Dual Solutions Research Articles

Optimization models for the installation planning of offshore wind farms

Challenges posed by global warming motivate the increasing interest in green energy sources, such as wind energy. To make wind energy attractive from an economic viewpoint, the decision-making problems faced in designing, installing, and maintaining wind farms have to be optimized. In this paper, we focus on the problem of optimally planning the installation process of offshore wind farms, as faced by Vattenfall, a leading European energy company. We first formulate the deterministic version of this Installation Planning of Offshore Wind Farms (IPOWF) problem as a Mixed Integer Linear Programming (MILP) model. From this model, we then derive other MILPs that can provide lower and upper bounds to the optimal installation cost. To cope with weather uncertainty, we also provide a light-robust MILP. These models are tested on real-life instances corresponding to two wind farms that Vattenfall has recently built in Denmark and Germany. Computational results show that tight primal and dual solutions can be computed for the deterministic case with the proposed models and that the light-robust model yields solutions that are robust to weather uncertainty.

Geopolymerisation and micro-pore assessment of soft soil used as flexible pavement material in road construction

In this study, inexpensive dual precursors namely rice husk ash (RHA) and quarry dust (QD) were mixed with an alkali solution (NaOH solution) in different mix proportions. Next, the effect of the dual precursors-alkali solutions (geopolymers) on the strength parameters of the different soil-geopolymer mixes subjected to various curing days were assessed. The result obtained from the assessment showed that the dual precursors-alkali solutions improved the strength properties of all the different soil-geopolymer mixes. The optimum soil-geopolymer mix proportion that resulted in the best improvement of the expansive subgrade soil properties was observed at a combination of 15%RHA+15%QD+3M NaOH. Furthermore, micro-pore analyses (porosity and pore size distribution), which were implemented with scanning electron micrographs and an image segmentation technique, local adaptive thresholding algorithm, showed progressive reduction in the porosity of the natural expansive subgrade soil and that of the optimal soil-geopolymer mix proportion cured for 7 and 28 days.

Analysis of Solution Stability in Al2O3-Cu/H2O over a Stretching/Shrinking Wedge

In this study, the stability analysis is conducted in order to observe the reliability of the generated solutions. The influence of several parameters is taken into consideration including heat source/sink parameter, suction, magnetic field, and heat transfer. The well-known heat transfer fluid that is Al2O3-Cu/H2O hybrid nanofluid past a stretching/shrinking wedge is implied. The similarity equations are achieved after implying suitable similarity transformation which then needed to be solved numerically using bvp4c, embedded in MATLAB software. Dual solutions are observed along the investigation within specified values of involved parameters. It is important to note that verification results show excellent concordance with pre-existing reports.

Velocity slip effect on magnetized casson nanofluid over shrinking/stretching cylinder: duality and stability analysis

The aim of current research is to assess heat transfer characteristics over a cylinder undergoing stretching or shrinking using Cason Nanofluid. Additionally, dual solutions stability has been analyzed. The governing equations for the two-phase model consist of partial differential equations (PDEs), which are subsequently transformed into ordinary differential equations (ODEs) using the similarity transformation. The MATLAB software’s bvp4c solver is then used to numerically solve the transformed ODEs. The graph also shows the results of skin friction coefficient, heat transfer coefficient, and the Sherwood number. Furthermore, the validity of the current data is confirmed and shown to be in good agreement by comparing them to the existing literature. The variation of Casson nano fluid velocity, which is the variation of reduced skin friction coefficient increases as the Casson fluid parameter values increase. The critical values against and are respectively. The temperature variation indicates that the local Nusselt number variation decreases as the curvature parameter values increase. The first solution is characterized by positive values of the smallest eigenvalues suggesting stability. Conversely, the second solution is associated with negative values of the smallest eigenvalues indicating instability. To identify the stable and physically feasible solution, a stability analysis is performed. Graphical analysis is used to examine the influence of relevant physical parameters on the profiles of temperature, concentration, velocity and momentum.

Flow and heat transfer in annuli owing to inner shrinking and outer stationary cylinder

The scope of the present work is to investigate the motion of the fluid flow and heat exchange within two concentric cylinders. It is assumed that the inner cylinder is moving (shrunk) and the outer cylinder is stationary. It derives the physical phenomenon, which can find industrial practical applications. The gap between the cylinders, curvature of cylinders, and Prandtl number are three physical parameters controlling the hydro-thermal flow motion. Asymptotic solutions are available when the gap size is moderately small. These and other solutions based on the numerical simulations clearly indicate that there are critical gap sizes beyond which either no solutions or dual solutions are possible. The common salient characteristic of solutions is that the axial fluid dragged into the cylinder as a result of surface shrinkage is thrown radially out through the outer cylinder. The non-existence of solutions after identifying critical gap sizes explains why the shrinking sheet and shrinking cylinder problems with an unbounded flow zone could not have been treated in the literature so far. This is the prime reason that with infinite gap size, other controlling mechanisms like surface transpiration, unsteadiness, velocity slip or magnetic field need to be included in the mathematical model to get a solvable problem under the surface shrinking process often studied among researchers. One of the solutions called branch I solution, has rational profiles with small pertinent physical quantities like skin friction, vertical/radial thrown velocity, and heat transfer rate. However, from a stability analysis, it can be shown that the branch II solution, possesses comparatively larger such quantities, making them questionable to appear in real physics with unstable eigenvalues.

Exploring Dual Solutions and Characterisation of Viscous Dissipation Effects on MHD Flow along a Stretching Sheet with Variable Thickness: A Computational Approach

This study explores forced convection in a viscous, electrically conducting fluid over a stretching sheet with variable thickness. It examines the effects of a varying free stream, magnetic field, and viscous dissipation on flow and heat transfer. The governing equations for mass, momentum, and energy are converted into ordinary differential equations using similarity transformations and solved numerically with the finite difference method. The results reveal dual solutions within specific parameter ranges: one stable and aligned with natural phenomena, the other unstable with higher skin friction and velocity. Viscous dissipation significantly alters velocity and temperature profiles. These insights are valuable for optimizing flow conditions in applications such as metal sheet cooling and polymer processing.

The mixed convection thermally radiated hybrid nanofluid flow through an inclined permeable shrinking plate with slip condition and inclined magnetic effect

In this paper, we consider the slip boundary condition to analyze the radiative inclined magnetohydrodynamics mixed convection hybrid nanofluid (Al2O3–Cu/H2O) flow across an inclined shrinking permeable plate. The following model’s governing PDEs are transformed into nonlinear ODEs with the help of similarity transformations. To achieve the numerical solution of ODEs, the boundary-value problem of 4th order accuracy (bvp4c) is applied. With appropriate values for the copper volume fraction, magnetic parameter, slip parameter, and radiation parameter, the numerical results are described and graphically represented in velocity and temperature profiles. Due to the shrinking plate, the dual solutions are to be observed. For higher values of copper volume fraction, slip parameter, and magnetic parameter the first solution in the velocity profile increases and the second solution in the velocity profile decreases by increasing the value of copper volume fraction and magnetic parameter. By increasing the magnetic parameter, and slip parameter first solution of the temperature profile decreases while both the solution increases by increasing the copper volume fraction and radiation parameter.

Dual solutions of water-based micropolar nanofluid flow over a shrinking sheet with thermal transmission: Stability analysis

Investigation of the nature of dual solutions of the water-based micropolar nanofluid-flow with thermal transmission due to a contracting surface has been done in the work. The flow is characterized by its shrinking velocity and imposed magnetic field. Also, this work is one of the contributions that illustrate the microrotation and microinertia descriptions of nanofluids. The effects of metallic nanoparticles Cu and CuO have been discussed throughout this study. A uniform magnetic field has been applied in the normal direction of the flow. A set of basic equations that supports the present problem are derived from the principle of conservation laws and have been modernized into a set of solvable forms by employing suitable similarity variables. The MATLAB built-in bvp4c solver scheme is engineered to solve this problem. In order to tackle boundary value problems that are highly non-linear, this numerical method largely relies on collocation and finite difference techniques. From this study, we have perceived that the speed of the motion of CuO-water nanofluid in both cases (the first and second solutions) is less than CuO-water nanofluid. The material parameter plays an important role by enhancing the heat transfer rate of the fluid at the surface of the sheet in both time-dependent and time-independent cases. From the stability analysis, the first solution has been found as the stable and physically attainable solution. Additionally, the material parameter aids in reducing the effects of couple stress and shear stress on the fluid in both situations near the surface.

Aspects of Dual Simulation for Modified Thermal Flux Advances in Non‐Newtonian Reiner–Philippoff Fluid Flow past a Shrinking Plate Embedded in a Porous Medium

This study investigates the effects of non‐Newtonian Reiner–Philippoff fluids on porous media, particularly in the context of steady radiative mixed convection flow and heat transfer near a shrinking plate surface in the presence of magnetohydrodynamics (MHD). The mathematical model is constructed using PDEs and transformed into ODEs via similarity transformations, with numerical solutions obtained using MATLAB’s bvp4c function. The results demonstrate that boundary layer separation occurs slowest for dilatant fluids and fastest for pseudoplastic fluids, with Newtonian fluids exhibiting moderate separation rates. Thermal radiation and media porosity parameters are found to reduce heat transfer by approximately 0.95% and 0.02%, respectively, while accelerating boundary layer separation. Conversely, magnetic effects and suction parameters increase heat transfer by about 0.08% and 4.25%, respectively, enhancing both fluid velocity and temperature. The mixed convection parameter indicates the possibility of dual solutions, with the opposing flow favoring this phenomenon more than the assisting flow. The time‐based stability analysis reveals that the first solution is stable, whereas the second solution is unstable. These findings provide significant insights into the behavior and control of Reiner–Philippoff fluids in practical applications involving porous media and magnetic fields.

Thermal characterization of hybrid nanofluid with impact of convective boundary layer flow and Joule heating law: Dual solutions case study

This study emphasizes the dual exposition of the impact of convective condition and the significant effects of magnetic field, heat source and velocity slip on hybrid nanofluid flow over the exponentially shrinking sheet. The hybrid nanofluid is regarded as a contemporary variety of nanofluid; consequently, it is utilized to enhance the efficiency of heat transfer. By applying the Tiwari–Das model, the key objective of the current research is to investigate the impact of involving factors Biot number, Eckert number, volume fraction, magnetohydrodynamic (MHD), slip and suction on the temperature and velocity profiles. In addition, the Nusselt number and skin friction variations have been explored against the suction effect on the solid volume fraction of copper [Formula: see text] [1–3%] and the Biot number [Formula: see text] [1–3%]. The nonlinear partial differential equations are converted to a set of an ordinary differential equations by incorporating exponential similarity vectors. Eventually, ordinary differential equations are rectified utilizing MATLAB bvp4c solver. The results demonstrate the presence of dual exposition for suction with varying values of copper volume fraction and the Biot number. The heat transmission rate is observed to escalate in both cases as the strength of the Biot and Eckert numbers intensifies in the range of 1–3%. In summary, the results show that duality and non-unique solutions occur in the aiding flow situation when the suction [Formula: see text], while there is no flow and unique solutions of hybrid nanofluid feasible when [Formula: see text].

Numerical Solution of Falkner-Skan Equation for a Moving Wedge in Hybrid Nanofluids

The flow and heat transfer past a moving wedge in a hybrid nanofluid is studied. A set of governing equations is transformed into ordinary differential equations using the similarity transformation. The resulting equations are then solved using bvp4c solver in MATLAB. The effects of the wedge angle parameter and nanoparticle volume fraction parameter of Al2O3-Cu/water on the skin friction coefficient and heat transfer characteristics are investigated. It is found that increasing the wedge angle parameter and Cu nanoparticles volume fraction gives rise to the skin friction coefficient and heat transfer on the surface. Further, dual solutions exist when the wedge moves in the opposite direction with the free stream.

Magnetized Casson SA-hybrid nanofluid flow over a permeable moving surface with stability analysis

This study employs the Tiwari and Das model to numerically evaluate the heat-transfer properties of a magnetized Casson sodium alginate-copper/alumina hybrid nanofluid flow over a permeable moving surface, taking into account the effects of viscous dissipation. Sodium alginate-alumina/copper hybrid nanofluids have garnered significant attention in modern society because of their potential uses in a variety of different industries. The key goal of the study is to examine the effect of the Casson parameter and copper solid volume fraction for skin friction coefficient and Nusselt number against velocity ratio parameter. Additionally, in the present study, velocity and temperature profiles for velocity ratio parameter, suction effect, magnetic parameter, Eckert number, and Prandtl number are also included. Using the bvp4c technique, the PDEs are converted into a set of non-linear ODEs with the use of suitable similarity variables. Dual approaches for controlling parameter using varied Casson fluid parameter and copper solid volume fraction quantities have been identified. Unique solutions, no solutions, and dual solutions exist in specific ranges of parameters. Consequently, beyond the critical values, no solution exists. The rate of heat transfer strengthened as the capacity of the Eckert number augmented. The temperature profile also diminished when the Prandtl number improved. The appearance of the boundary layer separation extension resulted from incorporating the copper solid volume fraction. Finally, stability analysis reveals that the first solution is stable, and the second solution is unstable.

Stability analysis of MHD stagnation point flow of Casson fluid past a shrinking sheet in porous medium considering heat sink or source, thermal radiation and suction effects

This study investigates the magnetohydrodynamic stagnation point flow of Casson fluid and heat transfer phenomena past a shrinking sheet in porous medium with thermal radiation, heat sink/source and suction effects. The Lie group analysis approach transforms the governing equations into self-similar equations, which are then solved through the shooting methodology. Asymptotic solutions for large stretching and suction parameters are reported. Dual solutions are found for a certain range of shrinking parameter. Stability analysis is performed in order to identify the stable and physically realizable solution. The stability is examined by calculating the lowest eigenvalues of the linearized boundary value problem associated with the self-similar equations. The analysis indicates that the first solution is stable, while the second solution is unstable. It is observed that as the suction parameter (s), magnetic field parameter (M) and Casson fluid parameter (β) increase, the range of dual similarity solutions increases.

Dual solutions for axisymmetric flow and heat transfer due to a permeable radially shrinking disk in copper oxide (CuO) and silver (Ag) hybrid nanofluids with radiation effect

Purpose This study aims to investigate the dual solutions for axisymmetric flow and heat transfer due to a permeable radially shrinking disk in copper oxide (CuO) and silver (Ag) hybrid nanofluids with radiation effect. Design/methodology/approach The partial differential equations that governed the problem will undergo a transformation into a set of similarity equations. Following this transformation, a numerical solution will be obtained using the boundary value problem solver, bvp4c, built in the MATLAB software. Later, analysis and discussion are conducted to specifically examine how various physical parameters affect both the flow characteristics and the thermal properties of the hybrid nanofluid. Findings Dual solutions are discovered to occur for the case of shrinking disk (λ < 0). Stronger suction triggers the critical values’ expansion and delays the boundary layer separation. Through stability analysis, it is determined that one of the solutions is stable, whereas the other solution exhibits instability, over time. Moreover, volume fraction upsurge enhances skin friction and heat transfer in hybrid nanofluid. The hybrid nanofluid’s heat transfer also heightened with the influence of radiation. Originality/value Flow over a shrinking disk has received limited research focus, in contrast to the extensively studied axisymmetric flow problem over a diverse set of geometries such as flat surfaces, curved surfaces and cylinder. Hence, this study highlights the axisymmetric flow due to a shrinking disk under radiation influence, using hybrid nanofluids containing CuO and Ag. Upon additional analysis, it is evidently shows that only one of the solutions exhibits stability, making it a physically dependable choice in practical applications. The authors are very confident that the findings of this study are novel, with several practical uses of hybrid nanofluids in modern industry.

Hiemenz stagnation point flow of a second-order micropolar slip flow with heat transfer

Purpose This paper aims to study a non-Newtonian micropolar fluid flow over a bidirectional flexible surface for multiple exact solutions of momentum boundary layer and thermal transport phenomenon subject to wall mass flux, second-order slip and thermal jump conditions. Design/methodology/approach The coupled equations are transformed into ordinary differential equations using similarity variables. Analytical and numerical techniques are used to solve the coupled equations for single, dual or multiple solutions. Findings The results show that the stretching flow, shrinking flow, the wall drag, thermal profile and temperature gradient manifest large changes when treated for special effects of the standard parameters. The role of critical numbers is definitive in locating the domains for the existence of exact solutions. The nondimensional parameters, such as mass transfer parameter, bidirectional moving parameter, plate deformation strength parameter, velocity slips, material parameter, thermal jump and Prandtl number, are considered, and their physical effects are presented graphically. The presence of governing parameters exhibits special effects on the flow, microrotation and temperature distributions, and various exact solutions are obtained for the special parametric cases. Originality/value The originality and value of this work lie in its exploration of non-Newtonian micropolar fluid flow over a bidirectional flexible surface, highlighting the multiple exact solutions for momentum boundary layers and thermal transport under various physical conditions. The study provides insights into the effects of key parameters on flow and thermal behavior, contributing to the understanding of complex fluid dynamics.

Stability analysis of dual solutions for mixed convection and thermal radiation with hybrid nanofluid flow past shrinking/stretching curved surface

This study portrays the stability analysis and dual solutions of mixed convection and thermal radiation of hybrid nano-fluid flow past stretching/shrinking a curved surface in the presence of injection/suction conditions. A hybrid nano-fluid, in which water is used as the base fluid, copper and alumina are used as nano-particles, and the magnetic field is taken into account. The present study’s findings will provide fruitful implications for future research in the field of fluid dynamics. The bvp4c method using Matlab software is implemented to get the numerical solution of the nonlinear partial differential equation transformed into the ordinary differential equation. The behavior of the first and second solutions under the governing parameters on the curved surface of dimensionless velocity g′(ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g'(\\xi )$$\\end{document}, shear stress profile g′′(ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g''(\\xi )$$\\end{document}, temperature profile θ(ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta (\\xi )$$\\end{document}, skin friction coefficient Cfs, and local Nusselt’s number Nus were visualized in figurative and tabular form. From this the following are investigated: as the values of ϕj\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi j$$\\end{document} increase, the velocity profile for the second solution decreases, and the opposite trend is observed for the first solution. For the values of K and λ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda 1$$\\end{document}, the shear stress profile increased for the first solution, and the opposite trend was observed for the second solution, though after some interval points, the inverse of this statement was observed.For the values of S,Pr,Rd,M,andλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S,Pr,Rd,M, \\;\ ext{and}\\; \\lambda$$\\end{document}, the upwind thermal boundary layer of the first solution is larger than the second solution.For the value of M uphill, the estimation of the absolute value of λci\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda ci$$\\end{document} increases for both the skin friction coefficient and the local Nusselt number. In the second solution, increasing the values of β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}, Pr, S, Ec, and M has a similar effect on g′′(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g''(0)$$\\end{document}, Cfs, -θ′(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta '(0)$$\\end{document}, and Nus. In the first solution, increasing the values of Ec, S, and Pr on g′′(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g''(0)$$\\end{document} and Cfs results in a decrease. The first solution has a positive eigenvalue, whereas the second solution has a negative eigenvalue. Agreement between the present analysis and literature is acceptable.

Dual solutions for the two-dimension copper oxide with silver (CuO–Αg) and zinc oxide with silver (ΖnO–Αg) hybrid nanofluid flow past a permeable shrinking sheet in a dusty fluid with velocity slip

PurposeThis paper aims to present dual solutions for the two-dimension copper oxide with silver (CuO–Ag) and zinc oxide with silver (ZnO–Ag) hybrid nanofluid flow past a permeable shrinking sheet in a dusty fluid with velocity slip.Design/methodology/approachThe governing partial differential equations for the two dust particle phases are reduced to the pertinent ordinary differential equations using a similarity transformation. Closed-form analytical solutions for the reduced skin friction and reduced Nusselt number, as well as for the velocity and temperature profiles, were presented, both graphically and in tables, under specific non-dimensional physical parameters such as the suction parameter, Prandtl number, slip parameter and shrinking parameter, which are also presented in both figures and tables.FindingsThe results indicate that for the shrinking flow, the wall skin friction is higher in the dusty fluid when compared with the clear (viscous) fluid. In addition, the effect of the fluid–particle interaction parameter to the fluid phase can be seen more clearly in the shrinking flow. Furthermore, multiple (dual, upper and lower branch solutions) are found for the governing similarity equations and the upper branch solution expanded with higher values of the suction parameter. It can be confirmed that the lower branch solution is unstable.Practical implicationsIn practice, the study of the stretching/shrinking flow is crucially important and useful. Both the problems of steady and unsteady flow of a dusty fluid have a wide range of possible applications in practice, such as in the centrifugal separation of particles, sedimentation and underground disposal of radioactive waste materials.Originality/valueEven though the problem of dusty fluid has been broadly investigated, very limited results can be found for a shrinking sheet. Indeed, this paper has succeeded to obtain analytically dual solutions. The stability analysis can be performed by following many published papers on stretching/shrinking sheets. Finally, the critical values and plotting curves for obtaining single or dual solution are successfully presented.

Dual solutions of unsteady flow of copper-alumina/water based hybrid nanofluid with acute magnetic force and slip condition

Suspending particles of tiny solid in a fluid used to transport energy can enhance its thermal conductivity and heat transport properties. Our main goal of this examination is to study the radiative unsteady two-dimensional (2D) flow on a continuously diminishing, horizontal sheet. with suction for the hybrid water-based nanofluid and an aligned field of magnetic, including the combined suction, magnetic, and velocity slip conditions effect. The Tiwari & Das model of nanofluid equations is used, which takes into consideration the solid volume percentage. Equations of similarity are derived by employing the transformations of similarity, and the associated equations have been simplified numerically by employing the bvp4c method in MATLAB software for a variety of values of the nanoparticle volume fraction, the unsteadiness, and the wall mass suction in water. It is discovered that, within the given the unsteadiness parameter range, two solutions exist. Moreover, it is found that the fluid velocity slows down in 1st solution as volume fraction of copper nanoparticles rises but speeds up in the second solution at first before slowing down again. Using a temporal stability analysis, it is found that only one of the dual branches is stable over the long run, while the other is unstable.

Analytical study on couple stress flow of GO-EG and GO-W nanofluid over an extending cylinder along with variable viscosity

The main goal of this research is to present the concept of enhancing heat transfer within emerging technology. To achieve this, tiny metal and nonmetal particles ranging from 1 to 100 nm in size are introduced into base liquids. These nanoscale particles are utilized to improve the thermal performance of the liquids, leading to what are termed nanofluids. The utilization of these fluids and the examination of the flow of thin films have valuable implications across various sectors such as engineering, technology, and industries. This research focuses on analyzing the convective flow behavior of nanofluids, specifically, graphene oxide-ethylene glycol (GO−EG) and graphene oxide-water (GO−W) on a moving surface. The study investigates the impacts of magnetic fields and varying viscosity. By making use of the thermophysical characteristics of the base fluid and the nanofluid, as well as implementing a similarity transformation within the fundamental equations that govern energy and momentum, we formulate a 5th order nonlinear ordinary differential equation (NODE) to describe the velocity profile. This is combined with a second-order NODE that describes the distribution of temperature. To solve this derived NODE, we employ a method known as the Homotopy Analysis Method (HAM) for analytical solution. The impact of the relevant factors, Prandtl number, including magnetic field parameter, thickness of the liquid, couple stress parameter, temperature distribution, dynamic viscosity, and Eckert number, on the skin friction, velocity profile, and Nusselt's number are interrogated through graphical representation. The velocity field exhibits a decline as the couple stress parameter, magnetic field parameter, liquid thickness, and dynamic viscosity experience an increase. Conversely, the temperature field displays a rise as the Eckert number and dynamic viscosity experience an increase. To ensure the convergence of the issue, dual solutions of the problem are employed, and this is verified through the utilization graphs and tables. Due to the considerable challenge encountered in heat transfer applications for cooling diverse equipment and devices across industries like automotive, microelectronics, defense, and manufacturing, there is a strong expectation that this theoretical methodology could make a favorable contribution towards enhancing heat transfer efficiency. This improvement is sought to meet the requirements of the manufacturing and engineering sectors.

Heat transfer in a dissipative nanofluid passing by a convective stretching/shrinking cylinder near the stagnation point

AbstractThis contemporary article examines the transfer of heat properties and the flow behavior of water‐based nanofluid suspended with silver nanoparticles. These silver nanoparticles have a very huge thermal conductivity and hence it is presumed that the resulting nanofluid shall have enhanced thermal conductance. This article is more focused on the study of nanofluid flowing past a cylinder that is modeled mathematically using the cylindrical coordinate system. The initial modeling is designed using a system of partial derivatives while at a later stage, this system is transformed into a nonlinear group of ordinary differential equations (ODEs). The equations in this system are solved to obtain the dual solutions by implementing the RKF‐45 method which has a greater rate of convergence and additionally, it is computationally very effective. The findings of the study are dealt by plotting graphs and the discussions are based on the appearance of graphs. It is further noticed that the critical point remains constant at for any changes made in the values of heat generation/absorption coefficient. Similarly, the critical value remains constant at for any change made in the values of the Eckert number. Meanwhile, it is also observed that the increase in the Eckert number increases the temperature absorbed by the nanofluid whereas it decreases the Nusselt number. Furthermore, the higher values of the velocity slip reduce the skin friction coefficient.