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The document is a correction notice for a scientific article titled "Darcy–Brinkman Model for Ternary Dusty Nanofluid Flow across Stretching/Shrinking Surface with Suction/Injection." It includes corrections to figures, text, equations, and nomenclature used in the original publication. The corrections aim to clarify and improve the accuracy of the scientific content presented in the article. The corrections were approved by the Academic Editor, and the original publication has been updated accordingly. [Extracted from the article]

Hybrid nanofluids have caught the attention of scholars and investigators in the present technological period due to their improved thermophysical features and the desire to boost heat transfer rates compared to those of conventional fluids. The present paper is mainly concerned with heat transmission in cone-disk geometry in the presence of a magnetic field, activation energy, and non-uniform heat absorption/generation. In this work, the cone-disk (CD) apparatus is considered to have a rotating cone (RC) and a stretching disk, along with iron oxide and cobalt ferrite-based hybrid nanofluid. Appropriate similarity transformations are employed to change the physically modeled equations into ordinary differential equations (ODEs). Heat transfer rates at both surfaces are estimated by implementing a modified energy equation with non-uniform heat absorption/generation. The outcomes illustrated that the inclusion of such physical streamwise heat conduction variables in the energy equation has a significant impact on the well-known conclusions of heat transfer rates. To understand flow profile behavior, we have resorted to the RKF-45 method and the shooting method, which are illustrated using graphs. The findings provide conclusive evidence that wall stretching alters the flow, heat, and mass profile characteristics within the conical gap. The wall deformation caused by disk stretching was found to have a potential impact of modifying the centripetal/centrifugal flow characteristics of the disk, increasing the flow velocity and swirling angles. A rise in activation energy leads to an improved concentration field. [ABSTRACT FROM AUTHOR]

The main objective of this study is to explore the inventive conception of the magnetohydrodynamic flow of a hybrid nanofluid over-porous stretching/shrinking sheet with the effect of radiation and mass suction/injection. The hybrid nanofluid advances both the manufactured nanofluid of the current region and the base fluid. For the current investigation, hybrid nanofluids comprising two different kinds of nanoparticles, aluminium oxide and ferrofluid, contained in water as a base fluid, are considered. A collection of highly nonlinear partial differential equations is used to model the whole physical problem. These equations are then transformed into highly nonlinear ordinary differential equations using an appropriate similarity technique. The transformed differential equations are nonlinear, and thus it is difficult to analytically solve considering temperature increases. Then, the outcome is described in incomplete gamma function form. The considered physical parameters namely, magnetic field, Inverse Darcy number, velocity slip, suction/injection, temperature jump effects on velocity, temperature, skin friction and Nusselt number profiles are reviewed using plots. The results reveal that magnetic field, and Inverse Darcy number values increase as the momentum boundary layer decreases. Moreover, higher values of heat sources and thermal radiation enhance the thermal boundary layer. The present problem has various applications in manufacturing and technological devices such as cooling systems, condensers, microelectronics, digital cooling, car radiators, nuclear power stations, nano-drag shipments, automobile production, and tumour treatments. [ABSTRACT FROM AUTHOR]

The current study adopts a Bernoulli wavelet technique to explore the magnetohydrodynamic (MHD) boundary layer flow of a Casson fluid through a wedge. A nanofluid across a wedge has been investigated for its stable laminar MHD flow, heat, and mass transport characteristics. By choosing worthwhile non‐dimensional parameters, the governing boundary layer equation is reformed into a dimensionless Falkner–Skan equation. The consequent nonlinear equation is addressed via the Bernoulli wavelet method. For a range of different values of the physical parameters, the nature of the boundary layer flow is visually observed. Fluid velocity rises with rise in the values of Casson parameter, wedge parameter, and magnetic parameter. Local wall skin friction increases with increase in wedge parameter and magnetic parameter values. To ensure the accuracy of the findings, local wall skin friction is estimated and tested with other techniques that are already reported in the existing research. [ABSTRACT FROM AUTHOR]

Understanding of dusty fluids for different Brinkman numbers in porous media is limited. This study examines the Darcy–Brinkman model for two-dimensional magneto-hydrodynamic fluid flow across permeable stretching/shrinking surfaces with heat transfer. Water was considered as a conventional base fluid in which the copper (Cu), silver (Ag), and titanium dioxide ( T i O 2 ) nanoparticles were submerged in a preparation of a ternary dusty nanofluid. The governing nonlinear partial differential equations are converted to ordinary differential equations through suitable similarity conversions. Under radiation and mass transpiration, analytical solutions for stretching sheets/shrinking sheets are obtained. Several parameters are investigated, including the magnetic field, Darcy–Brinkman model, solution domain, and inverse Darcy number. The outcomes of the present article reveal that increasing the Brinkman number and inverse Darcy number decreases the velocity of the fluid and dusty phase. Increasing the magnetic field decreases the momentum of the boundary layer. Ternary dusty nanofluids have significantly improved the heat transmission process for manufacturing with applications in engineering, and biological and physical sciences. The findings of this study demonstrate that the ternary nanofluid phase's heat and mass transpiration performance is better than the dusty phase's performance. [ABSTRACT FROM AUTHOR]

The study of inclined magnetohydrodynamics (MHD) mixed convective incompressible flow of a fluid with hybrid nanoparticles containing a colloidal combination of nanofluids and base fluid is presented in the current research. Al2O3-Cu/H2O hybrid nanofluid is utilized in the current analysis to enhance the heat transfer analysis. The impact of radiation is also placed at energy equation. The main research methodology includes that the problem provided equations are first transformed into non-dimensional form, and then they are obtained in ordinary differential equations (ODEs) form. Then using the solutions of momentum and transfers equations to solve the given ODEs to get the root of the equation. The main purpose includes the resulting equations are then analytically resolved with the aid of suitable boundary conditions. The results can be discussed with various physical parameters viz., stretched/shrinked-Rayleigh number, stretching/shrinking parameter, Prandtl number, etc. Besides, skin friction and heat transfer coefficient can be examined with suitable similarity transformations. The main significance of the present work is to explain the mixed convective fluid flow on the basis of analytical method. Main findings at the end we found that the transverse and tangential velocities are more for more values of stretched/shrinked-Rayleigh number and mass transpiration for both suction and injection cases. This is the special method it includes stretched/shrinked-Rayleigh number, it contributes major role in this analysis. The purpose of finding the present work is to understand the analytical solution on the basis of mixed convective method. [ABSTRACT FROM AUTHOR]

The present study investigates the effect of mass transpiration on heat absorption/generation, thermal radiation and chemical reaction in the magnetohydrodynamics (MHD) Darcy–Forchheimer flow of a Newtonian fluid at the thermosolutal Marangoni boundary over a porous medium. The fluid region consists of H2O as the base fluid and fractions of TiO2–Ag nanoparticles. The mathematical approach given here employs the similarity transformation, in order to transform the leading partial differential equation (PDE) into a set of nonlinear ordinary differential equations (ODEs). The derived equations are solved analytically by using Cardon's method and the confluent hypergeometric function. The solutions are further graphically analyzed, taking into account parameters such as mass transpiration, chemical reaction coefficient, thermal radiation, Schmidt number, Marangoni number, and inverse Darcy number. According to our findings, adding TiO2–Ag nanoparticles into conventional fluids can greatly enhance heat transfer. In addition, the mixture of TiO2–Ag with H2O gives higher heat energy compared to the mixture of only TiO2 with H2O. [ABSTRACT FROM AUTHOR]

The present work employs the single-wall carbon nanotube (SWCNT) and multiwall carbon nanotube (MWCNT) models on axisymmetric Casson fluid flow over a permeable shrinking sheet in the presence of an inclined magnetic field and thermal radiation. By exploiting the similarity variable, the leading nonlinear partial differential equations (PDEs) are converted into dimensionless ordinary differential equations (ODEs). The derived equations are solved analytically, and a dual solution is obtained as a result of the shrinking sheet. The dual solutions for the associated model are found to be numerically stable once the stability analysis is conducted, and the upper branch solution is more stable compared to lower branch solutions. The impact of various physical parameters on velocity and temperature distribution is graphically depicted and discussed in detail. The single wall carbon nanotubes have been found to achieve higher temperatures compared to multiwall carbon nanotubes. According to our findings, adding carbon nanotubes volume fractions to convectional fluids can significantly improve thermal conductivity, and this can find applicability in real world applications such as lubricant technology, allowing for efficient heat dissipation in high-temperatures, enhancing the load-carrying capacity and wear resistance of the machinery. [ABSTRACT FROM AUTHOR]

In this research, we investigated the wall jet problem for a hybrid nanofluid with MHD, thermal radiation, velocity slip, and convective boundary conditions. After converting the governing partial differential equations into nonlinear ordinary ones, by using the proper similarity transformations, they were theoretically solved in view of the hypergeometric function. Comparing the present results with those in the literature, a very good agreement was obtained for fixed values of the included parameters. For some included parameters, regions of physical solutions were presented. By deducing the critical and terminated values of the suction/injection parameter, dual solutions for the reduced skin friction coefficient have been determined. It was found that the hybrid nanofluid temperature is efficiently heated by increasing values of the moving wall, absolute value of the Biot number, and thermal radiation parameters. However, it is cooled by enlarging values of the velocity slip and suction parameters. [ABSTRACT FROM AUTHOR]

This paper presents an analytical approach on capturing the effect of incompressible, non-Newtonian, viscous, Casson nanofluid flow past a stretching/shrinking surface, under the influence of heat radiation and mass transfer parameter. The governing nonlinear partial differential equations are first transformed into a series of associated nonlinear ordinary differential equations with aid of predictable transformation, while numerical computations follow. The implied nanofluid here is aluminum oxide ( A l 2 O 3 ). The analytical solution is exploited to reveal the accompanying non-dimensional boundary value problem and output results are employed to verify the method's reliability, where it is shown that they agree with current findings in the field. An incomplete gamma function is used to solve temperature equation analytically. We present various instances of the solution, depicting effects of the essential flow factor, the stretching/shrinking parameter, the mass transfer parameter, radiation parameter, and Prandtl number. [ABSTRACT FROM AUTHOR]

The present article describes the unsteady flow of a couple stress via a ternary hybrid nanofluid on a stretching surface with porous media. The nanofluid exhibits important properties for increasing heat transmission, and it is widely used in manufacturing and industrial applications. The basic similarity equations have been discovered to accommodate both stretching/shrinking surfaces. Ternary hybrid nanofluid is a colloidal combination of three types of microspheres: Al2O3, single wall carbon nanotubes, and graphene. For investigating spherical, cylindrical, and platelet nanoparticles, the governing partial differential equations are converted into ordinary differential equations, expending appropriate transformations. The analytical solution can then be carried out using various forms of nanoparticles, such as spherical, cylindrical, and platelet, to obtain the solution domain. Heat transfer is used in an electrically conducting fluid and also including thermal radiation, as calculated with the Biot number. The focus of the present effort is the evaluation of the flow of ternary hybrid nanofluid over a porous media via thermal radiation, with couple stress, using an analytical process. For various physical parameters, the velocity and temperature conditions are shown graphically. [ABSTRACT FROM AUTHOR]

This article focuses on analytic solutions for Newtonian fluid flow with slip and mass transpiration on a porous stretching sheet using the differential transform method and Pade approximants of an exceptionally nonlinear differential equation. The impacts of different parameters including mass transpiration (suction/injection), Navier's slip, and Darcy number parameters on the velocity of the liquid and tangential stress are discussed. A comprehensive comparison of our results with the previous one in the literature is made, and the results showed good agreement. An investigation is conducted of a combination of magnetic liquids that are conceivably pertinent for wound medicines, skin repair, and astute coatings for natural gadgets. It is found that there is a decrease in the velocity profiles and the boundary layer thickness for the case of suction. [ABSTRACT FROM AUTHOR]

This study investigates the behavior of carbon nanotubes (CNT) approaching an unsteady flow of a Newtonian fluid over a stagnation point on a stretching surface employing porous media. It flows when the liquid begins to move with the progression of time. Heat exchange with the environment has an impact on the flow. The implicitly limited component technique is used to solve the nondimensional partial differential equation with an associated boundary layer, which is an unstable system. Analytically, the solutions, as well as the required boundary conditions, are obtained. The effects of mass transpiration, volume fraction, and heat radiation on Newtonian fluid flow through porous media are explored. Single‐ and multi‐walled CNTs are used as well as water, as base fluids in the experiment. The impact of thermal radiation and heat source/sink is shown in the energy equation, which is solved under four different cases: uniform heat flux case, constant wall temperature case, general power‐law wall heat flux case, and general power‐law wall temperature case. By supplying distinct physical characteristics, a theoretical analysis of the existence and nonexistence of unique and dual solutions may be explored. These physical parameters determine the velocity distribution and temperature distribution. Prescribed surface temperature (PST) and prescribed wall heat flux (PHF) heat transfer solutions can be written using confluent hypergeometric equations, and generic power‐law PST and PHF situations can also be expressed using confluent hypergeometric equations. The graphical representations assist in the discussion of the current study's findings. [ABSTRACT FROM AUTHOR]

This examination is carried out on the two-dimensional magnetohydrodynamic problem for a steady incompressible flow over a porous medium. The C u − A l 2 O 3 nanoparticles are added to the water base fluid in order to improve thermal efficiency. The transverse magnetic field with strength B 0 is applied. The governing equations formed for the defined flow form a system of partial differential equations that are then converted to a system of ordinary differential equations upon applying the suitable similarity transformations. On analytically solving the obtained system, the solutions for velocity profile and temperature distribution are obtained in terms of exponential and Gamma functions, respectively. In addition, the physical parameter of interest, the local Nusselt number, is obtained. The results are analyzed through plotting graphs, and the effect of different parameters is analyzed. Furthermore, we observe that the suction/injection parameter enhances the axial velocity. The porous and radiation parameters enhance the temperature distribution, and the suction/injection parameter suppresses the temperature distribution. [ABSTRACT FROM AUTHOR]

An examination is carried out for three-dimensional incompressible viscoelastic fluid flow over a porous stretching/shrinking sheet with hybrid nanoparticles copper-alumina (C u − A l 2 O 3) in base fluid water (H 2 O) . The uniform magnetic field of strength B 0 is applied perpendicular to the fluid flow and considered the Navier slip. The mass transfer is considered with the chemical reaction rate. The governing equation for the defined flow forms the system of partial differential equations, which are then transformed into a system of ordinary differential equations via similarity transformations. The goal is to find the exact analytical solution, and the unique solution is determined by considering the boundary layer theory. Furthermore, the obtained system is solved to get the exact analytical solution for velocity and concentration fields in exponential form and in hypergeometric form, respectively. The exact solutions are obtained for velocity and temperature profiles, Skin friction, and Nusselt number. These findings are beneficial for future research in the present area. The parameters magnetic field, Inverse Darcy number, slip parameter, chemical reaction parameter, stretching/shrinking parameter, and viscoelastic parameter, influence the flow. The effect of these parameters on fluid velocity and concentration field will be analyzed through graphs. Skin friction and Nusselt number are also analyzed. This work found many applications in machining and manufacturing, solar energy, MHD flow meters and pumps, power generators, geothermal recovery, flow via filtering devices, chemical catalytic reactors, etc. [ABSTRACT FROM AUTHOR]

The current article explains the 3-D MHD fluid flow under the impact of a magnetic field with an inclined angle. The porous sheet is embedded in the flow of a fluid to yield the better results of the problem. The governing PDEs are mapped using various transformations to convert in the form of ODEs. The yielded ODEs momentum equation is examined analytically to derive the mass transpiration and then it is used in the energy equation and solved exactly by using various controlling parameters. In the case of multiple solutions, the closed-form exact solutions of highly non-linear differential equations of the flow are presented as viscoelastic fluid, which is classified as two classes, namely the second order liquid and Walters' liquid B fluid. The results can be obtained by using graphical arrangements. The current work is utilized in many real-life applications, such as automotive cooling systems, microelectronics, heat exchangers, and so on. At the end of the analysis, we concluded that velocity and mass transpiration was more for Chandrasekhar's number for both the stretching and shrinking case. [ABSTRACT FROM AUTHOR]

The current study experimentally investigates the heat transfer augmentation on the novel axial corrugated heat exchanger tube in which the spring tape is introduced. Air (Pr = 0.707) is used as a working fluid. In order to augment the thermohydraulic performance, a corrugated tube with inserts is offered. The experimental study is further extended by varying the important parameters like spring ratio (y = 1.5, 2.0, 2.5) and Reynolds number (Re = 10 000–52 000). The angular pitch between the two neighboring corrugations and the angle of the corrugation is kept constant through the experiments at β = 1200 and α = 600 respectively, while two different corrugations heights (h) are analyzed. While increasing the corrugation height and decreasing the spring ratio, the impact of the swirling effect improves the thermal performance of the system. The maximum thermal performance is obtained when the corrugation height is h = 0.2 and spring ratio y = 1.5. Eventually, correlations for predicting friction factor (f) and Nusselt number (Nu) are developed. [ABSTRACT FROM AUTHOR]

Detailed The analytical solution of the unsteady boundary layer flow due to impulsive porous stretching sheet is solved by means of an Adomian Decomposition Method (ADM). The ADM alongside Pade approximants are connected to tackle the nonlinear partial differential equation with different boundary conditions got in demonstrating the unsteady boundary layer flow. Apparently, this yields better accuracy when compared to results from other prime methods. The present results are in good match with the pioneering work of other authors. The impact of parameters like mass suction/injection parameter and Darcy number on the velocity profile were examined. The present results can be found very useful in a wide range of applications in extrusion process and related process in fluid dynamics and heat transfer problems. [ABSTRACT FROM AUTHOR]

The main objective of this article is to explore analytical solution for fourth order highly nonlinear differential equation. The stretching/shrinking is a non-linear (at least quadratic) function along the direction of stretching/shrinking. The stretching/ shrinking of a sheet with a velocity profiles are comparatively dependent on the distance from the velocity axes. The system of partial differential equations is transformed into the system of highly nonlinear ordinary differential equation via similarity transformation. Velocity profiles as a function of stretching/shrinking related parameters, Chandrasekhar number and viscoelastic parameter are discussed graphically. The analytical solutions of system of boundary layer equations emerging from the stretching/shrinking sheet problems were investigated and also the outcomes are in good agreement with the classical results on this topic. The outcomes have conceivable mechanical applications in fluid based frameworks including stretchable materials, in polymer extrusion process, metal spinning and other industrial processes. [ABSTRACT FROM AUTHOR]

The problem of mixed convective flow of a couple stress fluid in a vertical channel in the presence of first order chemical reaction and heat generation/absorption is studied. Two walls of the channel are considered with asymmetirc heating conditions along with flux boundary conditions on temperature and concentrations. The governing nonlinear differential equations are solved numerically. The effect of physical parameters like couple stress, heat generation/absorption, chemical reaction, thermal buoyancy and concentration buoyancy are evaluated numerically. Also heat transfer parameters such as Nusselt number, Sherwood number and skin friction factors are analyzed. [ABSTRACT FROM AUTHOR]

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