to 10.686840, 10.671419 and 10.655996. While in the second solution, keeping the same parameters temperature variation reduced to 9.750777, 9.557349 and 9.364489, respectively. On the other hand, the results indicate that the skin friction coefficient increases with copper volume fraction. This study shows that the thermal boundary layer thickness rises due to the rise in the solid volume fraction. It is also observed that the magnetic parameter, copper volume fraction and Eckert number widen the range of the stretching/shrinking parameter for which the solution exists.Practical implicationsIn practice, the investigation on the flow and heat transfer of a hybrid nanofluid past an exponentially stretching/shrinking sheet with mixed convection and Joule heating is crucial and useful. The problems related to hybrid nanofluid have numerous real-life and industrial applications, such as microelectronics, manufacturing, naval structures, nuclear system cooling, biomedical and drug reduction.Originality/valueIn specific, this study focuses on increasing thermal conductivity using a hybrid nanofluid mathematical model. The novelty of this study is the use of natural mixed convection and Joule heating in a hybrid nanofluid. This paper can obtain dual solutions. The authors declare that this study is new, and there is no previous published work similar to the present study.

Purpose This study aims to investigate the heat transfer characteristic of the magnetohydrodynamic (MHD) hybrid nanofluid over the linear stretching and shrinking surface in the presence of suction and thermal radiation effects. Design/methodology/approach Mathematical equations are transformed into pairs of self-similarity equations using similarity transformation. Boundary value problem solver (bvp4c) in MATLAB was adopted to solve the system of reduced similarity equations. In this study, the authors particularly examine the flow and heat transfer properties for different values of suction and thermal radiation parameters using single-phase nanofluid model. A comparison of the present results shows a good agreement with the published results. Findings It is noticed that the efficiency of heat transfer of hybrid nanofluid (Cu-Al2O3/H2O) is greater than the nanofluid (Cu/H2O). Furthermore, it is also found that dual solutions exist for a specific range of the stretching/shrinking parameter with different values of suction and radiation parameters. The results indicate that the skin friction coefficient and the local Nusselt number increase with suction effect. The values of the skin friction coefficient increases, but the local Nusselt number decreases for the first solution with the increasing of thermal radiation parameter. It is also observed that suction and thermal radiation widen the range of the stretching/shrinking parameter for which the solution exists. Practical implications In practice, the investigation on the flow and heat transfer of MHD hybrid nanofluid through a stretching/shrinking sheet with suction and thermal radiation effects is very important and useful. The problems related to hybrid nanofluid has numerous real-life and industrial applications, for example microfluidics, manufacturing, transportation, military and biomedical, etc. Originality/value In specific, this study focused on increasing thermal conductivity using a hybrid nanofluid mathematical model. This paper is able to obtain the dual solutions. To the best of author’s knowledge, this study is new and there is no previous published work similar to present study.

This paper deals with a stagnation-point boundary layer flow and heat transfer of a Casson fluid towards a stretching/shrinking sheet. The main objective of the present study is to analyse the effects of the injection parameter and heat source on the velocity and temperature profiles as well as the skin friction coefficient and the Nusselt number. It is vital to study the heat transfer and fluid flow problems in the presence of injection and heat source effects due to a wide variety of applications in engineering and industry. The governing nonlinear partial differential equations are transformed into a system of nonlinear ordinary differential equations by using similarity transformation, before being solved numerically using the boundary value problem solver bvp4c routine in MATLAB. Dual solutions are found to exist for the shrinking sheet case, whereas the solution is unique for the stretching case. The stability analysis has been performed to determine the stable solution. It is shown that the first solutions are stable and physically reliable while the second solutions are not. Further, the present results have been compared with the previous published results for a particular case and the comparisons are found to be in good agreement. The local Nusselt number is decreases with an increase in heat source parameter. Rising values of the injection parameter has decreases both the skin friction coefficient and the local Nusselt number.

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.