Nonlinear Ordinary Differential Equations Research Articles

Analytical simulation of Darcy-Forchheimer nanofluid flow over a curved expanding permeable surface

Abstract This research paper presents an analytical simulation of Darcy-Forchheimer flow over a porous curve stretching surface. In fluid dynamics, the Darcy-Forchheimer model combines Forchheimer adjustment and high-velocity effects with Darcy's formula for porous media flow: two nanofluid particles, molybdenum disulfide, and graphene oxide, form nanofluid with the base fluid blood. The governing partial differential equations for momentum and energy are converted into a nonlinear ordinary differential equations system by applying the appropriate similarity transformations. The homotopy analysis method (HAM) is used to solve the transform equations analytically. The impact of essential factors includes the Forchheimer parameter, porosity parameter, slip parameter, Eckert number, nanoparticle volume friction, magnetic field parameter, and curvature parameter. The results have applications in the design of sophisticated cooling systems, where effective thermal control is essential.

The Role of Two Models of Thermal Radiation and Variable Properties on Thermal Explosion Critical and Transition Conditions of the Optically Thin Droplets

ABSTRACT Determining the criticality and disappearance of optically thin droplets (gas phase) by thermal explosions is crucial for maintaining thermal stability in various industrial and nonindustrial applications. The governing nonlinear ordinary differential equations of this problem make use of two distinct approximation models for thermal radiation loss, which were completed by Cogley et al. and Sohrab et al. Two methods of solving the problem – the Semenov method and the inflection point method – are used to identify the circumstances of criticality and disappearance of criticality (transition conditions). The governing equations utilize the use of the temperature-dependent Arrhenius frequency factor of the reaction and the thermal conductivity of the gas. The analytical solution of the equations yields the transitional values of the distinctive parameters at various initial conditions. Numerical methods are used to determine the equation’s exact solution. At specific initial conditions, the impact of θo, θa, γSh,F, and αSh,F on the temperature-time histories of ignition (supercritical region), non-ignition (subcritical region), and the critical boundary between them is derived. Notable effects on the ignition, non-ignition, and critical temperatures and times include the changing frequency of the reaction, the gas thermal conductivity, and two approximation models of radiation heat loss. The two radiation heat loss approximation models of the analytical and numerical findings are compared.

Parametric identification of coefficients for a model of fatigue stiffness degradation of a composite material

The problem of finding the fatigue characteristics of a composite material based on test results is considered. The results of endurance tests of unidirectional polymer composite materials with different initial stiffness, breaking stress and working cycle stress were used as the initial data. As a mathematical model of stiffness degradation, a nonlinear ordinary differential equation with five unknown parameters is used, reflecting characteristic changes in the properties of the material. It is required to find such parameter values that the solution of the differential equation should describe the available test results with sufficient accuracy. The solution procedure is reduced to the problem of optimizing the objective function, the value of which characterizes the achieved accuracy. As optimization methods, a method simulating the behavior of a flock of moths and a method of sequential reduction of the search set were used. A step-by-step algorithm for finding unknown model parameters is proposed, and numerical results of processing input data containing information on changing the elasticity modulus of the composite material in the course of applying load cycles are presented.

A Six-Point y-Function Hybrid Block Method for Direct Solution of Third Order Ordinary Differential Equations

In this article, a new continuous numerical method was developed for the numerical integration of general third-order initial value problems (IVPs) of ordinary differential equations (ODEs). The method was derived by adopting interpolation and collocation approach using a combination of power series and exponential functions as the basis function for the approximate solution to the ODEs. The approximate solution was interpolated at both nodal and off-nodal points while the differential systems from the approximate solution was collocated at all nodal points to obtain the set of methods with given step-numbers. This derived approach ensured that the hybrid points are obtained at the y-values of the methods. This class of hybrid linear multi-step method satisfied the conditions for usability and accuracy of any given method. These conditions were confirmed by testing the consistency, stability and convergence of the developed method. The method was applied to solve directly linear and non-linear third order ODEs without reduction to system of first-order ordinary differential equations. Results from the computation compared favorably with some existing numerical methods in literature with comparable properties to confirm the superiority of the newly developed method.

Bioconvective magnetized flow analysis of a tangent hyperbolic nanofluid with nonuniform source/sink over an exponentially porous stretching sheet

AbstractThis study use numerical method to investigate the transportation of a bioconvective magnetized tangent hyperbolic nanofluid across an exponentially porous stretched sheet. The flow model takes into account the important contributions of Joule heating, thermal radiation, activation energy, and nonuniform heat source/sink. Moreover, slip boundary conditions with gyrotactic microorganisms are taken into flow analysis. The flow model is converted into a system of nonlinear ordinary differential equations (ODEs) using suitable similarity variables. The bvp4c approach in MATLAB is used to get numerical solutions for this system. The study evaluates the influence of different parameters on the velocity, temperature, concentration, and microorganism profile via graphical representations. The results indicate that increasing the mixed convection parameters accelerates the velocity profile, whereas raising the magnetic parameter slows it down. The temperature profile increases with higher values of radiation parameter, however, it decreases as the Prandtl number and thermal slip parameter increase. Furthermore, the microorganism profile decreases as the bioconvection Lewis number and microorganism slip parameter increase.

Boost heat transfer efficiency through thermal radiation and electrical conductivity in nanofluids

AbstractThermal radiation and electrical conductivity in nanofluids are crucial for their heat transfer characteristics, especially in applications with elevated temperatures or significant radiative heat transfer. The suspension of nanoparticles in suspension contributes to the fluid's electrical conductivity, which is essential for efficient heat dissipation in electronic cooling systems. Understanding the behavior of electroconductive nanofluids is particularly important in electronic cooling applications, where effective heat dissipation is crucial to prevent overheating. A study using a similarity parameter transformed a system of coupled PDEs into a set of nonlinear ordinary differential equations, revealing significant changes in key physical characteristics, such as temperature distribution, cubic concentration field, and velocity field. The study also examined the effects of parameters like Brownian motion, Rayleigh number, Peclet number, thermophoresis parameter, and buoyancy ratio parameter. The research provides valuable insights into the complex dynamics of magnetized non‐Newtonian nanofluids and offers practical implications for applications like microbial fuel cells and heat transmission equipment.

Tetrahybrid nanoparticles through squeezed channel with inclined Lorentz forces

Heat exchangers, nuclear power plants, and other engineering and manufacturing operations all involve high temperatures. These systems have a very big temperature gradient, which has a substantial impact on the fluid's transport characteristics. Under such circumstances, using the linear Boussinesq approximation and taking into account the constant thermophysical parameters for the ambient liquid become meaningless. Thus, under the impact on the angled magnetic field and joule heating, the radiative convection flow of the blood-based tetrahybrid nanoliquid in a squeezing channel is examined in this work. The base fluid that contains the distributed alumina, gold, silver, and titania nanoparticles is blood. Through suitable transformations, the partial governing equations were decreased into nonlinear ordinary differential equation's, which are then resolved mathematically applying the fourth-order accurate BVP4C technique. The increase in the viscous dissipation parameter corresponds toward the notable elevation on the blood temperature profile. The rate of heat transfer 57.4519% improved in Au-Ag-Al2O3-TiO2/blood than that of pure blood.

Nonlinear vibro-acoustic analysis of an axially moving submerged circular cylindrical shell

In this study, the nonlinear vibro-acoustic dynamics and stability of doubly-clamped axially moving cylindrical shell are investigated. The exterior surface of the shell is in contact with fluid and subjected to oblique incident plane sound wave. Donnell’s nonlinear shallow shell theory is used to derive the nonlinear partial differential equation of the cylindrical shell for the radial motion. The Galerkin method is employed to discretize the equations of motion into the set of coupled nonlinear nonhomogeneous ordinary second order differential equations. Considering both driven and companion modes, Multiple Scales Method is used to obtain the response of the system. The effects of sound level, incident angle and axially velocity on the frequency response of the system are studied. The results show that, with increase in the shell velocity transmission loss of the circular shell increases. In addition, at low shell axial velocities simultaneous excitation of the driven and companion modes occurs.

A Novel Method for Analyzing Global Bifurcation and Global Hysteresis of FGM Truncated Conical Shell Structure

In the dynamic bifurcation and dynamic hysteresis of a system, our first consideration is the tiny changes in time that can lead to variations in the vector field, potentially causing the occurrence of system dynamic bifurcation and dynamic hysteresis. This paper presents a novel method for analyzing the global bifurcation and hysteresis of Functionally Graded Material (FGM) truncated conical shell structure under aerodynamics and in-plane force along meridian near internal resonances. This method involves introducing the ratio of vector fields as a periodic perturbation parameter in the system’s nonlinear second-order ordinary differential governing equations. This allows for the analysis of the nonlinear ordinary differential equations as algebraic equations, yielding algebraic expressions that only involve parameters and do not contain state variables. Subsequently, the global bifurcation set and global hysteresis set of FGM truncated conical shell structure are obtained, providing the parameters interval ranges for the stability and instability of FGM truncated conical shell structure. By utilizing MAPLE software for numerical simulation, the images of the global bifurcation set and global hysteresis set about the amplitude of in-plane load are first simulated numerically. Subsequently, the amplitude graphs of the equilibrium points are numerically simulated for validation. The results demonstrate a perfect alignment between the images of the global bifurcation set and global hysteresis set and the data from the amplitude graphs of the equilibrium point. Due to the periodicity of the periodic perturbation parameter, it shuttles between different persistent regions as other parameters change, leading to the generation of chaotic solutions in the governing equations. The validity of the obtained results is confirmed through comparisons with existing literature.

Assisting and opposing mixed convective flow over a solid sphere at its bottom stagnation point with a constant surface heat, mass and motile microorganism flux

This study examined the laminar mixed convective flow around the lower point of stagnation for a solid sphere with a steady flux of gyrotactic microbial motility, mass and temperature. Biological convection phenomena have been considered in this study due to their intermixing traits and capacity in order to enhance mass transit in numerous biotechnology and biological systems. The first objective of this study is to examine and establish the mathematical formulation comprising partial differential equations for energy, momentum, mass conservation and mobile microorganism conservation balances. To do numerical calculations, the controlling partial differential equations are initially transformed by similarity transformations into a collection of interconnected non-linear ordinary differential equations. These equations are then solved by using the MATLAB Bvp4c scheme. The results indicate that several controlling parameters, particularly bioconvection parameters such as the Lewis number ( Le ), Bioconvection Lewis parameter ( Lb ) and Bioconvection Peclet number ( Pe ), have significant effects on flow transfer rates. A dual solution is observed in a certain region of mixed convection λ , where the mixed convection parameter is within the range of 0.41–0.65. The findings reveal significant results from dual solution analysis, highlighting the occurrence of unstable and stable phenomena for specific values of bioconvection parameters exclusively in the case of opposing flow. The assisting flow demonstrates only a single solution that is physically stable and unique. Observing lower stagnation point flow is essential for predicting and analysing complex fluid–microorganism interactions in various scientific and engineering applications such as designing spherical-shaped microbial fuel cell. Optimising flow conditions can enhance the efficiency of processes like wastewater treatment, where microorganisms play a crucial role in breaking down pollutants.

Influence of non-linear thermal radiation on the dynamics of homogeneous and heterogeneous chemical reactions between the cone and the disk

Abstract Purpose The current work presents a theoretical framework to boost heat transmission in a ternary hybrid nanofluid with homogeneous and heterogeneous reactions in the conical gap between the cone and disk apparatus. Furthermore, the impacts of non-linear thermal radiation on the ternary hybrid nanofluid composed of white graphene, diamond, and titanium dioxide dispersed in water are analyzed. Originality/value The combination of cone and disk systems is crucial for designing efficient heat exchange devices in the field of biomedical science for various purposes. For instance, in medical devices, the cone–disk apparatus is used to study the flow and heat transfer characteristics for better design and functionality. Hence, a sincere attempt has been made to study the impact of homogeneous and heterogeneous reactions on the nanofluid flow between the cone and disk in the presence of non-linear thermal radiation. Design/methodology/approach The mathematical model’s governing equations are partial differential equations (PDEs) which are then transformed into non-linear ordinary differential equations through appropriate similarity transformations. These transformed resultant equations are approximated by the Runge–Kutta–Fehlberg fourth/fifth order (RKF45) technique. The influence of essential aspects on the flow field, heat, and mass transfer rates was analyzed using a graphical representation. Findings The interesting part of this research is to discuss the power of parameters in three cases, namely, (1) rotating cone/disk, (2) rotating cone/stationary disk, and (3) stationary cone/rotating disk. Furthermore, the thermal variation of the fluid is analyzed by an artificial neural network with the help of the Levenberg–Marquardt backpropagation algorithm. The regression analysis, mean square error, and error histogram of the neural network are analyzed using this algorithm. From the graph, it is perceived that the flow field climbed up significantly with an increase in the values of radiation parameters in all cases. Also, it is noticed that temperature upsurges significantly by upward values of solid volume fraction of the nanoparticles ( ϕ \phi ).

Modified Hermite Wavelet Discrete Matrix Approach for the Brusselator Chemical Model

AbstractThe primary goal of this study is to create a wavelet collocation technique that can be used to solve nonlinear fractional order systems of ordinary differential equations, which are equations that arise in modeling problems related to auto‐catalytic chemical reactions. Using the Hermite wavelet collocation method (HWCM), the system of nonlinear ordinary differential equations of integer and fractional order is numerically solved. The nonlinear Brusselator system is transformed into an algebraic equation system using the collocation method and the fractional derivative operational matrices. The Newton‐Raphson method is used to solve these algebraic equations, and the approximate values of the derived unknown coefficients are substituted. Through the numerical examples, the method's computational effectiveness and correctness are illustrated with various model constraints. A numerical comparison is made between the current approach ND solver, RK method, and Haar wavelet method (HWM). The efficiency and reliability of the developed strategy's performance are shown in graphs and tables. The created Hermite wavelet collocation method is resilient and has good accuracy compared to current methods found in the literature. Numerical computations are performed through Mathematica, a mathematical software.

MHD double diffusive convective squeezing ternary nanofluid flow between parallel plates with activation energy and viscous dissipation

Purpose This study aims to present the consequences of activation energy and the chemical reactions on the unsteady MHD squeezing flow of an incompressible ternary hybrid nanofluid (THN) comprising magnetite (FE3O4), multiwalled carbon nano-tubes (MWCNT) and copper (Cu) along with water (H2O) as the base fluid. This investigation is performed within the framework of two moving parallel plates under the influence of magnetic field and viscous dissipation. Design/methodology/approach Due to the complementary benefits of nanoparticles, THN is used to augment the heat transmit fluid’s efficacy. The flow situation is expressed as a system of dimensionless, nonlinear partial differential equations, which are reduced to a set of nonlinear ordinary differential equations (ODEs) by suitable similarity substitutions. These transformed ODEs are then solved through a semianalytical technique called differential transform method (DTM). The effects of several changing physical parameters on the flow, temperature, concentration and the substantial measures of interest have been deliberated through graphs. This study verifies the reliability of the results by performing a comparison analysis with prior researches. Findings The enhanced activation energy results in improved concentration distribution and declined Sherwood number. Enhancement in chemical reaction parameter causes disparities in concentration of the ternary nanofluid. When the Hartmann number is zero, value of skin friction is high, but Nusselt and Sherwood numbers values are small. Rising nanoparticles concentrations correspond to a boost in overall thermal conductivity, causing reduced temperature profile. Research limitations/implications Due to its firm and simple nature, its implications are in various fields like chemical industry and medical industry for designing practical problems into mathematical models and experimental analysis. Practical implications Deployment of the squeezed flow of ternary nanofluid with activation energy has significant consideration in nuclear reactors, vehicles, manufacturing facilities and engineering environments. Social implications This study would be contributing significantly in the field of medical technology for treating cancer through hyperthermia treatment, and in industrial processes like water desalination and purification. Originality/value In this problem, a semianalytical approach called DTM is adopted to explore the consequences of activation energy and chemical reactions on the squeezing flow of ternary nanofluid.

A new approach to determine occupational accident dynamics by using ordinary differential equations based on SIR model

The motivation of this study is to develop and establish an occupational accident dynamical model (OA model) based on Susceptible-Infected-Recovered framework. In order to investigate the dynamics of the OA model, monthly occupational accident data from Turkey between 2013 and 2020 has been selected as dataset. The OA model is defined by a coupled first-order ordinary nonlinear differential equation with four variables. In addition, the relationships between these variables are described with ten parameters. The OA model’s characterization of the equilibrium points is analyzed by investigating the behaviors of these points according to the eigenvalues derived from the Jacobian matrix. Also, the stability of these points is obtained according to the eigenvalues. These results show the behavior of the system near equilibrium points. After that, the reproduction number is computed by using the next-generation matrix method. The calculated reproduction number for given parameters reveals that the OA model is unstable. The OA model is numerically solved in 96 steps, with a time interval of 1 month, using the ODE45 Matlab routine based on the explicit Runge–Kutta algorithm. In addition, a modified OA model is developed by adding the Occupational Health and Safety (OHS) re-training parameter to the OA model to observe a reducing effect on occupational accident numbers. The main results of this study provide a new approach about the future estimation of the number of occupational accidents. Furthermore, through the comparison of numerical results from both models, the study demonstrates that national safety policies, particularly those enhancing the efficacy of OHS training, can effectively mitigate accidents.

Modelling for a new mechanical passive damper and its mathematical analysis

To improve ride comfort, an oil damper should restrain its damping force in a high-velocity range. A lot of dampers with such properties were developed. For example, the one controlling the flux of oil by a leaf valve is widely adapted and reasonable. However, it is difficult to represent its dynamics with a simple mathematical model, and the cost of a computational fluid dynamics is too expensive. To overcome the disadvantages, the first author in [15] developed the other mechanical oil damper with sub-pistons instead of the leaf valve, which enabled us to propose a simple mathematical model with linear ordinary differential equations, thanks to the simple mechanism to control the oil flow. In this article, we give a more detailed mathematical model for the damper taking the dynamic pressure resistance into account, which is represented by nonlinear ordinary differential equations. In addition, a numerical scheme for the model is also proposed and its mathematical analysis such as the validity of the numerical solutions is shown.

Dynamic strategies and optimal control analysis for hepatitis C management: non-invasive liver fibrosis diagnosis

This study proposes a novel model employing nonlinear ordinary differential equations to dissect HCV dynamics. Six distinct population groups are delineated: Susceptible, Treatment, Responder, Non-Responder, Cured, and Fibrosis. A detailed numerical analysis of this model was conducted, tracking the predicted trends over a span of 20 years. The primary objective of this analysis is to assess and confirm the model’s predictive accuracy and its potential to supplant invasive diagnostic methods in monitoring the progression of liver fibrosis. By incorporating various control parameters, namely u 1 ( t ) , u 2 ( t ) , and u 3 ( t ) , the model offers a nuanced perspective on disease progression and treatment outcomes. Parameter u 1 ( t ) modulates treatment-induced fibrosis progression, providing a crucial lever for mitigating treatment-related side effects. u 2 ( t ) reflects treatment effectiveness, capturing the proportion of responders within the treatment cohort. Meanwhile, u 3 ( t ) governs fibrosis progression in non-responders, shedding light on the disease’s natural trajectory without effective treatment.

A mathematical model of Cholera–Typhoid coinfection dynamics with a hygiene-driven contact rate

Waterborne infections such as Cholera and Typhoid remain a huge burden on the public health systems of poor countries in Africa and Asia. The highest disease burden is concentrated in sub-Saharan Africa. In most regions, the recent spike in cases of both infections is attributed to dilapidated infrastructure and poor hygiene. This paper examines the role of poor hygiene in the surge of both infections. We use a system of non-linear ordinary differential equations to model the movement of people between the different classes of infections, and we use a sigmoidal function to model the different levels of hygiene in a community. The findings demonstrate that managing hygiene, even if limited to the direct transmission route, can significantly reduce the prevalence of both infections. This paper presents some of the public health implications of these findings.

Significance of mixed convection and heat transfer flow past a vertical Riga plate in the presence of alumina nanoparticles: Analysis of streamlines for oblique stagnation point

AbstractThe aim of the current study to inspect the magnetic flow properties and heat transport features near an oblique stagnation point of a nanofluid with mixed convection through a vertical Riga plate are examined. The Riga plate is a familiar actuator made out of permanently fixed electrodes and magnets that moved away from the plate due to an exponential decline in the Lorentz force. Nanofluid is taken into consideration due to its peculiar properties, such as remarkable thermal conductivity is the significance of the study. These properties are important in heat exchangers, electronics, advanced nanotechnology, and material sciences. Using usual similarity transformations, the group of leading partial differential equations is distorted into a group of nonlinear ordinary differential equations. Then, a very proficient procedure namely bvp4c is utilized to discover the solution. For particular values of the different influential fluid parameters, the characteristics of the dimensionless temperature and velocity along with drag force and heat transfer are investigated graphically. In addition, the symmetrical results were initiated for both cases of the slip and without slip parameters. It is demonstrated that for greater influence of the volume fraction of nanoparticles, the normal and tangential velocity profile drops down for the instances of aiding and opposing flows due to fluctuations in the presence of slip factor, and absence of slip factor. In contrast, the temperature profile intensifies in both the cases of slip parameters and without the slip parameter owing to the phenomenon of assisting flow and opposing flow subject to superior impressions of the nanoparticle volume fractions. It is also observed that depending on the equilibrium between buoyancy effects, obliqueness, velocity slip parameters, and straining motion, the location of the point of zero shear stress (friction factor on the surface of the wall) is displaced to the right or the left of the origin.

Utilizing numerical techniques for modeling radiating Casson nanofluid flow with thermophoretic phenomenon on a heated stretching surface

This work aims to use mathematical modeling to investigate the mechanics of heat and mass transport in dissipative Casson nanofluid flows over a linear rough sheet. This study considers various elements such as thermal radiation, magnetic fields, heat generation, the varied properties of porous media, and the thermophoretic impact. Besides that, it looks into the variations in viscosity, diffusivity, and thermal conductivity, as well as the energy dissipation from viscous internal friction and fluid temperature modifications. The method involves coming up with and changing the boundary layer equations into a group of linked nonlinear ordinary differential equations that use variables that do not have any dimensions. The foundation of the solution strategy for these equations is the Hermite collocation method (HCM), which is renowned for its precision and adaptability. It offers an organized approach to solving the complex differential equations, allowing for accurate numerical solutions. The use of graphical representations ensures thorough data analysis and clarity, while also providing insightful information about the computed outcomes. The code validation method uses numerical comparisons with recent research to confirm the algorithm’s accuracy and dependability, as well as its resilience in comparison to the existing literature. Important conclusions from the study show that thermal boundary layers and nanofluid velocity decrease with increases in the porosity parameter, slip parameter, and magnetic field parameter.

Smoluchowski temperature and Maxwell velocity slip impacts on magneto-radiative Fe 3 O 4-Al 2 O 3-Cu/H 2 O tri-hybrid nanofluid flow over a shrinking surface with entropy generation: a stability performance

ABSTRACT This investigation aims to find entropy production and to determine the existence of a dual solution in magnetohydrodynamic (MHD) thermal radiative tri-hybrid nanofluid flow over an exponentially shrinking surface by considering suction effects at the surface. Furthermore, the effects of Smoluchowski temperature and Maxwell velocity slip are considered. The products on various flow fields are examined for three different types of fluid: mono nanofluid ( F e 3 O 4 / H 2 O ), hybrid nanofluid ( F e 3 O 4 - A l 2 O 3 / H 2 O ), and tri-hybrid nanofluid ( F e 3 O 4 - A l 2 O 3 - Cu / H 2 O ). The similarity transformation translates the nonlinear partial differential equations (PDEs) into a set of ordinary differential equations (ODEs). The numerical MATLAB function bvp4c is utilized to construct and solve governing higher-order nonlinear ODEs. A tabular and graphical analysis is conducted on the impact of emergent factors on velocity, temperature, skin friction coefficient, Nusselt number, and entropy generation. The acceptable parameter ranges for the computation are as follows: 0.01 < ϕ 1 < 0.1 , 0.01 < ϕ 2 < 0.1 , 0.01 < ϕ 3 < 0.1 , − 0.7 < ξ < 1.6 , 2.0 < S < 3.0 , 0.0 < M < 0.2 , 0.5 < Nr < 1.5 , 0.1 < ω < 0.5 , and 0.1 < ε < 0.5 . Due to the contraction of the surface, dual solutions are found, but dual solutions cannot be found beyond the critical values. The critical values ξ λ are − 1.44285 , − 1.44804 , and 1.51550 for mono nanofluid, hybrid nanofluid, and tri-hybrid nanofluid, respectively. Including A l 2 O 3 and Cu nanoparticles inside F e 3 O 4 nanoparticles containing nanofluid depicted an accelerating effect on skin friction and Nusselt number rates. The skin friction coefficient and heat transference rate in both the solution branches improved as the suction parameter S was increased. It has been demonstrated that the fluid’s temperature is enlarged by the thermal radiation parameter Nr and increasing nanoparticle volume fractions ϕ 1 , ϕ 2 , ϕ 3 . However, the temperature is reduced greatly by the suction parameter S and Smoluchowski temperature slip parameter ε . Since dual solutions are present, stability analysis is performed by finding the minimum eigenvalue. A positive minimum eigenvalue ( β 1 ) denotes the upper stable solution branch, whereas a minimal eigenvalue that is negative indicates the bottom branch is unstable. The entropy of the flow was found to increase with thermal radiation parameter Nr and magnetic field parameter M in a physically stable solution. Still, it decreased with a higher nanoparticle volume fraction ( ϕ 1 , ϕ 2 , ϕ 3 ). The current optimization technique offers a new and beneficial insight through numerous applications in various fields, i.e. polymer engineering, extrusion of polymer, plastic sheet compression, glass production, fiber, metallic furnace, medicine, and the field of biomedical technology.