Existence Of Dual Solutions Research Articles

Magnetohydrodynamic flow of carbon nanotubes and heat transfer over a moving thin Needle: A numerical and research surface methodology

A steady flow of carbon nanotubes (CNTs) nanofluids and heat transfer past a horizontally moving thin needle are investigated under the influence of magnetohydrodynamic (MHD). Single-walled CNTs (SWCNTs) and multi-walled CNTs (MWCNTs) are the two main nanoparticles that represent CNTs. The slender needle moves relative to the flow with parallel velocity in either the same or opposite direction. Using the similarity method, a system of partial differential equations (PDEs) subject to boundary conditions is converted into nondimensional ordinary equations (ODEs). The ODEs are then reduced to a first-order ODEs system and solved using the MATLAB R2022b bvp4c solver. On a numerical scale, the impacts of varying potential parameters, such as the magnetic, CNTs’ volume fraction, and moving parameters, on the velocity and temperature profiles, the skin friction, and heat transfer coefficients are investigated. Utilising response surface methodology (RSM), the optimisation of the response based on numerical experimentation of physical quantities is performed. The outcomes are depicted using tables and a graphical approach. Results indicate the existence of dual solutions when the needle travels in the opposite direction. Moreover, the increase in the magnetic parameter by 100% in the flow will increase both the skin friction and heat transfer coefficients by nearly 30% and 4%, respectively. Furthermore, when the value of the CNTs volume fraction increases by 100%, the heat transfer rate increases substantially by almost 33%.However, doubling the size of the thin needle cansignificantly reduce the skin friction coefficient by nearly 32%. RSM results demonstrate that the maximal heat transfer coefficient is generated at the highest values of the magnetic and CNTs’ volume friction parameters and the lowest value of the needle size parameter. Findings also show that SWCTNs are superior to MWCNTs both in the skin friction and heat transfer coefficients. When comparing the performance of water and kerosene, we find that water is less effective as a base fluid than kerosene.

Exact solutions for magnetohydrodynamic nanofluids flow and heat transfer over a permeable axisymmetric radially stretching/shrinking sheet

We report on the magnetohydrodynamic impact on the axisymmetric flow of Al2O3/Cu nanoparticles suspended in H2O past a stretched/shrinked sheet. With the use of partial differential equations and the corresponding thermophysical characteristics of nanoparticles, the physical flow process is illustrated. The resultant nonlinear system of partial differential equations is converted into a system of ordinary differential equations using the suitable similarity transformations. The transformed differential equations are solved analytically. Impacts of the magnetic parameter, solid volume fraction and stretching/shrinking parameter on momentum and temperature distribution have been analyzed and interpreted graphically. The skin friction and Nusselt number were also evaluated. In addition, existence of dual solution was deduced for the shrinking sheet and unique solution for the stretching one. Further, Al2O3/H2O nanofluid flow has better thermal conductivity on comparing with Cu/H2O nanofluid. Furthermore, it was found that the first solutions of the stream are stable and physically realizable, whereas those of the second ones are unstable.

Multiple solutions and their stability for a Ti-alloy nanofluid saturated Darcy porous medium: Flow separation and reversal

In this article, the authors provided the stability analysis for Darcy porous medium saturated by a Ti-alloy nanofluid in the presence of tilted magnetic field, Joule heating, heat generation, and velocity slip effects. The Tiwari-Das model is implemented and the relevant similarity transformations are identified for mathematical modeling of the proposed fluid flow problem. The set of resultant ODEs has been solved using the shooting method in conjunction with the RK-4th order scheme. The existence of dual solutions is observed along with their critical values for the present setup at and after the bifurcation point but for a limited range of certain fixed parameters. Through the eigenvalue approach, it is shown that the first solutions are stable, but not the second solutions. The streamlines are depicted to understand the flow patterns in addition to the identification of flow separation point and flow reversal. The important findings are as follows: (a) the delay of boundary layer separation is pointed out with the enhancing values of volume fraction of Ti-alloy nanoparticles and the inclined magnetic parameter in the presence of suction, (b) the Ti-alloy nanofluids velocity rises but the temperature decays with rising values of the Ti-alloy nanoparticle volume fraction, and (c) the value of smallest eigenvalues is growing with the growing values of magnetic parameter. Finally, this type of research may be extremely beneficial in the fields of aerodynamics (i.e. production of engine components, aircraft turbines, and high-performance automatic parts) and medicine.

DUAL SOLUTION OF CONVECTIVE BIOMAGNETIC FLUID THROUGH PERMEABLE MOVING FLAT PLATE CONSIDERING WALL TRANSPIRATION AND MAGNETIZATION

A two-dimensional (2D) steady boundary layer flow along with heat transfer of a self-similar biomagnetic fluid over a permeable moving flat plate has been taken into consideration in this work. The flow is contemplated to be embedded by a magnetic dipole of sufficient magnetic strength. Transpiration as well as movement along the wall is also regarded. By imposing the appropriate similarity technique, the governing equations are converted into a system of coupled nondimensional equations. An efficient numerical technique has been incorporated to solve these dimensionless coupled nonlinear ordinary differential equations. The existence of dual solutions along with their stability has been established with the consideration of stability analysis. We discovered from our analysis that two solutions exist (one stable and another unstable) for the arbitrary values of transpiration, movement velocity and biomagnetic interaction parameters on flow and physical parameters. The attained results are demonstrated graphically and in tabular form. For the validity of our numerical scheme, we compared our findings with others previously published and found significant agreement.

Lagrange Multipliers, Duality, and Sensitivity in Set-Valued Convex Programming via Pointed Processes

We present a new kind of Lagrangian duality theory for set-valued convex optimization problems whose objective and constraint maps are defined between preordered normed spaces. The theory is accomplished by introducing a new set-valued Lagrange multiplier theorem and a dual program with variables that are pointed closed convex processes. The pointed nature assumed for the processes is essential for the derivation of the main results presented in this research. We also develop a strong duality theorem that guarantees the existence of dual solutions, which are closely related to the sensitivity of the primal program. It allows extending the common methods used in the study of scalar programs to the set-valued vector case.

Stability Analysis of Unsteady Hybrid Nanofluid Flow Past a Permeable Stretching/Shrinking Cylinder

The study of boundary layer flow has gained considerable interest owing to its extensive engineering applications. Thus, this numerical study aims to investigate the stability analysis of unsteady flow in the hybrid Al2O3-Cu/H2O nanofluid past a shrinking permeable cylinder. The impacts of suction and unsteadiness parameters are considered in this study. The partial differential equations are converted into a system of nonlinear ordinary differential equations by selecting suitable similarity transformation and solved using the bvp4c code in the MATLAB program. The findings revealed that the existence of dual solutions is visible. The skin friction coefficient and the local Nusselt numbers of Al2O3-Cu/H2O increase with the inclusion of the suction parameter. The presence of the unsteadiness parameter actively promotes heat transfer degradation on the shrinking cylinder. Stability analysis indicates that a stable and physically realizable solution appeared in the first solution, whereas the second solution is unstable.

Unsteady Boundary Layer Flow Over a Permeable Stretching/Shrinking Cylinder Immersed in Nanofluid

In this study, the unsteady boundary layer flow over a stretching/shrinking cylinder immersed in nanofluid with the presence of suction effect is analyzed. The governing partial differential equations are converted to ordinary differential equations by introducing similarity transformation variables. The shooting method is applied to solve the system where the numerical solutions are obtained and presented graphically. The study's objective is to investigate the effect of nanoparticle volume fraction, the unsteadiness parameter, the stretching/shrinking parameter on the velocity and temperature gradients. It is found that the dual solutions are obtained in a specific range of these parameters for both stretching and shrinking cylinders. Besides, a high volume of the nanoparticle in the base fluid increases the velocity gradient and decreases the temperature gradient at the surface. Also, increasing nanoparticle volume fraction in the base fluid expands the solution's range, which denotes the boundary layer separation from the surface has been delayed. The existence of dual solutions allows stability analysis performance by introducing a new dimensionless variable and is solved using bvp4c function in Matlab software. This phase obtains the smallest eigenvalue, showing that the first solution is stable and physically realizable while the second solution is not stable.

Numerical duality of MHD stagnation point flow and heat transfer of nanofluid past a shrinking/stretching sheet: Metaheuristic approach

In the present era of Industry 4.0, automation, and nanotechnology, many industrial applications require the simultaneous control of heat and fluid flow for better efficiency of machines. The polymeric excursion process, manufacturing of plastic films, glass, metallic equipment, all need control over friction forces and heat flow rates. Considering the importance of the subject, the present study deals with the stagnation point flow and heat transfer of water-based nanofluid over a stretching/shrinking sheet with a non-aligned axis of symmetry. The permeable sheet is considered in the presence of a uniform magnetic field. The non-linear partial differential equations governing the flow and heat transfer are converted into non-linear coupled ordinary differential equations using similarity transformations. The resultant equations are solved using “Particle Swarm Optimization” and Runge-Kutta method based hybrid shooting technique. Considering the nanoparticle size, concentration and shape-dependence on physical and thermal properties of nanofluid more realistic aggregation based mathematical models are used for both viscosity and thermal conductivity. Numerical results for the velocity profile, non-alignment function, temperature distribution and local skin friction are discussed with respect to the physical parameters like nanoparticle concentration, nanoparticle size, magnetic field, suction, and stretching/shrinking velocity. The existence of dual solutions are observed for the shrinking velocity parameter λ<−1, and the practically realizable and stable solutions are filtered out by computing the eigenvalues for the formulated perturbed Eigen problem. The results for the stable solutions unveiled that, by changing the stretching velocity parameter λ from 0 to 0.5, there is approximately 38% enhancement in heat transfer rate and 42% decrement in skin friction coefficient. Moreover, in the case of shrinking velocity parameter λ=−1.25, with an increase in nanoparticle concentration from 0 to 5%, the skin friction coefficient reduces by 53%, whereas the change in nanoparticle size from 10 nm to 50 nm has a skin friction reduction of only 10%.

Existence of dual solutions and three-dimensional instability in helical pipe flow

Three-dimensional (3D) direct numerical simulations (DNS) of the viscous incompressible fluid flow through a helical pipe with circular cross section were performed. The flow is governed by three parameters: the Dean number (or the Reynolds number), curvature, and torsion. First, we obtained steady solutions by steady 3D calculations, where dual solutions were found, one was uniform in the pipe axial direction and the other varied very slowly, if torsion exceeded a critical value. Then, the instability of the steady solutions obtained was studied by unsteady 3D calculations. We obtained critical Reynolds numbers of steady to unsteady transition by observing the behaviors of the unsteady solutions. The present results of the critical Reynolds number nearly agreed with those by the 2D linear stability analysis (Yamamoto et al. [9]) except for the lowest critical Reynolds number region, where the present study gave the critical Reynolds number much less than that obtained by the 2D linear stability analysis. We found the vortical structures in the form of a standing wave slightly above the marginal instability state, which is a trigger of explosive 3D instability.

Stability analysis on dual solutions of second- grade fluid flow with heat and mass transfers over a stretching sheet

Stability on dual solutions of second-grade fluid flow over a stretching surface with simultaneous thermal and mass diffusions has been studied. The fluid flow is governed by Lorentz force and energy dissipation due to viscosity. Lorentz force is generated due to the application of magnetic field along the transverse direction. In methodology, suitable similarity transformation and MATLAB built-in bvp4c solver technique have been adopted. Effects of some flow parameters are exhibited through figures and tables and a special emphasis is given on the existence of dual solutions. A stability analysis is executed to determine the stable and physically achievable solutions. For the laminar flow, the drag force on the surface for the time-independent case is reduced due to amplifying values of But, it enhances the drag force for the time-dependent case. This shows the effectiveness of the first solution (during steady case) over the unsteady case.

On the stability of the flow over a shrinking cylinder with prescribed surface heat flux

This study investigates the steady stagnation point flow and heat transfer passes a horizontal shrinking permeable cylinder. The free stream velocity and the prescribed surface heat flux are assumed to vary linearly with the distance from a fixed point on the cylinder. The partial differential equations governing the flow and heat transfer are transformed into a system of ordinary differential equations via similarity transformation. These equations are solved numerically for several values of the governing parameters, such as suction parameter s, curvature parameter γ, and shrinking parameter λ. The equations are solved numerically by employing the boundary value problem solver package available in MATLAB software, bvp4c. The effects of the governing parameters on the skin friction coefficient, surface temperature, velocity, and temperature profiles are examined. Given the existence of dual solutions in the present study for a certain range of the curvature parameter, stability analysis is carried out to determine which one of the solutions is stable as time passes. The outcome of the stability analysis demonstrates that only the first solution, with lower boundary layer thickness, appeared to be stable and thus physically reliable, while the other is not. It is also discovered that the boundary layer separation is delayed by reducing the curvature parameter.

Magnetohydrodynamics Stagnation-Point Flow of a Nanofluid Past a Stretching/Shrinking Sheet with Induced Magnetic Field: A Revised Model

The revised Buongiorno’s nanofluid model with the effect of induced magnetic field on steady magnetohydrodynamics (MHD) stagnation-point flow of nanofluid over a stretching or shrinking sheet is investigated. The effects of zero mass flux and suction are taken into account. A similarity transformation with symmetry variables are introduced in order to alter from the governing nonlinear partial differential equations into a nonlinear ordinary differential equations. These governing equations are numerically solved using the bvp4c function in Matlab solver, a very adequate finite difference method. The influences of considered parameters ( P r , M, χ , L e , N b , N t , S, and λ ) on velocity, induced magnetic, temperature, and concentration profiles together with the reduced skin friction and heat transfer rate are discussed. Results from these criterion exposed the existence of dual solutions when magnetic field and suction are applied for a specific range of λ . The stability of the solutions obtained is carried out by performing a stability analysis.

Oblique Stagnation Point Flow of Nanofluids over Stretching/Shrinking Sheet with Cattaneo–Christov Heat Flux Model: Existence of Dual Solution

In the present work we consider a numerical solution for laminar, incompressible, and steady oblique stagnation point flow of Cu − water nanofluid over a stretching/shrinking sheet with mass suction S . We make use of the Cattaneo–Christov heat flux model to develop the equation of energy and investigate the qualities of surface heat transfer. The governing flow and energy equations are modified into the ordinary differential equations by similarity method for reasonable change. The subsequent ordinary differential equations are illuminated numerically through the function bvp4c in MATLAB. The impact of different flow parameters for example thermal relaxation parameter, suction parameter, stretching/shrinking parameter, free stream parameter, and nanoparticles volume fraction on the skin friction coefficient, local Nusselt number, and streamlines are contemplated and exposed through graphs. It turns out that the lower branch solution for the skin friction coefficient becomes singular in shrinking area, although the upper branch solution is smooth in both stretching and shrinking domain. For oblique stagnation-point flow the streamlines pattern are not symmetric, and reversed phenomenon are detected close to the shrinking surface. Also, we observed that the free stream parameter changes the direction of the oncoming flow and controls the obliqueness of the flow. The existing work mostly includes heat and mass transfer as a mechanism for improving the heat transfer rate, which is the main objective of the authors.

Stability analysis of Cu–H2O nanofluid over a curved stretching–shrinking sheet: existence of dual solutions

The effect of mass suction with temperature jump and velocity slip of viscous, unsteady nanofluid flow past a curved shrinking–stretching surface is analyzed in this work. Copper (Cu) and water are considered nanoparticles and base fluids, respectively. The complicated coupled system of differential equations is converted into non-dimensional form with some suitable similarity variables. The solution of the nonlinear problem is produced by use of numerical scheme available in the form of bvp4c package in MATLAB. In the case of shrinking towards the surface, a reverse flow situation is also developed and requires careful selection of solution by examining the stability of the solution. Detailed stability analysis is done and critical values are determined for the possible existence of dual solutions. Variation in parameters is analyzed by plotting graphs and tables. The numerical values are also calculated for the reduced Nusselt number and skin friction due to variation in values of different flow parameters. Results have shown that for the curved shrinking surfaces, one should expect multiple solutions for a set of parameter values such as mass suction, curvature, nanoparticles volume fraction, and unsteadiness.

MHD oblique stagnation point flow of nanofluid over an oscillatory stretching/shrinking sheet: existence of dual solutions

An unsteady magnetohydrodynamic oblique stagnation point flow of an incompressible nanofluid towards a stretching/shrinking surface with the effect of mass suction on the oscillatory surface is analyzed. Water is considered as a base fluid while single wall carbon nanotubes are assumed as nanoparticles. The governing flow equations are transformed into a system of coupled nonlinear ordinary differential equations via suitable transformation. Dual solutions of these nonlinear coupled equations are obtained by utilizing the bvp4c package in MATLAB. It is also found that these solutions survive in a certain domain of controlling parameters. The physical influence of several controlling parameters like nanoparticle volume fraction parameter, Hartmann number, stretching/shrinking parameter and suction parameter on heat transfer characteristics and flow field are discussed in detail and are demonstrated through graphs. We find that dual solutions exist in both the stretching case and the shrinking case. Also, the lower solution branch shows singular behavior for the skin friction coefficient in the shrinking domain whereas the upper solution branch has smooth behavior in both the stretching and shrinking domains.

Existence of Dual Solutions and Melting Phenomenon in Unsteady Nanofluid Flow and Heat Transfer over a Stretching Surface

ABSTRACTThe problem of unsteady boundary layer flow of a nanofluid over a stretching surface is studied. Heat transfer due to melting is analyzed. Using a similarity transformation the governing coupled nonlinear partial differential equations of the model are reduced to a system of nonlinear ordinary differential equations, and then solved numerically by a Runge-Kutta method with a shooting technique. Dual solutions are observed numerically and their characteristics are analyzed. The effects of the pertinent parameters such as the acceleration parameter, the Brownian motion parameter, the thermophoresis parameter, the Prandtl number and the Lewis number on the velocity, temperature and concentration fields are discussed. Also the effects of these parameters on the skin friction coefficient, the Nusselt number and the Sherwood number are analyzed through graphs. It is observed that the melting phenomenon has a significant effect on the flow, heat and mass transfer characteristics.

Analysis of Unsteady Flow and Heat Transfer of Nanofluid Using Blasius–Rayleigh–Stokes Variable

This article investigates the unsteady flow and heat transfer analyses of a viscous-based nanofluid over a moving surface emerging from a moving slot. This new form of boundary layer flow resembles with the boundary layer flow over a stretching/shrinking surface depending on the motion of the moving slot. The governing partial differential equations are transformed to correct similar form using the Blasius–Rayleigh–Stokes variable. The transformed equations are solved numerically. Existence of dual solutions is observed for a certain range of moving slot parameter. The range of dual solution is strongly influenced by Brownian and thermophoretic diffusion of nanoparticles.

Existence and stability of heat and fluid flow in the presence of nanoparticles along a curved surface by mean of dual nature solution

This study examines the possible existence and stability of the heat transfer solutions due to unsteady nanofluid flow over a curved surface. Alumina oxide (Al2O3) is considered as one of the important nanoparticles due to its potential use in various fields like purification of water, ceramics and insulation products. It is considered that the small quantity of nanoparticles is incorporated in the Ethylene Glycol to form the nanofluid. Mathematical form of the problem is obtained by using the fundamental laws of motion and heat transfer. These equations are modified for the presence of solid concentrations in an ordinary liquid. The complicated coupled unsteady system is transformed into non-dimensional form by the use of relevant transformations. The solution of nonlinear problem is produced by the use of numerical scheme, available in the form of BVP4C package in MATLAB. In the case of surface shrinking towards the surface, a reverse flow situation is also developed, and requires careful selection of solution by examining the stability of solution. Detailed stability analysis is carried, and critical values are determined for the possible existence of dual solutions. Moreover, various parameters variation is also analyzed by plotting graphs and tables. The numerical values are also calculated for the reduced Nusselt number and skin friction values caused due to variation in various flow parameters. Results have shown that for the curved shrinking surfaces, one should expect multiple solutions for a set of parameter values like mass suction, curvature, nanoparticles volume fraction and unsteadiness. The isotherms suggest linear decay of heat away from the surface.

Discretisation and duality of optimal Skorokhod embedding problems

We prove a strong duality result for a linear programming problem which has the interpretation of being a discretised optimal Skorokhod embedding problem, and we recover this continuous time problem as a limit of the discrete problems. With the discrete setup we show that for a suitably chosen objective function, the optimiser takes the form of a hitting time for a random walk. In the limiting problem we then reprove the existence of the Root, Rost, and cave embedding solutions of the Skorokhod embedding problem.The main strength of this approach is that we can derive properties of the discrete problem more easily than in continuous time, and then prove that these properties hold in the limit. For example, a consequence of the strong duality result is that dual optimisers exist, and our limiting arguments can be used to derive properties of the continuous time dual functions. These arguments are applied in Cox and Kinsley (2017), where the existence of dual solutions is required to prove characterisation results for optimal barriers in a financial application.

Dual solutions for nonlinear boundary value problems by the Adomian decomposition method

Purpose The purpose of this paper is to use the Adomian decomposition method (ADM) for solving boundary value problems with dual solutions. Design/methodology/approach The ADM has been previously demonstrated to be eminently practical with widespread applicability to frontier problems arising in scientific applications. In this work, the authors seek to determine the relative merits of the ADM in the context of several important nonlinear boundary value models characterized by the existence of dual solutions. Findings The ADM is shown to readily solve specific nonlinear BVPs possessing more than one solution. The decomposition series solution of these models requires the calculation of the Adomian polynomials appropriate to the particular system nonlinearity. The authors show that the ADM solves these models for any analytic nonlinearity in a practical and straightforward manner. The conclusions are supported by several numerical examples arising in various scientific applications which admit dual solutions. Originality/value This paper presents an accurate work for solving nonlinear BVPs that possess dual solutions. The authors have demonstrated the widespread applicability of the ADM for solving various forms of these nonlinear equations.