Lobatto IIIA Research Articles

Numerical simulation of nanofluid flow between two parallel disks using 3-stage Lobatto III-A formula

Abstract The development of nanofluid technology has become a key research area in physics, mathematics, engineering, and materials science. Nowadays, in many industrial applications, nanofluids are widely used to enhance thermophysical properties such as thermal diffusivity, thermal conductivity, and convective heat transfer. Scientists and engineers have established interests in the direction of flow problems developed via disk-shaped bodies. There are various logics to discuss flow phenomenon due to rotating bodies, but its applications include in thermal power engineering system, gas turbine rotors, air cleaning machines, aerodynamics, etc. Nowadays manufacturing industries have inaugurated to select liquid based on heat transfer properties. Therefore, this article focuses on studying the laminar incompressible nanofluid between two parallel disks. Mathematical formulations of the law of conservation of mass, momentum, and heat transfer are investigated numerically. By using suitable similarities, the flow equations are converted into nonlinear ordinary differential equations. The resulting equations were solved numerically via MATLAB software. The effects of physical parameters of interest, such as Reynolds number, magnetic factor, Brownian parameter, and thermophoresis parameter on normal velocity, streamwise velocity, temperature, and concentration profiles are computed and presented using the graphs. The results revealed that the energy profile significantly rises, and the profile moves closer to the upper disk by enhancing the Brownian motion and thermophoresis parameter. The dynamics behind this is that by increasing the Brownian motion, the boundary layer wideness increases which increases the temperature. Moreover, streamwise velocity increases for large values of Reynolds number. Besides, the thermophoresis profile increases for large values of the thermophoresis factor. It could be observed that shear stress at nonporous/porous disk is adjusted by selecting a suitable value of injection velocity at the porous disk. Also, normal velocity decreases by increasing the parameter M.

Stability aspect of magnetized hybrid nanofluid with suction and injection phenomenon: Modified thermal model

Nanoparticles with ultra-high thermal efficiency and stability have dynamic applications in numerous eras of thermal sciences, including energy production, heat transmission devices, cooling and heating systems, manufacturing applications, aircraft, and solar energy, among others. This study's primary objective is to investigate the mathematical modeling using a Tiwari and Das nanofluid model, taking into account the effects of magnetic, suction/injection, and thermal radiation, as well as the stability analysis of a hybrid nanofluid containing copper (Cu) and alumina (Al2O3) nanoparticles in a water-based liquid. Using similarity transformations, self-similarity solutions of the system of governing ordinary differential equations (ODEs) were obtained, and the resulting ODEs were simulated using implementations of the three-stage Lobatto IIIa technique. The numerical results indicate that the energy characteristics such as thermal conductivity increase rapidly when copper nanomaterials are used. It is also noticed that the combination of both nano-materials results in an excellent energy enhancement. For the solution validation, novel stability performances for the obtained simulations are determined.

Neural artificial networking for a nonlinear darcy–forchheimer chemically reactive flow: Levenberg marquardt analysis

In this paper, the 3D Darcy–Forchheimer flow model (TDDF-FM) is discussed by utilizing the artificial intelligence-based backpropagated neural networks with the algorithm of Levenberg–Marquardt (BNN-ALM) along with the convective conditions. The effects of Brownian diffusion and thermophoresis are also examined. The governing partial differential equations are transformed into a system of ODEs. Lobatto IIIA method is used to interpret the reference dataset of BNN-ALM for different scenarios of TDDF-FM by variation of porosity parameter, Forchheimer parameter, ratio parameter, Prandtl number, Biot number, Brownian motion parameter, Schmidt number, strength parameter of homogeneous reaction and thermophoresis parameter. The solutions are computed for the designed TDDF-FM by testing, training and validation processes of BNN-ALM. The performance analysis of the designed BNN-ALM is validated by histogram analysis, regression studies and the results of mean square error (MSE). Graphs are presented for temperature profile, concentration profile and concentration rate profile. The increasing values of porosity parameter, Forchheimer parameter and Biot number enhance the temperature distribution, whereas the temperature profile shows a decreasing behavior with the increase in Prandtl number and ratio parameter. The concentration profile increases with the increase in porosity parameter, Forchheimer parameter and thermophoresis parameter and it decreases with the increasing behavior of Brownian motion parameter, ratio parameter and Schmidt number. The concentration rate profile decreases with the increase in porosity parameter, Forchheimer parameter and strength parameter of homogeneous reaction and it increases with the ratio parameter. The best performance values for different cases of scenarios are 1.42E−08, 1.70E−08, 1.78E−08, 1.61E−08, 2.43E−08, 1.05E−08, 1.30E−08, 1.59E−08 and 2.31E−08, attained at 215, 186, 178, 99, 125, 213, 187, 190 and 191 epochs. The values for the Gradient factor are 9.93E−08, 9.86E−08, 9.99E−08, 2.97E−07, 1.23E−07, 9.94E−08, 9.94E−08, 9.87E−08 and 9.94E−08, where the value of regression is [Formula: see text] for the training, testing and validation data.

Hall Current and Viscous Dissipation Impact on MHD Mixed Convection Flow towards a Porous Exponentially Surface with its Engineering Applications

Power generators, Hall accelerators, and flight MHD all require high levels of Hall current. The influence of Hall current and viscous dissipation on time-independent hydro-magnetic mixed convective radiative flow across a porous heated surface has thus been investigated using numerical computing and mathematical modeling in the current study. The fluid is electrically conducted and varies exponentially. It is assumed that the wall temperature and elongation rate will vary with specific exponential shapes. A solid uniform magnetic field BSUB0/SUB is employed normally to the surface. The mathematical model of PDEs for incompressible flow is transformed into ODE by applying a numerical technique based on a finite-difference structure which includes a three-stage Lobatto IIIa scheme with the help of MATLAB. The obtained solution depends on the convergence constraints involving the radiation parameter R, magnetic parameter M, porosity parameter Ω, Hall parameter m, buoyancy parameter ε, temperature distribution parameter a, Eckert number ESUBc/SUB, Prandtl number PSUBr/SUB, and convective term bh. Graphs of the velocity and temperature profiles are explained via pertinent parameters. Skin friction factor, and Nusselt number are also evaluated and presented graphically and in tabular form. Results clarify that temperature profile reduces by increasing values of temperature distribution parameter whereas opposite behavior is noted for positive values of the buoyancy parameter.

Numerical investigation on the thermal transportation of MHD Cu/Al 2 O 3-H 2 O Casson-hybrid-nanofluid flow across an exponentially stretching cylinder incorporating heat source

This paper investigates the stagnation point flow and heat transfer of MHD boundary layer Casson hybrid nanofluid through a porous space across an exponentially stretched cylinder. The heat source impact is also considered in the model. A 3-stage Lobatto IIIa formula, Bvp4c, is operated to obtain the numerical findings. The influences of dimensionless parameters, namely Prandtl number, Reynolds number, heat source parameter, and Casson fluid parameter on Skin-friction coefficient, velocity, thermal profile, and Nusselt number, are portrayed graphically and depicted through tables. The significant findings indicate that the hybrid Casson nanofluids have a higher thermal conductivity when compared to the Casson nanofluids. As a result, the Casson hybrid nanofluid fluid has a significant impact on increasing heat processes. It has been investigated that the coefficient of the absolute Skin friction for Casson hybrid nanofluid is raised up to 29% compared to Casson nanofluid. The effective heat transfer rate of Casson hybrid nanofluid is improved over 9% compared to Casson nanofluid. As the heat source and Casson fluid parameters are incremented, the numeric values of the Nusselt number reduce.

MHD Casson Nanofluid in Darcy-Forchheimer Porous Medium in the Presence of Heat Source and Arrhenious Activation Energy: Applications of Neural Networks

ABSTRACT The present study investigates the significance of activation energy during chemical reactions, thermal radiations and temperature gradient of 3-D steady Magnetohydrodynamic Casson nanofluid flow (MHDCNFM) in Darcy−Forchheimer medium over an oscillating disk by using an intelligent numerical-based computing solver through the Levenberg-Marquardt backpropagation neural network scheme (LMBNNS). The adjusted Buongiorno model is utilized to develop the system of partial differential equations (PDEs) for MHDCNFM, and further, by invoking Von Karman similarity transformation, the system of governing PDEs is turned out into ordinary differential equations (ODEs). First, a data set for the magnetohydrodynamic Casson nanofluid flow model (MHDNNFM) is generated for a range of sundry parameters for the radial velocity, tangential velocity, heat distribution and concentration profile by the variations of Casson parameter, magnetic parameter, Brownian motion parameter, thermophoresis parameter, Forchheimer parameter, activation energy parameter, chemical reaction parameter, stretching parameter, porosity parameter and Schmidt number through the Lobatto IIIA technique, and after this, by applying an intelligent computing algorithm through nftool training, testing and validation steps are taken into account to find out the approximate solution for various cases. The designed solver LMBNNS is used to solve the MHDCNFM through mean squared error (MSE), regression, gradient analysis and histogram studies.

Combined effects of Coriolis force and nanoparticle properties on the dynamics of gold–water nanofluid across nonuniform surface

AbstractInvention of nanofluid has proved revolutionary in the enhancement of fluid thermal and electrical conductivity. Industrial applications of gold–water nanofluids over rotating surface include, but not limited to, heat transfer fluids, as a solar absorber, in medieval medicine for the diagnosis of syphilis, and so forth. Gold–water nanofluid is useful in colorant of glass and silk, nonlinear optics, and molecular recognition. Studies have been carried out mostly across stationary or stretching flat surfaces. This paper studies the flow of gold–water nanofluids over the rotating upper horizontal surface of a paraboloid of revolution. The relevant body forces are added to the Navier–Stokes equations to formulate appropriate equations for the flow of gold–water nanofluids over a surface with nonuniform thickness under the action of Coriolis force. Appropriate Blasius similarity transformation is used to nondimensionalize the governing equations and thereby reducing the nonlinear partial differential equations to nonlinear ordinary differential equations. The numerical method used is the Runge–Kutta–Gills method with shooting technique and the three‐stage Lobatto IIIa collocation method, and the results are illustrated graphically. Coriolis force is found to have reduced the coefficient of skin friction in the x‐direction but enhances the coefficient of skin friction in the z‐direction. The haphazard motion of the nanoparticles and the nanoparticle volume fraction are found to enhance the skin friction coefficient in the x‐ and y‐directions.

Impact of thermal energy on MHD Casson fluid through a Forchheimer porous medium with inclined non-linear surface: A soft computing approach

In this work, the influence of thermal energy in term of heat source, thermal radiation and chemical reaction on magneto hydrodynamic Casson fluid flow model (MHD-CFM) over a nonlinear slanted extending surface with slip velocity in a Forchheimer permeable medium is numerically studied using the Levenberg Marquardt methodology with backpropagated learning mechanism. It is valuable to evaluate the flow of Cason fluids based on materials (such as drilling muds, clay coatings, various suspensions and certain lubricating oils, polymeric melts, and a wide range of colloids) in the occurrence of heat transfer. Using efficient data, PDEs of (MHD-CFM) were converted to ordinary differential equations. These obtained non-linear ODEs are then rectified using the computational power of the Lobatto IIIA approach to obtain a dataset of Levenberg Marquardt algorithm based trained neural networks (LMA-TNN) for six scenarios of this presented model, which were graphically represented using nftool to obtain regression, efficiency, fit curve, error bars, and trained state analysis. The velocity, temperature, and concentration profiles were computed, and the findings were presented. Additionally, the skin friction coefficient, Nusselt number, & local Sherwood number are explored. The graphs show that when values of radiation parameter and the Forchheimer porous media parameter increase, the temperature of the plate drops. In the existence of a chemical process and a high Schmidt number, the concentration drops. The accuracy achieved in terms of relative error demonstrates the validity and significance of the solution process.

Further analysis of double-diffusive flow of nanofluid through a porous medium situated on an inclined plane: AI-based Levenberg–Marquardt scheme with backpropagated neural network

The present article exploits a novel application of AI-based Levenberg–Marquardt scheme with backpropagated neural network (LMS–BPNN) to analyze the double-diffusive free convection nanofluid flow model (DDFC-NFM) over an inclined plate in the existence of Brownian motion and thermophoresis properties embedded in a porous medium. The governing PDEs representing DDFC-NFM are transformed into system of nonlinear ODEs by applying suitable transformation. The reference data set is generated from Lobatto III-A numerical solver by variation of magnetic field parameter (M), thermal Grashof number (Gr), angle of inclination (α), Brownian motion parameter (Nb), Dufour-solutal Lewis number (Ld), modified Dufour parameter (Nd) and thermophoresis parameter (Nt) for all scenarios of the designed LMS–BPNN. The approximate solution and its comparison with standard solution are analyzed by execution of training, testing and validation procedure of the designed LMS–BPNN. The effectiveness and reliable performance of LMS–BPNN are endorsed with MSE-based fitness curve, regression analysis, error histogram analysis and correlation index. Results reveal that velocity increases with the rise in Gr, whereas reverse trend has been noticed for angle of inclination and magnetic field parameter and the temperature profile increases with the increase in Nb, Nd and Nt. The solutal concentration profile increases with the increment in Ld, while an increase in Nd causes a decrease in it. When Nt increases, the enhancement in the nanoparticle volume frictions occurs, but an opposite behavior is depicted for Brownian motion parameter.

Thermal entrance problem for blood flow inside an axisymmetric tube: The classical Graetz problem extended for Quemada's bio-rheological fluid with axial conduction.

The heat-conducting nature of blood is critical in the human circulatory system and features also in important thermal regulation and blood processing systems in biomedicine. Motivated by these applications, in the present investigation, the classical Graetz problem in heat transfer is extended to the case of a bio-rheological fluid model. The Quemada bio-rheological fluid model is selected since it has been shown to be accurate in mimicking physiological flows (blood) at different shear rates and hematocrits. The steady two-dimensional energy equation without viscous dissipation in stationary regime is tackled via a separation of variables approach for the isothermal wall temperature case. Following the introduction of transformation variables, the ensuing dimensionless boundary value problem is solved numerically via MATLAB based algorithm known as bvp5c (a finite difference code that implements the four-stage Lobatto IIIa collocation formula). Numerical validation is also presented against two analytical approaches namely, series solutions and Kummer function techniques. Axial conduction in terms of Péclet number is also considered. Typical values of Reynolds number and Prandtl number are used to categorize the vascular regions. The graphical representation of mean temperature, temperature gradient, and Nusselt numbers along with detail discussions are presented for the effects of Quemada non-Newtonian parameters and Péclet number. The current analysis may also have potential applications for the development of microfluidic and biofluidic devices particularly which are used in the diagnosis of diseases in addition to blood oxygenation technologies.

Zinc oxide–silver/water hybrid nanofluid flow toward an off‐centered rotating disk using temperature‐dependent experimental‐based thermal conductivity

AbstractHerein, the off‐centered stagnation flow and heat transfer of zinc oxide–silver/water hybrid nanofluid over a rotating disk according to the mass‐based algorithm is studied. It is assumed that the nanoparticles have a spherical shape. Also, the velocity slip between the base fluid and nanoparticles is negligible. The Prandtl number is kept constant at 6.2. In addition, it has been used an experimental relation for effective thermal conductivity which is a function of volume fraction and temperature. The governing partial differential equations are converted to dimensionless ordinary differential equation (ODE)s by the similarity transformation method. The simplified ODEs are solved numerically by the bvp4c function from MATLAB which is an efficient and reliable code according to the three‐stage Lobatto IIIa formula. The influence of rotational parameters and both nanoparticles masses on the profiles and quantities of engineering interest are presented and discussed in detail. It is shown that the flow becomes complicated when there is a distance between the flow axis and the disk axis. Under determined conditions for a hybrid nanofluid with 30‐g mass for both nanoparticles and 100‐g mass for pure water, adding 30 g of the second nanoparticle's mass into the base fluid leads to enhance all hydrodynamic quantities of engineering interest by about 4.3%, while dispersing 30 g of the first nanoparticle's mass inside water results in decreasing the similarity temperature gradient at the surface about 3.6%. Also, when the disk rotates faster, the maximum radial velocity near the disk, s′(0) and f″(0) increases.

A numerical study of rotating Bödewadt flow of micropolar fluid over porous disk

Micropolar fluids are microstructure fluids that represent liquids consisting of rigid, randomly oriented particles suspended in a viscous medium, ignoring the deformation of the fluid particles. The present paper treats steady micropolar fluid flow over a permeable stationary disk numerically. It is supposed that the three-dimensional Bödewadt flow with stabilized radial pressure gradient rotates about [Formula: see text] having consistent angular velocity [Formula: see text]. The developed flow model contains a suction and micro-rotational parameter. Governing equations representing steadily rotating micropolar fluid, are presented before being transformed into non-dimensional forms. The numerical solutions are verified with the bvp4c package based on the Lobatto IIIa formula. This collocation technique uses a mesh of points to divide the integration interval into subintervals. A numerical solution is determined by solving a system of differential equations resulting from the boundary conditions. Finally, the collocation conditions are imposed on all the subintervals. In this paper, the contribution of micro-rotation on the stationary disk is carefully scrutinized. Furthermore, the impact of varying choices of parameters on radial, axial and tangential velocity profiles are evaluated numerically. The influence of these parameters, following Bödewadt boundary conditions, on temperature profile, are also analyzed.

The role of enthalpy and activation energy on thermophysical properties of third grade fluid near a magnetized radioactive irregular wavy surface

The present article communicates mathematical and theoretical study of three dimensional mixed convection flow of an electrical conducting third grade fluid flow near a non-uniform irregular wavy surface. The flow field is incorporated with an applied magnetic field. Heat and mass transfer characteristics are scrutinized with combined impact of enthalpy change along with energy activation and non-linear thermal radiation. The prime structure of differential equations is achieved in momentum, energy, magnetic and concentration forms. We defined some new dimensionless functions which transformed non-linear PDEs into dimensionless ODEs. Lobatto(iiiA) formula based technique Bvp4c is used for graphical and numerical solution of governing equations. Features of different dimensionless parameters are studied for velocity, concentration, magnetic and energy distributions by using graphical approach. Skin friction coefficients, rate of heat and mass fluxes are calculated by changing the values of some dimensionless parameters. Comparative study of Nusselt number with previous published results are demonstrated to investigate the accuracy of current problem. Final outcome justifies that better agreement is achieved with existing published work.

Numerical investigation of thin-film flow over a rotating disk subject to the heat source and nonlinear radiation: Lobatto IIIA approach

The analysis of thin liquid film over a horizontal rotating disk has attained fabulous attention in the field of the coating industry. Therefore, the present study aims to communicate the nature of the thin-film flow of Maxwell magnetized nanomaterials over the surface of the rotating disk. The important aspects of nano liquid known as the Brownian motion and thermophoresis are considered. This liquid film is explored under the influence of a heat source/sink. Dimensionless variables have been implemented to acquire the nonlinear system of ODEs. The obtained system has been tackled numerically via Lobatto IIIA technique for the variation of unsteadiness parameter ranging from 0.3 to 0.45, thin-film thickness parameter ranging 2.20–2.35, rotation parameter ranging 0.7–1.0, magnetic parameter ranging 4.0–7.0, Prandtl number ranging 1.3–1.6, and radiation parameter ranging 0.5–0.8. The outcomes for the flow problem have been estimated against various embedding variables. In addition, tables are made for sundry cases of different scenarios, mesh points, maximum residual errors, Nusselt and Sherwood numbers. Analysis elaborates that an increase in rotation parameter causes a reduction in the azimuthal component of velocity while the temperature is enhancing the function of the radiation parameter.

Scrutinization of thermodynamic second law for the steady flow of couple stress nanofluid in an inclined microchannel by varying thermal conductivity

AbstractThis study comprises the flow of Ti6Al4N nanoparticle alloy allowed to flow within the microchannel kept at a certain angle. Engine oil is the base fluid in which nanodimensional particles are immersed. The flow is caused by the influence of linear and exponential heat sources. The microchannel is a permeable media with the suction/injection at the microchannel extremes. During the flow, the impact of the microstructure of these particles is concerned, thus reporting the rotational interaction amidst the particles. The obtained nonlinear equations are solved using the effective numerical method, namely, finite difference code improved by using the Lobatto IIIA formula and the findings are discussed using the graphs. The results attained claim that linear and exponential space‐dependent heat source parameter has augmented the thermal profile. Also, consideration of the couple stress of the fluid is known to retard the flow.

Numerical investigation of a convective hybrid nanofluids around a rotating sheet

Hybrid nanofluids are formulated with various kinds of base fluids. They are designed to provide good heat transfer performance. They can achieve this by dispersing various kinds of nanoparticles in the base materials. This new technology of formulating hybrid nanofluids has a wide range of applications in various industries such as solar energy, medical equipment, and aerospace. Keeping these applications in view, this study provides an insight into the effects of convective heat transport on a Hybrid nanofluid, across a rotating sheet with a variable heat source. In this investigation, the governing boundary layer partial differential equations were modified into the ordinary differential equations, by using the proper similarity transformations. Later, they were solved numerically, with the support of the Lobatto IIIA technique in MATLAB. The influence of the Richardson number on flow parameters was studied, and it was discovered that increasing Ri increases the velocity while decreasing temperature and concentration profiles. The impact of various other flow parameters on the flow fields and also on the behavior of Nusselt number, coefficient skin friction, and Sherwood number were studied and represented graphically. The outcomes were found to be in excellent accord when compared with quoted studies.

Off-centered stagnation point flow of an experimental-based hybrid nanofluid impinging to a spinning disk with low to high non-alignments

PurposeThe purpose of this study is to model and solve numerically the three-dimensional off-centered stagnation point flow and heat transfer of magnesium oxide–silver/water hybrid nanofluid impinging to a spinning disk.Design/methodology/approachThe applied effective thermophysical properties of hybrid nanofluid including thermal conductivity and dynamics viscosity are according to the reported experimental relations that would be expanded by a mass-based algorithm. The single phase formulations coupled with experimental-based hybrid nanofluid model is implemented to derive the governing partial differential equations which are then transferred to a set of dimensionless ordinary differential equations (ODEs) with the use of the similarity transformation method. Afterward, the reduced ODEs are solved numerically by bvp4c function from MATLAB that is a trustworthy and efficient code according to three-stage Lobatto IIIa formula.FindingsThe effect of spinning parameter and nanoparticles masses (mMgO, mAg) on the hydrodynamics and thermal boundary layers behavior and also the quantities of engineering interest are presented in tabular and graphical forms. The recent work demonstrates that the analysis of flow and heat transfer becomes more complicated when there is a non-alignment between the impinging flow and the disk axes. From computational results demonstrate that, the radial and azimuthal velocities are, respectively, the increasing and decreasing functions of the disk spinning parameter. Further, for the greater values of the spinning parameter, an overshoot of the radial velocity owing to the centrifugal forces of the spinning disk is observed. Besides, the quantities of engineering interest gently enhance with first and second nanoparticle masses, while comparing their absolute values illustrates the fact that the effect of second nanoparticle mass (mAg) is greater. Further, it is inferred that the second nanoparticle’s mass enhancement results in the amplification of the heat transfer; although, the high skin friction and the relevant shear stress should be controlled.Originality/valueThe combination of experimental thermophysical properties with theoretical modeling of the problem can be the novelty of the present work. It is evident that the experimental relations of effective thermophysical properties can be trustable and flexible in the theoretical/mathematical modeling of hybrid nanofluids flows. Besides, to the best of the authors’ knowledge, no one has ever attempted to study the present problem through a mass-based model for hybrid nanofluid.

Numerical investigation for melting heat transport of nanofluids due to stretching surface with Cattaneo-Christov thermal model

The goal of this work is to present the numerical results of MHD flow of nanofluid with Cattaneo-Christov heat flux model and thermally radiation by a stretching surface with melting boundary conditions. Nanofluids are a new kind of nanofluid but their commercial application is still in the works. It is also expected that hybrid nanomaterials should be used in more effective related technologies. These nanofluids would be used in cancer treatment, electronic systems, packaging materials and scientific research. It was established that the physical illustration exists for the examined boundary value problem. For a stretching surface, it was realized that a nanofluid with γ-Al2O3 nanoparticles mechanism is a cooler on growing some of the examined flow parameters. Blood, water and C2H6O2 are used as base fluids and nanoparticles γ-Al2O3 were measured as a nanofluid. By employing appropriate similarity transformations, the main governing PDEs are composed to a set of nonlinear ODEs which are then numerically solved in the computational tool MATLAB by a three-stage Lobatto III-A formula. The velocity is declined for the greater estimations of the permeability parameter. The temperature is boomed up for the greater estimations of Biot number and radiation parameter. Moreover, the flow ranges for each flow parameter, such as 0.1⩽λ⩽1.0, 0.1⩽ϕ⩽0.3, 0.1⩽K⩽1.0, -0.5⩽α⩽0.5, 0.1⩽Rd⩽1.0, 0.4⩽Me⩽1.2, 1.0⩽Ω⩽1.2, 0.4⩽Bi⩽1.0 and 1.5⩽θw⩽1.7 are scrutinized. In this research work, a numerical scheme was used to explore the aspects of γ-Al2O3 nanomaterials across a starching surface. The authenticity of the aforementioned research work guarantees that the new study was never identified before and is wholly new; thus, in the case of results authenticity, a Lobatto III-A formula is designed to simulate in mathematical software MATLAB by built-in routine bvp4c (shooting) method and it is observed to be in good agreement with the previous works of literature.

A numerical study of hybrid nanofluids near an irregular 3D surface having slips and exothermic effects

The current article presents a comprehensive investigation of MHD viscous flow of hybrid-nanofluids (Al2O3 − Ag/ water and (Al2O3 − Cu/) over a horizontally irregular 3D plane with non-uniform thickness combined with slip effects. The foremost aim of conducting this study is to enhance thermal transportation. Based on the following novelties, the subject study holds tremendous significance: i. A comparative analysis of two hybrid nanofluids with hybrid-base fluid together with slip effects ii. An exclusive study where the Tiwari and Das nanofluid model is employed combined with Fourier's heat flux model iii. Development of finite-difference code which implements the three-stage Lobatto IIIa approach for the designed problem. We have used suitable scaling transformations to convert the three-dimensional conservation equations of mass, momentum, and energy into a dimensionless system of boundary layer equations. The numerical solution of the coupled non-linear boundary layer problem is determined using the built-in finite-difference code designed to employ the three-stage Lobatto IIIa formula. A comprehensive assessment is carried out in how the velocity components, temperature, skin friction, and heat transfer rate are affected by the physical parameters of interest. The same is presented through graphs and in tabular form to offer a pictorial overview. The fluctuating trends of skin friction coefficients (x, y-directions) and Nusselt number are investigated to explore the physical landscape of the current study. The findings of this study offer a noticeable contrast to their existing counterparts.

Theoretical Analysis of Activation Energy Effect on Prandtl–Eyring Nanoliquid Flow Subject to Melting Condition

Abstract This study models the convective flow of Prandtl–Eyring nanomaterials driven by a stretched surface. The model incorporates the significant aspects of activation energy, Joule heating and chemical reaction. The thermal impulses of particles with melting condition is addressed. The system of equations is an ordinary differential equation (ODE) system and is tackled numerically by utilizing the Lobatto IIIA computational solver. The physical importance of flow controlling variables to the temperature, velocity and concentration is analyzed using graphical illustrations. The skin friction coefficient and Nusselt number are examined. The results of several scenarios, mesh-point utilization, the number of ODEs and boundary conditions evaluation are provided via tables.