Dimensionless Nonlinear Ordinary Differential Equations Research Articles

Non-linear primary resonance analysis of stiffened FGM conical panels resting on viscoelastic foundation

This article investigates the non-linear forced vibrations and primary resonance analysis of stiffened FGM conical panels (OCPs) resting on a viscoelastic foundation utilizing a semi-analytical method. The classical shells theory and von Kármán nonlinear strain–displacement relations are used to model of OCPs. The equations of motion are derived by applying Hamilton’s principle. Then, the dimensionless equations of motion are established and converted into dimensionless nonlinear ordinary differential equations by applying Galerkin’s method. The solution of the ordinary differential equations is carried out based on the multiple-scale method for analyzing the vibration behavior. In this regard, the necessary relations for the nonlinear primary resonance and internal resonance of stiffened FGM-OCPs are obtained. To validate the suggested methodology, first, the results are verified with various previous research. Second, a numerical method based on the fourth-order Runge-Kutta’s approach is used for verifying the nonlinear vibration analysis. Finally, changes in various parameters such as viscoelastic foundation coefficients, stiffeners, temperature, semi-vertex angle and subtended angle are examined by helping the frequency response plots.

Simulation of hybrid boiling nano fluid flow with convective boundary conditions through a porous stretching sheet through Levenberg Marquardt artificial neural networks approach

In the domain of artificial neural networks, the Levenberg-Marquardt technique stands out for its innovative approach and convergent stability. It generates a numerical approach for the progression of a hybrid Cu–Al2CO3/water nanofluid on a porous stretched sheet (HNF-PSS), employing regression plots, state transition measures, histogram representations, and mean squared errors. This work explores the analysis of the fluid flow problem associated with HNF-PSS, introducing a novel application of an intelligent computing system that utilizes the effectiveness of neural networks trained by the Levenberg-Marquardt algorithm (NN-TLMA). The initial mathematical expression, expressed in terms of partial differential equations (PDEs) for HNF-PSS, undergoes transformation into dimensionless nonlinear ordinary differential equations (ODEs). The collection of data for the envisioned NN-TLMA involves parameters influencing the velocity of the HNF-PSS system model, generated using the Lobatto IIIa formula. The efficacy of the suggested NN-TLMA for solving the HNF-PSS is robustly verified by examinations of state transition dynamics, regression analyses, mean square error, and error histogram studies. The consistent alignment of the obtained results with corresponding solutions attests to the legitimacy of the structure, achieving a high level of accuracy within the range of 10–6 to 10–8. Insightful findings demonstrate that the thermal Biot number contributes significantly to an upsurge in fluid temperature. Furthermore, an augmentation in nanofluid concentration and thermal profile is found to increase the porosity strength of the medium. Nevertheless, an attenuation in the velocity profile is noted under these conditions. The application of this simulation work can be found in optimizing the designs of heat exchanger, in biomedical application and in renewable energy processes.

Numerical analysis of mathematical model for heat and mass transfer through Bioconvective Maxwell nanofluid flow subject to Darcy-Forcheimer and Lorentz forces

The objective of this study to evaluate heat and mass transport rate of Maxwell nanofluid flow through a stretched medium in the existence of Darcy-Forcheimmer and Lorentz forces. In addition to above by involving motile microorganisms the heat and mass transport done through the process of bioconvection. Thermal radiation, activation energy, Brownian motion, and thermophoresis effects on the flow rate through this are also investigated. The governing model of the flow problem consist of momentum, thermal, concentration, and micro-organisms density equations, these equations are in the form of non-linear partial differential equations. To transform these non-linear partial differential equations (PDE‘s) to dimensionless non-linear ordinary differential equations (ODE‘s), we introduced couple of similarity variables. For numerical treatment of governing non-linear ODE‘s, MATLAB built in ‘bvp4c‘ solver along with shooting technique utilized. In graphical outcomes, the impact of various physical variables over the momentum, temporal, concentration, and micro-organism descriptions are elaborated briefly. The impact of deborah number on the velocity profile, by increasing the count of Deborah number the velocity profile become decrease. The effect of Lewis number over the concentration depiction, the rise in profile caused by maximizing the values of parameter. The rate of flow of mass and heat transport become increase while the values of thermophoretic parameter increases. Microorganism‘s profile decreases on increasing the Peclet number. In tabular analysis, the impacts of physical variables on the local skin friction, Nusselt number, Sherwood number, and microorganism density number are evaluated. All the physical quantities numerics computed by fixing the values of parameters as 0.2<λ<0.6, 0.3<Nb<0.5,2.0<Pr<4.0, 5.0<Le<5.3, 3.0<Lb<5.0, 0.4<Pe<0.8, 0.2<Nt<0.4, 0.2<Rb<0.4. The computed results are fine as compared to previous published results.

Bvp4c approach and duality of hybrid nanofluid over extending and contracting sheet with chemical reaction and cross-diffusion effects

Heat transfer has a major effect on material selection and mechanical efficiency. The importance of peristaltic fluid motion in transmitting heat is obvious in the domains of biomedical research through the processes of metabolic heat generation, blood transportation, capillary dehydration, thermal control, and bio-heat exchange pathways. The present work aims to explore the computational and theoretical evaluation of hybrid nanofluid (HNF)-suspended Cu and Al2O3 nanoparticles in water in the presence of chemical reaction, Lorentz force and ross-diffusion. The fundamental flow system based on Navier-Stokes in terms of partial differential equations is converted to dimensionless nonlinear ordinary differential equations via similarity transformations. The transformed ODEs are computationally solved by bvp4c approach. Dual solutions have been observed for emerging factors, so stability examinations are implemented to find the stable solution. Based on eigenvalues, it is witnessed that smallest positive eigenvalues indicates the stable while negative depicts the unstable solutions. The influences of involved factors on the flow characteristics are depicted through graphs and tables. The confirmation of the present analysis is done with the published study. Decreasing behavior is investigated for both branches of velocity as the M is improved while f′(η) is an increasing function of K. From this analysis it is reported that θ(η) is directly proportional to R, Q and Q1. Increasing phenomena is reported for θη and ϕη profiles as the quantities of Ec are augmented from 0.1 to 1. The conclusions of this theoretical analysis have impending applications in numerous fields such as technological devices, solar cooling and heating systems for cars, surfactants, lubricating qualities, hybrid generators, metallic welding process, and the production of medical equipment.

Analysis of irreversibility on nanofluid flow through porous medium considering shape, dispersion and melting effects

This study is conducted to explore the impact of thermal dispersion and melting on the natural convective nanofluid (nickel zinc ferrite + SAE 20W-40 motor oil) flow over a stretching sheet in a Darcy porous medium. An irreversibility analysis has also been performed using the second law of thermodynamics. The fundamental nonlinear partial differential equations (PDEs) are converted into a system of dimensionless nonlinear ordinary differential equations (ODEs) through Lie group similarity transformations and then solved by employing the spectral local linearization method implemented in MATLAB software. It is noteworthy to highlight that the validation results indicate a strong concurrence with existing findings in previously published literature. After simulation, it is specified that the thermal dispersion disrupts the entropy (by 63.7%) and enhances the surface friction. Likewise, the heat transfer rate improves by 1.5% with the melting effect and 58.7% with the Darcy number. The results indicate that the brick-shaped particles exhibit a higher rate of heat transfer and lower entropy production. Further, this work has many uses such as, core, microwave, and biomedical applications like tumor treatments, bone plate surgeries, EMI (electromagnetic interfaces), electronics cooling, and aerodynamic extrusion processes, in the design of passive ventilation systems.

Existence of dual solution for MHD boundary layer flow over a stretching/shrinking surface in the presence of thermal radiation and porous media: KKL nanofluid model

Present study is dedicated to analyze the closed form solution of nanofluid flow over a stretching/shrinking sheet with dual availability. Flow is developed through two-dimensional boundary layer theory. Appropriate tensor is used to generate the continuity, energy, and momentum equations. Converted governing partial differential equations (PDEs) into dimensionless non-linear ordinary differential equations (ODEs) by adoption of favorable similarity variables. The dimensionless ODEs of energy and momentum produced a dual nature solution in closed form under certain conditions. To deal with the nanofluid, the Koo-Kleinstreuer and Li (KKL) model is used, and the equations are solved using well-known software Maple. The effect of porosity Φ, suction/injection fw, stretching/shrinking λ, and magnetic effect M on skin friction, velocity, temperature, and streamlines are well explored and showcased. The results for the stable solutions have been showed that the upper branch's fluid velocity is increasing as the magnetic parameter M rises whereas the lower branch's fluid velocity is decreasing as M rises. Additionally, the CuO-nanofluid's velocity is impacted by the volume fraction of nanoparticles, with an increase in volume fraction causing a decrease in velocity. On both the lower and upper branches, the temperature profile is seen to improve as the Biot number increases. On the other hand, as the magnetic parameter varies and the magnetic field increases, the local Nusselt number against suction/injection decreases, as well as the rate of heat transfer in the upper branch decreases.

Numerical study of MHD Prandtl Eyring fuzzy hybrid nanofluid flow over a wedge

The prime aim of this effort is to investigate the steady, incompressible flow of magnetohydrodynamics (MHD) Prandtl Eyring fuzzy hybrid nanofluid through a wedge surface when viscous dissipation, nonlinear thermal radiation, and fuzzy volume fraction are presented. The flow phenomena of a hybrid nanofluid are investigated for the two-dimensional fluid flow with the inclusion of copper and alumina particles added into sodium alginate (SA). By employing a dimensionless similarity transformation, the presented model’s nonlinear governing partial differential equations (PDEs) are redefined into a set of dimensionless nonlinear ordinary differential equations (ODEs), which are then solved utilizing numerical methods named built-in MATAB BVP4C (Finite Difference Method). Also, nanoparticle volume fractions are said to be triangular fuzzy numbers (TFN). has control over the TFNs and triangular membership function. The importance of physical factors is emphasized through tables and graphs. The findings of the present investigation indicate that compared to Prandtl Eyring fluid, hybrid nanofluid transmits heat at a faster rate. Thermal distribution is improved by the solid volume fraction percentage. As thermal radiation and second fluid parameter levels rise, heat transfer rate also accelerated. The triangular fuzzy membership analysis specifies that fuzzy temperature rises more quickly for hybrid nanofluids when compared with simple nanofluids.

Stability analysis of radiative-magnetic hybrid nanofluid slip flow due to an exponentially stretching/shrinking permeable sheet with heat generation

The present article aims to find the duality of solutions and investigate the magneto-radiative hybrid nanofluid flow over an exponentially stretching/shrinking permeable sheet with heat generation and multiple slip boundary conditions. The governing nonlinear partial differential equations are transformed into dimensionless nonlinear ordinary differential equations which are computed by bvp4c in Matlab platform. The hybridisation of alumina and copper nanoparticles enhances the system's heat transfer performance. The dual nature solution is obtained only if the suction is applied to the stretching/shrinking surface. From stability analysis, we can conclude that the first solution is stable, but the other is unstable. It is realistic to have the solution stable with the smallest positive eigenvalues, while the lower solution branch is unstable with the smallest negative eigenvalues. It is found that the upsurges of solid volume fraction of the copper nanoparticle enhanced the skin friction coefficient and the local Nusselt number.

Outlining the features of nanoparticle diameter and solid–liquid interfacial layer and Hall current effect on a nanofluid flow configured by a slippery bent surface

AbstractA study was executed to explore the flow characteristics of an electrically conducted, magnetized nanofluid across a curved stretching surface. This flow is accomplished in water‐based nanofluid escorting Aluminum oxide nanoparticles. The novelty of the work is primarily to capture the consequence of solid–liquid interfacial layer and diameter of Aluminum oxide nanoparticles on flow mechanism. Second, the significance of Hall‐current, nonlinear thermal radiation, and velocity slip of second order are engaged in flow and also assume that heat transmission of the flow is attained through convection. The foremost mathematical equations are reformed into dimensionless nonlinear ordinary differential equations through similarity performance. Then, the subsequent equations are resolved by engaging RK‐4 based shooting procedure. Inspiration of stimulating flow elements is achieved accurately through various graphs and tables. Some streamline plots and surface plots are provided to enrich the results section. Temperature circulation enhances for curvature factor and thermal biot number whereas the opposite consequences are perceived when the diameter of the nano‐sized particles enhances. Here, numerical consequences confirm that the utmost rate of heat transport is enriched by 82.79% and 78.28%, respectively, in appearance and nonappearance of radiative heat flux in the flow.

Application of heat transfer on MHD shear thickening fluid flow past a stretched surface with variable heat source/sink

AbstractThe purpose of this study is to explore the viscous dissipation stimulus on the steady convective magnetohydrodynamic shear thickening liquid stream across a vertically stretched sheet. The impact of thermic heat, first‐order velocity slip, and variable heat generation/absorption are considered and also ignored the effect of magnetic Reynold's number. We converted flow controlling equations into the set of dimensionless nonlinear ordinary differential equations by employing similarity variables to solve these coupled equations by R–K and shooting technique. The effect of different dimensionless variables on velocity, heat, friction factor, and local Nusselt numbers are presented through graphs and tables. Depreciation in velocity and growth in temperature distribution is detected when the Casson fluid parameter is increased. Temperature is the increasing function of the Eckert number.

The significance of exponential space-based heat generation and variable thermophysical properties on the dynamics of Casson fluid over a stratified surface with non-uniform thickness

Stratification is characterized as having applications in a variety of energy storage disciplines, including solar thermal energy usage systems. The effect of heat production, magnetic field, and varied fluid characteristics on the flow of Casson fluid across unstratified and stably stratified paraboloid of revolution is investigated in this work. The velocity index is suitable to describe the fluid flow on an upper horizontal surface of a paraboloid of revolution. It is imperative to mention that the upper horizontal surface of a paraboloid of revolution is likened to the pointed surface of an aircraft which is also described as a surface with variable thickness. The governing partial differential equations are transformed into the nonlinear ordinary differential equations using the similarity transformation. The dimensionless nonlinear ordinary differential equations are solved with the help of the Bvp4c numerical algorithm. The effects of relevant parameters are revealed on various profiles and discussed. It is observed that both the temperature and concentration distributions decrease with increase in thermal stratification and solutal stratification parameters when the velocity index parameter

MHD Double-Diffusive Carreau Fluid Flow through a Porous Medium with Variable Thermal Conductivity and Suction/Injection.

In this article, we consider the effects of double diffusion on magnetohydrodynamics (MHD) Carreau fluid flow through a porous medium along a stretching sheet. Variable thermal conductivity and suction/injection parameter effects are also taken into the consideration. Similarity transformations are utilized to transform the equations governing the Carreau fluid flow model to dimensionless non-linear ordinary differential equations. Maple software is utilized for the numerical solution. These solutions are then presented through graphs. The velocity, concentration, temperature profile, skin friction coefficient, and the Nusselt and Sherwood numbers under the impact of different parameters are studied. The fluid flow is analyzed for both suction and injection cases. From the analysis carried out, it is observed that the velocity profile reduces by increasing the porosity parameter while it enhances both the temperature and concentration profile. The temperature field enhances with increasing the variable thermal conductivity and the Nusselt number exhibits opposite behavior.

Entropy generation optimization of unsteady radiative hybrid nanofluid flow over a slippery spinning disk

Entropy generation investigation of hybrid nanofluidic transport over an unsteady spinning disk is reported in this analysis. The magnetic influence, velocity slips, and thermal radiative effects are included within the flow. Ferrous oxide (Fe3O4) and graphene oxide (GO) are used as tiny nano ingredients, and water (H2O) is the base medium. The dimensional leading equations are settled to dimensionless nonlinear ordinary differential equations (ODEs) by significant similarity transformations. Then, classical RK-4 scheme with a shooting process has been initiated to execute the numerical simulation. The software MAPLE-18 is used to run the entire simulation with an indispensable accuracy rate. Several streamlines, graphs, and requisite tables are executed to divulge the parametric impact on the nanofluidic stream. Entropy generation–related figures are depicted for diverse parameters, and parametric effects on Bejan number are also analyzed. Moreover, the corresponding physical consignments like the measure of the frictional hindrance, heat transport are calculated and reviewed. The entropy generation augments for higher magnetic value but reduces for velocity slip, radiation, and nanoparticle concentration. Hybrid nanofluid gives a lower magnitude in entropy production as compared to the usual nanofluid. Magnetic parameter reduces the Bejan number, while slip factor and nanoparticle concentration amplify such effects. Heat transfer ultimately seems to increase for nanoparticle volume fraction, and the increase rate is 4.01685 for usual nanofluid, but it is 6.7557 for hybrid nanofluid. Also, the numerical fallouts address the possibility of using magnetized spinning disks in space engines and nuclear propulsion, and such a model conveys significant applications in heat transport improvement in so many industrial thermal management equipment and renewable energy systems.

Thermal Transport of Hybrid Liquid over Thin Needle with Heat Sink/Source and Darcy–Forchheimer Porous Medium Aspects

The structure of needle is mostly similar as paraboloid of upset parallel to flow possesses and has huge applications in building and modern procedures including microstructure electronic gadgets and microscale cooling gadgets for thermal evacuation application. Based on these applications, an axisymmetric Darcy–Forchheimer flow and energy transport of hybrid nanoliquid over a slender moving needle is considered. Impact of heat sink/source and Newtonian heating is inspected. The dimensional nonlinear administering partial differential equations (PDEs) are modified to dimensionless nonlinear ordinary differential equations (ODEs) with assistance of similarity technique which is then explained numerically utilizing Runge–Kutta–Fehlberg scheme from Dsolve code MAPLE. It ought to be noticed that approval of results shows a decent concurrence with previously existing reports. It is noticed that the enhancing Forchheimer number and porosity parameter increase the temperature.

Thermal performance of Ag–water nanofluid flow over a curved surface due to chemical reaction using Buongiorno's model

AbstractAn analysis of heat and mass transfer is carried out under the influence of chemical reaction, friction heating, and heat generation/absorption over a curved surface. The impacts of random motion attributes of nanoparticles and thermophoresis are also applied in the expressions of energy and concentration. With the help of assigned transformations, the nonlinear partial differential equations are changed to dimensionless nonlinear ordinary differential equations. Then, the numerical solution is obtained using fourth‐fifth order Runge‐Kutta‐Fehlberg method via the shooting technique. The impacts of relevant parameters on velocity, temperature, and concentration are depicted through graphs and tables. The results illustrate that the lowest concentration distribution of nanofluid is related to the higher value of chemical reaction parameter. Moreover, it is found that thermophoresis and Brownian motion parameters have a propensity to increase the temperature profile while curvature parameter decreases the velocity profile. Also, velocity and temperature fields show a similar behavior for the increasing values of volume fraction of the nanoparticles, while a reverse trend is observed in the concentration profile under the same condition. To authenticate the results of the current study, the obtained data were compared with previously published data.

Mathematical Solution on MHD Stagnation Point Flow of Ferrofluid

This study presents a numerical investigation on the magnetohydrodynamic (MHD) stagnation point flow of a ferrofluid with Newtonian heating. The black oxide of iron, magnetite (Fe3O4) which acts as magnetic materials and water as a base fluid are considered. The two dimensional stagnation point flow of cold ferrofluid against a hot wall under the influence of the uniform magnetic field of strength is located some distance behind the stagnation point. The effect of magnetic and volume fraction on the velocity and temperature boundary layer profiles are obtained through the formulated governing equations. The governing equations which are in the form of dimensional non-linear partial differential equations are reduced to dimensionless non-linear ordinary differential equations by using appropriate similarity transformation. Then, they are solved numerically by using the Keller-box method which is programmed in the Matlab software. It is found that the cold fluid moves towards the magnetic source that is close to the hot wall. Hence, leads to the better cooling rate and enhances the heat transfer rate. Meanwhile, an increase of the magnetite nanoparticles volume fraction, increases the ferrofluid capabilities in thermal conductivity and consequently enhances the heat transfer.

Effects of Newtonian heating and thermal radiation on micropolar ferrofluid flow past a stretching surface: Spectral quasi‐linearization method

AbstractThis study article addressesthe flow and heat transfer characteristics of a magnetite Fe3O4 micropolar ferrofluid flow past a stretching sheet. For practical interest, thermal radiation, Newtonian heating, and a heat source or sink are considered in this investigation. A useful Tiwari‐Das nanofluid model is considered to analyze the microstructure and inertial characteristics of the water‐based nanofluids containing iron oxide. The dimensionless nonlinear ordinary differential equations are solved by employing suitable similarity variables. The resulting nonlinear system is solved by the spectral quasi‐linearization method. The effects of different nondimensional parameters on various profiles are shown graphically and explored in detail. It is found that the micropolar ferrofluid exhibits a higher energy distribution than that of a classical micropolar fluid. Compared to the classical micropolar liquid, local skin‐friction is more significant for the micropolar magnetite ferrofluid. In the presence of Newtonian heating, the thermal behavior of the micropolar nanofluid is remarkably better than that of the classical micropolar fluid.

Consequences of chemical reaction in temperature-dependent thermal conductivity fluid flow by a rotating disk with variable thickness

This paper examines the steady flow due to a rotating disk with variable thickness. Equations are modelled by considering the homogeneous–heterogeneous reactions and variable thermal conductivity. The modified Von Karman transformations are utilised to convert the governing partial differential equations into dimensionless nonlinear ordinary differential equations. Convergent series solutions are computed. The impact of relevant parameters on flow fields is computed and interpreted. It is predicted that an increase in disk thickness index decreases the axial velocity while increases the radial and tangential velocities. The Nusselt number enhances by increasing the thickness parameter of a disk.

HEAT GENERATING HYDROMAGNETIC THIRD GRADE FLUID FLOW THROUGH ISOTHERMAL PLATES WITH OHMIC HEATING EFFECT

This paper studies the combined effect of a transverse magnetic field and internal heat generation on the flow of an incompressible third-grade fluid through parallel plates of uniform temperature. Analytical solutions of the dimensionless nonlinear ordinary differential equations are obtained using regular perturbation method implemented on a symbolic package-Mathematica. Variations of flow pa- rameters on the velocity and temperature profiles are presented and discussed.

Finite Element Analysis of Radiative Mixed Convection Magneto‐Micropolar Flow in a Darcian Porous Medium with Variable Viscosity and Convective Surface Condition

A mathematical study is presented for the collective influence of the buoyancy parameter, convective boundary parameter and temperature dependent viscosity on the steady mixed convective laminar boundary flow of a radiative magneto‐micropolar fluid adjacent to a vertical porous stretching sheet embedded in a Darcian porous medium. The fluid viscosity is assumed to vary as an inverse linear function of temperature. Using appropriate transformations, the governing equations of the problem under consideration are transformed into a system of dimensionless nonlinear ordinary differential equations, which are then solved with the well‐tested, efficient finite element method. The results obtained are depicted graphically to illustrate the effect of the various important controlling parameters on velocity, microrotation, and temperature functions. The skin friction coefficient, wall couple stress, and the rate of heat transfer have also been computed and presented in tabular form. Comparison of the present numerical results with earlier published data has been performed and the results are found to be in good agreement, thus validating the accuracy of the present numerical code. The study finds applications in conducting polymer flows in filtration systems, trickle bed magnetohydrodynamics in chemical engineering, electro‐conductive materials processing, and so on.