Dual Solutions Research Articles

Exploring dual solutions and thermal conductivity in hybrid nanofluids: a comparative study of Xue and Hamilton-Crosser models.

Hybrid nanofluids show great potential for heat transport applications such as solar thermal systems, car cooling systems, heat sinks, and thermal energy storage. They possess better thermal stability and properties compared to standard nanofluids. In this study, a base fluid, methanol, is injected into an electrically conducting heat-generating/absorbing disk of permeable boundary, and dual solutions are obtained. Two alternative models, Xue and Hamilton-Crosser are considered, and their thermal conductivities are contrasted. Furthermore, thermal radiation and ohmic heating are also considered, and convective boundary conditions are utilized to simulate overall heat gains or losses resulting from conduction, forced or natural convection between nearby objects of nearly constant temperature. Using a similarity transform, the governing equations are obtained and numerically solved via bvp4c, a finite difference method. It is observed that the presence of a magnetic field and the shrinking of the disk elevate the energy transport rate and wall stress. Additionally, the skin friction coefficient and thermal distribution rate increase with wall transmission constraint while fluid flow and energy transport diminish. Furthermore, particle clustering and nano-layer creation suggest that the Hamilton-Crosser model exhibits better thermal conductivity than the Xue model.

Magneto-hydrodynamic boundary layer flow and heat transfer of hybrid carbon nanotube over a moving surface

The boundary layer flow and heat transfer of hybrid carbon nanotubes over a moving surface with magneto-hydrodynamic effect are studied numerically in this paper. Single-wall (SWCNT) and multi-wall (MWCNT) carbon nanotubes are combined with water as the base fluid to form hybrid carbon nanotubes. The governing partial differential equations were transformed into a set of nonlinear ordinary differential equations using the similarity transformation, which were then numerically solved in the Matlab software using bvp4c. The influence of the nanoparticle volume fraction, magnetic parameter and velocity ratio parameter, on velocity and temperature profiles, local skin friction and local Nusselt number are discussed and presented in graphical forms. The results show that dual solutions appear when the free stream and plate move in the opposite direction, and the rate of heat transfer for hybrid carbon nanotubes is higher than viscous fluid and carbon nanotubes.

Dual Solutions of Unsteady Mixed Convection Hybrid Nanofluid Flow Past a Vertical Riga Plate with Radiation Effect

A mathematical model for the unsteady, two-dimensional mixed convection stagnation point flow over a Riga plate is presented in this study. Convective boundary conditions, time-dependent derivatives, mixed convection, radiation effects, and the Grinberg term were all incorporated into the formulation of the governing equations and boundary conditions. By incorporating similarity transformations, ordinary differential (similarity) equations (ODEs) are derived from the partial differential equations (PDEs) of the flow model. The boundary value problem of the fourth-order accuracy code (bvp4c) was implemented in MATLAB (2017b, The MathWorks, Inc., Natick, MA. USA, 2017) to solve the mathematical model numerically. Due to the plate’s shrinking motion, two (dual) solutions are possible (first and second solutions). Based on the stability analysis, it was found that the first solution is stable and physically realizable in practice, while the second solution is not stable and not physically realizable in practice. It was found that the increase in the mixed convection parameter, modified Hartmann number, and unsteadiness parameter improved the hybrid nanofluid’s temperature profile. In addition, increasing the unsteadiness parameter decreased the velocity profile and the skin friction coefficient. Thus, the numerical results suggested that the augmentation of the modified Hartmann number, mixed convection parameter, and unsteadiness parameter can enhance the heat transfer performance in this flow model. This study offers valuable insight into fundamental transport phenomena such as the transmission of momentum, heat, or mass. Hence, it provides valuable information on the gradients of essential factors to control the boundary layer flow pattern.

Dual Drazin inverses of dual matrices and dual Drazin-inverse solutions of systems of linear dual equations

In this paper, we study a kind of dual generalized inverse, which is called the dual Drazin inverse. Unlike the real matrices case, the dual Drazin inverse of a square dual matrix may not exist. It is shown that the dual Drazin inverse is unique when it exists. Some necessary and sufficient conditions for the existence of the dual Drazin inverse are presented. A compact formula for the computation of the dual Drazin inverse is given when it exists. Moreover, we find an unexpected result that the dual Drazin inverse can be obtained by computing the Drazin inverse of a 2 ? 2 upper triangular block matrix. We also introduce the dual Drazin-inverse solution of systems of linear dual equations. Some characterizations of the dual Drazin-inverse solution are given. In addition, some numerical examples are provided to illustrate the results.

Numerically investigating the effects of slip and thermal convective on nanofluid boundary layer past a stretching/shrinking surface

The study is focusing on the steady boundary layer flow, heat and mass transfer passing through stretching/shrinking sheet immersed in nanofluid in the presence of the second order slip velocity and thermal convective at the boundary. The governing partial differential equations are converted into ordinary differential equations by applying the similarity variables before being solved computationally using bvp4c function in Matlab software. The results of skin friction, heat transfer as well as mass transfer coefficient on the governing parameter such as the first order slip parameter, the second order slip parameter, Biot number, Brownian motion parameter and thermopherosis parameter are shown graphically in the discussion. The dual solutions exist in all range of stretching and shrinking parameter. Therefore the stability analysis is performed and concluded that the first solution is stable and physically relevant while the second solution acts in opposite way.

Unsteady Flow and Heat Transfer of a Casson Micropolar Nanofluid over a Curved Stretching/Shrinking Surface

We present the results of an investigation into the behavior of the unsteady flow of a Casson Micropolar nanofluid over a shrinking/stretching curved surface, together with a heat transfer analysis of the same problem. The body force acting perpendicular to the surface wall is in charge of regulating the fluid flow rate. Curvilinear coordinates are used to account for the considered curved geometry and a set of balance equations for mass, momentum, energy and concentration is obtained accordingly. These are turned into ordinary differential equations using a similarity transformation. We show that these equations have dual solutions for a number of different combinations of various parameters. The stability of such solutions is investigated by applying perturbations on the steady states. It is found that high values of the Micropolar and Casson parameters cause the flow to move more slowly. However, when compared to a shrunken surface, a stretched surface produces a greater Micro-rotation flux.

Communication‐aware distributed rebalancing for cooperative car‐sharing service

AbstractThis study presents a cooperative car‐sharing system in which various service providers jointly operate a sharing service. A distributed algorithm is proposed for rebalancing control to reduce deadhead vehicles as much as possible under the constraints of dispatchable vehicles and parking slots at car stations. Each provider cooperatively searches for the optimal solution by exchanging the estimated values of the optimal dual solutions over the communication networks between providers. Event‐driven communication is considered in which information exchange with the neighboring providers is performed only when the amount of change of the estimated optimal dual solution exceeds a threshold. Numerical instances explain that the estimations of all service providers meet the optimal rebalancing solutions.

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.

Action complexity of charged black holes with higher derivative interactions

Quantum complexity of CFT states can be computed holographically from the dual gravitational solutions. In this paper, we have studied the late time growth of holographic complexity of a charged black hole in five-dimensional, Anti-de Sitter spacetime in the presence of quartic derivative interaction terms using the Complexity = Action conjecture. These interaction terms in the gravitational action can lead to the violation of Llyod's bound. The dual CFT is known to admit a hydrodynamic description where the KSS bound is also violated due to the presence of higher derivative terms in the bulk action. The origin of terms which violate both the bounds are the same for the gravitational action of consideration. We have also discussed the late time complexity growth rate of the Jackiw-Teitelboim (JT) model with higher derivative corrections.

Analysis of unsteady boundary layer flow of nanofluids with heat transfer over a permeable stretching/shrinking sheet via a shifted Chebyshev collocation method

In this article, the detailed study of unsteady boundary layer flow over a permeable stretching/shrinking sheet problem of a nanofluid is carried out. The appropriate similarity transformations are utilized to transform the governing equations into nonlinear ordinary differential equations with variable coefficients over an infinite domain. The numerical solution of these equations is achieved by using semi-numerical technique viz. Shifted Chebyshev Collocation Method (SCCM). The variations in velocity, temperature, and concentration profiles under the influence of various physical parameters are analysed and outcomes are expressed in terms of tables and graphs. The obtained results are compared with previous finding for validation along with error analysis. For particular range of stretching/shrinking parameters the dual solutions are exhibited. Also, the range increases with increment in mass suction and unsteadiness parameter, which results into enhancement of skin friction coefficient, local Nusselt number and local Sherwood number. From the graphs it is anticipated that, both temperature and nanoparticle fraction profiles decrease with raise in the values of parameters viz. unsteadiness, suction/injection, Lewis number. Reverse nature of these profiles is observed with the enhancement in thermophoresis parameter. Whereas, temperature increases with increase of Brownian motion and reverse behavior is observed for nanoparticle fraction.

Stability Analysis of Buoyancy Magneto Flow of Hybrid Nanofluid through a Stretchable/Shrinkable Vertical Sheet Induced by a Micropolar Fluid Subject to Nonlinear Heat Sink/Source

The utilization of hybrid nanofluids (HNs) to boost heat transfer is a promising area of study, and thus, numerous scientists, researchers, and academics have voiced their admiration and interest in this area. One of the main functions of nanofluids is their dynamic role in cooling small electrical devices such as microchips and associated gadgets. The major goal of this study is to perform an analysis of the buoyancy flow of a shrinking/stretching sheet, whilst considering the fascinating and practical uses of hybrid nanofluids. The influence of a nonlinear heat source/sink induced by a micropolar fluid is also inspected. Water-based alumina and copper nanoparticles are utilized to calculate the fine points of the fluid flow and the features of heat transfer. The governing equations are framed with acceptable assumptions and the required similarity transformations are used to turn the set of partial differential equations into ordinary differential equations. The bvp4c technique is used to solve the simplified equations. Dual solutions are presented for certain values of stretching/shrinking parameters as well as the mixed convective parameter. In addition, the shear stress coefficient in the first-branch solution (FBS) escalates and decelerates for the second-branch solution (SBS) with the superior impact of the magnetic parameter, the mass transpiration parameter, and the solid nanoparticles volume fraction, while the contrary behavior is seen in both (FB and SB) solutions for the larger values of the material parameter.

Magnetohydrodynamics flow of Ag-TiO2 hybrid nanofluid over a permeable wedge with thermal radiation and viscous dissipation

Hybrid nanofluids, which are made by suspending non-identical nanoparticles, have been a prominent research area because of their high efficiency in heat transfer. The analysis of the magnetohydrodynamics flow of Ag-TiO2 hybrid nanofluid over a permeable wedge with heat radiation and viscous dissipation is mathematically examined in this paper. Ordinary differential equations are deduced by applying the corresponding similarity transformations to the mathematical modelling of the governing partial differential equations. The dimensionless governing equations are solved using the built-in bvp4c function in the MATLAB package to compute the dual solutions and the stability analysis. A respectable degree of agreement has been obtained after comparing the current results with the earlier study. Prandtl number, magnetic parameter, radiation parameter, Eckert number, and other governing factors have all been studied, along with their physical impacts on fluid flow. The graphical results have been demonstrated and described in relation to the profiles of temperature and velocity distribution, skin friction as well as the Nusselt number. It has been established that the higher volume percentage of titania nanoparticles has the potential to improve thermal conductivity, and the first solution has been found to be stable in this flow.

Stagnation point flow of a water-based graphene-oxide over a stretching/shrinking sheet under an induced magnetic field with homogeneous-heterogeneous chemical reaction

Water has received a great deal of attention as lubrication for production due to its efficiency, affordable, and friendly to the environment. GO (graphene oxide) ingredients have the capability to upsurge the efficiency of water lubrication owing to theire extraordinary machine-driven/mechanical properties, water dispensability, and variety of practical uses. With such a motivation, it is anticipated the effect of the induced magnetic field (IMF) on the fluid flow close to a standstill point created by a stretching/shrinking sheet with a homogeneous-heterogeneous chemical reaction and dispensed with water-based GO nanoparticles. The leading governing partial differential equations (PDEs) of the problem are converted into ordinary differential equations (ODEs) via employing similarity variables. After that, the resulting equations are worked out numerically with help of the bvp4c (boundary-value problem of fourth-order) solver. Figures are employed as representations to demonstrate the stimulation of pertinent parameters such as the reciprocal magnetic parameter, the magnetic parameter, the Prandtl number, the Schmidt number, the ratio of the diffusion coefficients, radiation, nanoparticle volume fraction, heat source/sink, and heterogeneous reaction parameter on the profile of induced magnetic, velocity, temperature, concentration, rate of heat transfer, and friction factor. It is noted that both the gradients are enhanced by the nanoparticle fraction volume. Moreover, dual solutions can be established in the limited values of stretching/shrinking parameter. Moreover, the shear stress upsurges by almost 4.18% and 0.52% for the upper and lower solution branches with higher values of the solid nanoparticles volume fraction, respectively, whilst it shrinkages up to 7.56% in the upper branch outcome owing to the higher influence of the magnetic parameter but increases by almost 14.56% for the lower branch solution.

A Function Approximation Approach for Parametric Optimization

We present a novel approach for approximating the primal and dual parameter-dependent solution functions of parametric optimization problems. We start with an equation reformulation of the first-order necessary optimality conditions. Then, we replace the primal and dual solutions with some approximating functions and find for some test parameters optimal coefficients as solution of a single nonlinear least-squares problem. Under mild assumptions it can be shown that stationary points are global minima and that the function approximations interpolate the solution functions at all test parameters. Further, we have a cheap function evaluation criterion to estimate the approximation error. Finally, we present some preliminary numerical results showing the viability of our approach.

Prediction of heat and mass transfer in radiative hybrid nanofluid with chemical reaction using the least square method: A stability analysis of dual solution

This analysis aims to investigate the stability of dual solutions for the heat and mass transfer flow of hybrid nanofluid over a stretching/shrinking surface with a uniform shear flow. The impacts of thermal radiation, transpiration, and chemical reaction are also considered in the flow. Gold (Au) and zinc oxide (ZnO) are selected as nanomaterials while engine oil (EO) is chosen to be the base fluid. By applying the similarity transformation technique, the nondimensional differential equations are changed into dimensionless coupled differential equations. The least square method (LSM) is applied to solve the system analytically and obtained the dual solutions in a particular range of stretching/shrinking parameter Due to this, the stability analysis is implemented to ensure that only the first solution is stable. The smallest eigenvalues are computed by utilizing the bvp4c function in Matlab software, where positive eigenvalues are linked with the stable solution while negative eigenvalues are linked with the unstable solution. The impacts of several physical parameters on the governing problem are discussed in detail and displayed in graphical form.

Applications of the Bernoulli wavelet collocation method in the analysis of MHD boundary layer flow of a viscous fluid

This study focuses on the flow of viscous, electrically conducting incompressible fluid over a stretching plate. The Falkner–Skan equation is a nonlinear, third-order boundary value problem. No closed-form solutions are available for this two-point boundary value problem. Here, we developed a new functional matrix of integration using the Bernoulli wavelet and also generated a new technique called Bernoulli wavelet collocation method (BWCM) to solve the nonlinear differential equation that arises in the fluid flow over a stretching plate. The boundary layer model is transformed to a nonlinear ordinary differential equation called the Falkner-Skan type equation using suitable transformation. Using BWCM, we have solved the unbounded governing equations of different types that arise in the MHD boundary-layer flow of a viscous fluid over a stretching plate. Several aspects of this problem are justified using the Haar wavelet and the previously obtained theoretical results. It is observed that the boundary-layer thickness decreases as the pressure gradient and magnetic field parameters increase. The overshoots and undershoots are observed for some particular parameters using BWCM. Furthermore, our research yields dual solutions for some physical parameters, which are investigated for the first time in the literature using the Bernoulli wavelet approach. The nature of the flow problem is discussed through the graphs by varying the physical parameters.

Excellent catalytic performance of mechanically alloyed AlCrFeMnTiZr0.5 high-entropy alloy for malachite green degradation

• AlCrFeMnTiZr 0.5 HEA powder photocatalyst is prepared by mechanical alloying. • AlCrFeMnTiZr 0.5 catalyst exhibits good malachite green degradation under light. • Nano-reactants are mainly oxides or hydroxides of Al and Mn after catalysis. • Nanoparticles and nanosheets covering surface promote adsorption and degradation of MG. • Exfoliated nano-reactants and exposed fresh surface enhance degradation reusability. This work reports that the mechanical alloyed AlFeMnCrTiZr 0.5 high-entropy alloy (HEA) powder was used as catalyst for malachite green (MG) degradation. The HEA with dual solid solutions shows good catalytic performance under visible light. Its efficiency of reaching degradation saturation and maximum degradation rate are much greater than those of Fe 0 powder. The oxides or hydroxides of Al and Mn nucleate and grow on the powder surface, existing in the form of initial nanoparticles and growing nanosheets, which promote the adsorption and catalytic degradation of MG. The exfoliated nano-reactants and exposed fresh powder surface of catalyst enhance the reusability and sustainability of degradation. The catalytic mechanism is mainly visualized by the synergy of chemical dye decomposition and physical dye adsorption.

Duality and sensitivity analysis of multistage linear stochastic programs

In this paper we investigate the dual of a Multistage Stochastic Linear Program (MSLP). By writing Dynamic Programming equations for the dual, we can employ an SDDP type method, called Dual SDDP, which solves these Dynamic Programming equations. allows us to compute a sequence of nonincreasing deterministic upper bounds for the optimal value of the problem. Since the Relatively Complete Recourse (RCR) condition may fail to hold for the dual (even for simple problems), we design two variants of Dual SDDP, namely Dual SDDP with penalizations and Dual SDDP with feasibility cuts, that converge to the optimal value of the dual (and therefore primal when there is no duality gap) problem under mild assumptions. We also show that optimal dual solutions can be obtained using dual information from Primal SDDP (applied to the original primal MSLP) subproblems.As a byproduct of the developed methodology we study sensitivity of the optimal value of the problem as a function of the involved parameters. For the sensitivity analysis we provide formulas for the derivatives of the value function with respect to the parameters and illustrate their application on an inventory problem. Since these formulas involve optimal dual solutions, we need an algorithm that computes such solutions to use them, i.e., we need to solve the dual problem.Finally, as a proof of concept of the tools developed, we present the results of numerical experiments computing the sensitivity of the optimal value of an inventory problem as a function of parameters of the demand process and compare Primal and Dual SDDP on the inventory and a hydro-thermal planning problems.

Steady MHD Flow in Hybrid Nanofluids Across through a Moving Plate with Regression Analysis

The steady two-dimensional flow and heat transfer past a moving plate in a hybrid nanofluid with magnetic fileds slip and thermal radiation effect were studied. The continuity, momentum and energy eqautions for this study are transformed into similarity equations via similarity transformations. Then, the similarity equations are solved numerically using bvp4c solver in Matlab software. The effects of several parameters on the skin friction coefficient and the local Nusselt number are presented and discussed. The results showed that dual solutions exist for a certain range for the nanoparticle volume fraction parameters. It also found that the heat transfer rate increased with the increasing magnetic field effect parameters. Regression analysis was also performed to estimate the value of the local Nusselt number.

Magnetohydrodynamic Boundary Layer in Carbon Nanotube past a Moving Plat with Slip and Thermal Radiation Effects

Momentum and thermal boundary layers stimulated by magnetohydrodynamic flow (MHD) on moving plate surface in single and multiwall carbon nanotubes (SWCNT and MWCNT) with slip and thermal radiation effects were studied. The mathematical modelling is developed to derive the governing partial differential equations. These models were transformed into a system of nonlinear ordinary differential equations via similarity transformation, which are then solved numerically using the boundary value problem with fourth-order accuracy (bvp4c) solver in MATLAB software. The effects of relevant parameters such as nanoparticle volume fraction, magnetic field, slip and thermal radiation on heat transfer are analyzed and discussed. It is found that dual solutions exist in opposing flows. The study also found that the heat flux on the moving plate increased with the increasing in slip effect. Also, the heat transfer rate in SWCNT-water is higher than that of MWCNT-water.