Homotopic Solutions Research Articles

Homotopic dynamic solution of hydrodynamic nonlinear natural convection containing superhydrophobicity and isothermally heated parallel plate with hybrid nanoparticles

Abstract The purpose of this work is to investigate the effect of thermal radiation on convective heat transfer for an electrically conducting hybrid nanofluid moving perpendicularly through a microchannel with parallel plates heated under isothermal conditions, while subjected to a transverse magnetic field. In this case, one surface exhibited a superhydrophobic slip and temperature jump, whereas the others did not. The objective of the study was to determine the impacts of thermal radiation, heat generation, and magnetism on the volume flow rate, velocity, and bulk temperature when either surface is heated by a constant wall temperature. Our findings reveal that the radiation parameter significantly influences both the Nusselt number and skin friction differently depending on the surface conditions, while also reducing flow rate and bulk temperature. The heat generation parameter similarly affects these variables but varies with the type of surface being heated. Additionally, both the velocity and temperature profiles increase with the porosity parameter and heat generation coefficient, regardless of the heated surface. Statistical analysis confirms the significance of magnetic and heat generation parameters in determining skin friction and the Nusselt number, and streamlines and isotherms illustrating the effects of thermal radiation and magnetism on two different hybrid nanofluids when heated on nonslip and superhydrophobic surfaces were provided. These insights provide valuable information for the design and maintenance of mini- and microdevices in the fields of nanoscience and nanotechnology.

Engineering applications development through OHAM in heat and mass transfer during stagnant flow of second-grade nanomaterial

This paper investigates the stagnation point flow of a second-grade nanomaterial, considering nanofluid effects via thermophoresis and Brownian motion. The study addresses convective heat and mass transfer conditions within the flow induced by a stretching cylinder. The transformed systems are solved using the optimal homotopy solutions (OHAM) method, revealing the impact of various parameters on the quantities of interest. The results indicate that an increase in curvature, viscoelasticity, velocity ratio and injection parameters leads to an enhancement in velocity, while the opposite trend is observed due to suction. The viscoelasticity and curvature parameters also increase skin friction. Additionally, thermophoresis enhances the temperature and concentration fields, while Brownian motion reduces mass diffusion.

Time-independent three-dimensional flow of a water-based hybrid nanofluid past a Riga plate with slips and convective conditions: A homotopic solution

Abstract In the present analysis, we have analyzed the three-dimensional flow of an electromagnetohydrodynamic copper–aluminum/water hybrid nanofluid flow on a Riga plate. The heat and mass flux model proposed by Cattaneo-Christov is deliberated here. Thermal radiation, thermophoretic diffusion, Brownian motion, and chemical reaction phenomena are considered in analyzing the flow problem. Thermal convective, mass convective, and velocity slip conditions are adapted in this analysis. Suitable resemblance variables are implemented for the conversion of the model equations to dimension-free form. The homotopy analysis method is adopted to solve the modeled equations. The obtained results show that the velocity profiles are reduced with an increasing estimation of the slip factors. Additionally, the nanoparticles’ concentration and the temperature of the hybrid nanofluid increase with higher values of thermal and solutal Biot numbers. The Nusselt number is increased with an increase in the radiation factor and thermal Biot number.

Analytical solution for unsteady Walters-B fluid flow by a deforming surface with acceleration using OHAM based package BVPh2.0

Current study aims at simulating fluid flow due to a deformable heated surface in an otherwise static viscoelastic fluid obeying Walters-B model. Velocity of the surface is supposed to grow as time from its initiation of motion progress. Simulations in this work are based on the assumption of quadratic surface temperature distribution. Temperature rise attributed to the frictional heating effect is accounted for in the analysis. By choosing appropriate base functions, homotopy solutions are developed for reasonably large values of material fluid parameter. Reliability of the analytical results is established by computing averaged squared residual of the system. The contributions of the surface acceleration and elasticity on the boundary layer formation are enlightened through the plots of velocity components and temperature. Skin friction measuring the stress experienced by the surface is evaluated and examined under different controlling parameters. The paper also presents a numerical solution using NDSolve of MATHEMATICA in a special case of steady flow, and such solution agrees very well with the corresponding homotopy solution.

Numerical exploration of slip effects on second-grade fluid motion over a porous revolving disk with heat and mass transfer

Revolving-disk systems are employed in various industrial settings including turbine engineering, chemical and food processing industries and others. The current article scrutinizes a second-grade fluid motion generated by an infinite porous disk having partial slip character. Heat transfer induced by heating of the disk surface and by viscous and ohmic heating effects is modeled and analyzed under thermal slip condition. Accompanying mass transfer process with thermophoretic diffusion is also formulated. A self-similar system is obtained akin to the case of no-slip case discussed in a previously published study. The adoption of velocity slip assumption induces non-linearity in the boundary conditions in velocity components. Computational procedure embedded in MATLAB bvp4c platform is opted to simulate the system for full range of slip parameters. In contrast to a previously published work pertaining to the no-slip case, present numerical methodology gives accurate results for wide ranges of Prandtl number and elasticity parameter. Boundary layer formations above the disk are examined under various controlling parameters. A comparative assessment of slip and no-slip cases is presented through both graphical illustrations and tabulated results for the resisting torque, the Nusselt number and the Sherwood number. Current numerical findings match very well with the existing homotopy solutions for the no-slip case. The presence of a wall slip mechanism leads to a clear suppression of all the velocity components. Furthermore, an augmentation in the thermal/concentration slip coefficient significantly reduces the thermal/solutal penetration depth. Additionally, we observe a noticeable increase in the driving torque as the elasticity parameter enhances. The slip action of the surface is also predicted to raise the torque required by the disk.

Functional magnetic Maxwell viscoelastic nanofluids for tribological coatings- a model for stretching flow using the generalized theory of heat-mass fluxes, Darcy-Forchheimer formulation and dual convection

A theoretical and computational study of an improved Fourier and Fickian model for locally non-similar magnetohydrodynamic Maxwell (non-Newtonian) nanofluid convective flow under thermo-solutal buoyancy forces in a non-Darcian (Darcy-Forchheimer) porous medium is presented. Heat sink/source is included. Buongiorno’s two-component nanofluid model is used to simulate nanoscale characteristics featuring thermophoresis and Brownian diffusions. The problem under consideration is formulated utilizing fundamental relations of fluid dynamics. The primitive partial differential expressions (PDEs) are transfigured to ordinary ones with apposite transformations. The emerging locally non-similar boundary value problem is solved via series expansions utilizing a homotopy algorithm. Characteristics of sundry parameters on velocity, temperature, nanoparticle concentration and skin-friction coefficients are interpreted graphically. Convergence results acquired via homotopy algorithm are presented. Comparison results are included for the authentication of the homotopy solutions. The velocity distribution is increased with greater mixed convection and non-Newtonian material parameters whereas velocity distribution is reduced with increment in Hartmann (magnetic) number, porosity and buoyancy ratio parameters. The temperature distribution is reduced when Prandtl number and thermal-relaxation (non-Fourier) parameter are augmented whereas temperature distribution is increased for larger Brownian diffusion and thermophoresis. Additionally, it is observed that increment in the heat absorption variable diminishes temperature whereas an enhancement in the heat generation variable augments temperature. Nanoparticle concentration is enhanced subjected to higher values of thermophoresis factor whereas it reduces with larger Schmidt number, Brownian movement and solutal-relaxation (non-Fickian) parameters. Furthermore, it is noticed that elevation in Hartmann number, porosity and inertial (non-Darcy) coefficient parameters increase the skin friction coefficient whereas elevation in Deborah number and buoyancy ratio is found to suppress skin friction. The simulations are relevant to hydromagnetic nano-materials processing operations for coatings deployed in multi-functional tribological systems and surface protection.

Simulation for bloodstream conveying bi-nanoparticles in an endoscopic canal with blood clot under intense electromagnetic force

In the current era, the electromagnetic pumping flow of hybrid nano-biofluid features in myriad magneto-biomedical engineering applications. In this scenario, the current disquisition is centralized to unfold the electro-magneto-hemodynamic distinctive features of ionized bloodstream conveying silver and aluminium oxide hybridized nanoparticles driven by electroosmosis via an endoscopic conduit (space between two coaxial tubes) formed by a uniform and rigid endoscope and a complaint walled artery. The formulation involves the dominance of Hall and ion-slip factors, internal energy generation, Joule warming, a blood clot (coagulation), and convective wall condition. The contribution of nanoparticles' shape is dissected in this examination. The Poisson-Boltzmann equation is utilized to emulate the conduit's electric double layer (EDL). The lubrication and Debye-Hückel linearization principles are opted to simplify the normalized complicated leading equations. The homotopic series solutions of the consequent coupled nonlinear dimensionless equations are computed. A critical examination of significant flow-controlling parameters over the relevant hemodynamical characteristics is executed via graphs and tables. From the obtained outcomes, it is worthy of imparting that a discernible lesson is viewed in the blood velocity profile against the intensified estimates of Hall and ion-slip, and electro-osmotic factors. Blood is warmed with a positive Joule heating factor, whereas it is cooled with negative values of this factor. Upraised volumetric proportions of hybridized nanoparticles cool the blood in the conduit. Streamline patterns are also graphically displayed to see the blood flow pattern and the formation of entrapped boluses in the endoscopic domain. This research study considering multiple physical aspects such as electromagnetic phenomena, electromagnetic force, inclusion of hybridized nanoparticles, and coagulation is an innovative approach. Our findings in this simulation are expected to open up a new opportunity in biomedical engineering applications, including magneto-endoscopic operation, cheap devices for drug distribution, bio-magnetic therapy, electromagnetic hyperthermia treatment for cancer, etc.

Lie symmetry and exact homotopic solutions of a non-linear double-diffusion problem

The Lie symmetry method is applied, and exact homotopic solutions of a non-linear double-diffusion problem are obtained. Additionally, we derived Lie point symmetries and corresponding transformations for equations representing heat and mass transfer in a thin liquid film over an unsteady stretching surface, using MAPLE. We used these symmetries to construct new (Lie) similarity transformations that are different from those that are commonly used for flow and mass transfer problems. These new (Lie) similarity transformations map the partial differential equations of a mathematical model under consideration to ordinary differential equations along with boundary conditions. Lie similarity transformations are shown to lead to new solutions for the considered flow problem. These solutions are obtained using the homotopy analysis method to analytically solve the ordinary differential equations that resulted from the reduction of considered flow equations through Lie similarity transformations. With the aid of these solutions, effects of various parameters on the flow and heat transfer are discussed and presented graphically in this study.

Homotopic solution of the chemically reactive magnetohydrodynamic flow of a hybrid nanofluid over a rotating disk with Brownian motion and thermophoresis effects

AbstractDue to the synergetic effects of several types of nanomaterials, the primary goal of the hybrid nanofluid is to enhance the energy transport capabilities over a base fluid. Hybrid nanofluids have a wide range of applications in the industrial, technical, and medical industries, including solar heating systems, food processing, microchannel heat sinks (MCHS), and medicines. In this article, the researchers have investigated the water‐based hybrid nanofluid flow comprising silver and alumina nanoparticles past a spinning disk. The effect of Brownian motion, activation energy, magnetic field, and thermophoresis are taken into account. The PDEs are transformed into ODEs by means of suitable correspondence transformations. The modeled equations are solved by using a semi‐analytical method known as HAM. Graphical representations of the nanofluid and hybrid nanofluid profiles are used. The current findings are contrasted with those that have already been published and are confirmed to be remarkably comparable. The outcomes showed that the radial and tangential velocities of the nanofluids and hybrid nanofluid reduced as the magnetic factor augmented. Nanofluids and hybrid nanofluid surface drag is increased by magnetic factor. Hybrid nanofluid exhibits higher growth due to the magnetic factor than nanofluids do. The heat transmission rates of nanofluids and hybrid nanoliquid have grown as a result of the thermophoresis factor and nanoparticle volume fractions. In comparison to nanofluids, the hybrid nanofluid also possesses a better thermal conductivity.

Homotopic solutions of an unsteady magnetohydrodynamic flow of Casson nanofluid flow by a vertical cylinder with Brownian and viscous dissipation effects

The present study investigates the mathematical and computational framework of an unsteady magnetohydrodynamic (MHD) Casson nanofluid flow subjected to convective heat transfer enhancement in a vertically moving cylinder. The two main important aspects of nanosized particles movement in nanofluids are the Brownian motion and thermophoresis force which have substantial impacts on the thermophysical characteristics of nanofluids. The mathematical formulation regarding the assumed diversified and control parameters are modeled in constitutive partial differential equations (PDEs). The boundary layer theory is used to reduce the nonlinearity of subsequent dimensionless partial differential equations. The homotopic solution of these dimensionless PDEs is obtained via an analytical algorithm. The impact of sundry influential physical parameters comprises flow and heat transfer individualities. Computational results of the homotopic scheme toward several governing parameters of interest, such as the skin friction, heat, and mass transfer at the diverse condition and dependency of different parameters are offered by means of plotted graphs and elaborated.

An Optimal Investigation of Convective Fluid Flow Suspended by Carbon Nanotubes and Thermal Radiation Impact

This study is focused towards analyzing the heat and flow movement among two stretching rotating disks inside water-based carbon nanotubes. The idea of thermal boundary conditions and heat convection is used and the system is expressed in partial differential equations. Using the similarity techniques, the model is successfully converted to a nonlinear ordinary differential equation. A familiar collocation method is used to simulate the outcomes of the governed system while the method is validated through a set of tables and assessed with existing literature. The physical aspects of the proposed model have been studied in detail and assisted via graphical diagrams against the variation of different parameters. It is found that the multiple-wall carbon nanotubes intensify the system quickly and improve the rate of heat transmission. It is also noted that the proposed method is in excellent in agreement with already published studies and can be extended for other physical problems. Moreover, when values of Re parameter increase, a drop is noted in the magnitude of radial velocity near the faces of the disks. It is very clear from the tabular comparison that collocation scheme is in good agreement with already published studies and homotopic solutions.

Visualization of multiple slip effects on the hydromagnetic Casson nanofluid past a nonlinear extended permeable surface: a numerical approach

This study intends to explore the interaction effects of the magnetic field and the slip effect of the MHD flow of the Casson nanofluid by a nonlinear stretching surface embedded in a porous medium. The nanofluids’ flow over a stretching surface is critically important due to their application in the broad spectrum of industries and scientific fields. The mathematical formulation for mass conservation, momentum, energy and concentration profiles is expressed in the form of partial differential equations. The dimensionless system of ordinary differential equations has been diminished from modeled equations via a transformation framework. A numerical algorithm bvc4c method is implemented for a local similar solution of highly nonlinear equations. The characteristics of material parameters on the flow field, temperature, concentration, surface drag force and Nusselt number are tackled via plotted graphs. The authentication of the numerical code has been verified with a homotopic solution and an excellent agreement is perceived as a limiting case. Higher values of the Casson parameter , magnetic parameter , porosity parameter and stretching parameter correspond to a reduction in the velocity profile gradually. As noticed from this graph, the deescalating temperature profile is subject to an increment in parameter . From this study, it is observed that the velocity profile increases with the velocity slip number , and the temperature profile decreases with the temperature slip number It is also investigated that the temperature profiles enhance with increasing and decline with . The concentration profiles increase with the increasing values of and .

Motion planning by learning the solution manifold in trajectory optimization

The objective function used in trajectory optimization is often non-convex and can have an infinite set of local optima. In such cases, there are diverse solutions to perform a given task. Although there are a few methods to find multiple solutions for motion planning, they are limited to generating a finite set of solutions. To address this issue, we present an optimization method that learns an infinite set of solutions in trajectory optimization. In our framework, diverse solutions are obtained by learning latent representations of solutions. Our approach can be interpreted as training a deep generative model of collision-free trajectories for motion planning. The experimental results indicate that the trained model represents an infinite set of homotopic solutions for motion planning problems.

Homotopic Solution for 3D Darcy–Forchheimer Flow of Prandtl Fluid through Bidirectional Extending Surface with Cattaneo–Christov Heat and Mass Flux Model

The 3D Prandtl fluid flow through a bidirectional extending surface is analytically investigated. Cattaneo–Christov fluid model is employed to govern the heat and mass flux during fluid motion. The Prandtl fluid motion is mathematically modeled using the law of conservations of mass, momentum, and energy. The set of coupled nonlinear PDEs is converted to ODEs by employing appropriate similarity relations. The system of coupled ODEs is analytically solved using the well-established mathematical technique of HAM. The impacts of various physical parameters over the fluid state variables are investigated by displaying their corresponding plots. The augmenting Prandtl parameter enhances the fluid velocity and reduces the temperature and concentration of the fluid. The momentum boundary layer boosts while the thermal boundary layer mitigates with the rising elastic parameter ( α 2 ) strength. Furthermore, the enhancing thermal relaxation parameter ( γ e )) reduces the temperature distribution, whereas the augmenting concentration parameter ( γ c ) drops the strength of the concentration profile. The increasing Prandtl parameter declines the fluid temperature while the augmenting Schmidt number drops the fluid concentration. The comparison of the HAM technique with the numerical solution shows an excellent agreement and hence ascertains the accuracy of the applied analytical technique. This work finds applications in numerous fields involving the flow of non-Newtonian fluids.

Homotopy solutions of the acoustic eikonal equation for strongly attenuating transversely isotropic media with a vertical symmetry axis

Many applications in seismology involve the modeling of seismic-wave traveltimes in anisotropic media. We have developed homotopy solutions of the acoustic eikonal equation for P-wave traveltimes in attenuating transversely isotropic media with a vertical symmetry axis. Instead of the commonly used perturbation theory, we use the homotopy analysis method to express the traveltimes by a Taylor series expansion over powers of an embedding parameter. For the derivation, we first perform homotopy analysis of the eikonal equation and derive the linearized ordinary differential equations for the coefficients of the Taylor series expansion. Then, we obtain the homotopy solutions for the traveltimes by solving the linearized ordinary differential equations. Results of our investigation with approximate formulas demonstrate that the analytical expressions are efficient methods for the computation of traveltimes from the eikonal equation. In addition, these formulas are also effective methods for benchmarking approximate numerical solutions in strongly attenuating anisotropic media.

Irreversibility analysis for axisymmetric nanomaterial flow towards a stretched surface

• Hydromagnetic axisymmetric flow of nanomaterial by a nonlinear stretched sheet is addressed. • Dissipation and radiation effects are scrutinized in heat equation. • Thermophoresis and Brownian motion are scrutinized. • Irreversibility analysis is scrutinized. Magnetohydrodynamic axisymmetric flow of viscous nanoliquid towards a variable stretching sheet is scrutinized. Flow is generated due to nonlinear stretching. Joule heating, heat flux and dissipation are analyzed in heat expression. Random and thermophoresis diffusions are considered. Physical description of entropy generation is also accounted. Entropy generation and heat transfer analysis are scrutinized through thermodynamics laws. Furthermore chemical reaction along with Arrhenius activation energy is addressed. Ordinary differential system is obtained through suitable variables. Homotopic convergent solutions for nonlinear system is developed. Influence of flow variables on entropy rate, velocity, Bejan number, concentration and temperature are analyzed. Further velocity gradient and heat and mass transfer rates are discussed. Reduction in velocity is noticed for magnetic variable. Thermal field has an enhancing trend for magnetic and Eckert number. Larger thermophoresis variable rises the temperature. An increment in entropy rate is observed for magnetic parameter. An increment in drag force is seen for magnetic variable.

Impact of Nanofluid Flow over an Elongated Moving Surface with a Uniform Hydromagnetic Field and Nonlinear Heat Reservoir

The increasing global demand for energy necessitates devoted attention to the formulation and exploration of mechanisms of thermal heat exchangers to explore and save heat energy. Thus, innovative thermal transport fluids require to boost thermal conductivity and heat flow features to upsurge convection heat rate, and nanofluids have been effectively employed as standard heat transfer fluids. With such intention, herein, we formulated and developed the constitutive flow laws by utilizing the Rossland diffusion approximation and Stephen’s law along with the MHD effect. The mathematical formulation is based on boundary layer theory pioneered by Prandtl. Governing nonlinear partial differential flow equations are changed to ODEs via the implementation of the similarity variables. A well-known computational algorithm BVPh2 has been utilized for the solution of the nonlinear system of ODEs. The consequence of innumerable physical parameters on flow field, thermal distribution, and solutal field, such as magnetic field, Lewis number, velocity parameter, Prandtl number, drag force, Nusselt number, and Sherwood number, is plotted via graphs. Finally, numerical consequences are compared with the homotopic solution as a limiting case, and an exceptional agreement is found.

Mixed convective 3D flow of Maxwell nanofluid induced by stretching sheet: Application of Cattaneo-Christov theory

The present study invokes the application of Cattaneo-Christov theory for the thermal analysis in the buoyancy driven three dimensional flow of Maxwell nanofluid. The flow is induced above the vertical bidirectional stretching sheet. The phenomena of thermophoresis and Brownian diffusion of nanoparticles in the flow Maxwell liquid are deliberated with the help of Buongiorno model for nanofluid. The physical problem is formulated in the form of boundary layer partial differential equations (PDEs). Moreover, suitable ansatz for flow mechanism are employed to reduce the governing PDEs into the non-linear ordinary differential equations (ODEs). The flow mechanism of Maxwell fluid along with energy transport is analyzed in the form of homotopic solutions of the governing ODEs. The outcomes are presented graphically and discussed with physical explanation. The analysis revealed that both buoyancy and mixed convection parameters enhanced the [Formula: see text]-component of velocity field but declined the [Formula: see text]-component. Moreover, in assisting mode these two parameters also increased the thermal and solutal energy transport in nanofluid. It is noted that the thermophorectic force boosts up the thermal energy transport in the flow in the presence of thermal relaxation phenomenon. The validation of the present results are confirmed through tabular data with some previous studies.

Irreversibility analysis in squeezing nanofluid flow with thermal radiation

PurposeMagnetohydrodynamic (MHD) nanoliquid are significant for thermal conductivity enhancement. The examination of heat transfer of crushing time-subordinate liquid flow past isometric surfaces has throughout the decades been a field of consideration for its wide scope of physical necessities: nourishment preparation, pressure, grease setup and hydrodynamic machines. Entropy generation in the squeezing flow of viscous nanomaterial is developed. MHD, Brownian motion and thermophoresis are considered. Porous space between the disks is taken. The analysis is carried out in the presence of radiation and viscous dissipation.Design/methodology/approachNonlinear systems are reduced to an ordinary one through similarity variables. The convergent solution is developed by employing the homotopy analysis technique (HAM).FindingsConvergent homotopic solutions are developed for the velocity, temperature and concentration. Entropy generation and Bejan number are explained. Skin friction and Nusselt number and Sherwood number are analyzed. For a higher approximation of porosity, parameter velocity is augmented. Temperature upsurges for larger thermophoresis and Brownian diffusion parameters. Concentration has an increasing effect on thermophoresis and Brownian diffusion parameters. For the rising value of the radiation parameter, both the Bejan number and entropy rate have increasing behaviors.Originality/valueNo such work is yet published in the literature.

On Behavioral Response of 3D Squeezing Flow of Nanofluids in a Rotating Channel

In this article, a behavioral study of three-dimensional (3D) squeezing flow of nanofluids with magnetic effect in a rotating channel has been performed. Using Navier–Stokes equations along with suitable similarity transformations, a nonlinear coupled ordinary differential system has been derived which models the 3D squeezing flow of nanofluids with lower permeable stretching porous wall where the channel is also rotating. The base fluid in the channel is considered to be water that contains different nanoparticles including silicon, copper, silver, gold, and platinum. The homotopy perturbation method (HPM) is employed for the solution of highly nonlinear coupled system. For validation purpose, system of equations is also solved through the Runge–Kutta–Fehlberg (RK45) scheme and results are compared with homotopy solutions, and excellent agreement has been found between analytical and numerical results. Also, validation has been performed by finding average residual error of the coupled system. Furthermore, the effects of various parameters such as nanoparticle volume fraction, suction parameter, characteristic parameter of the flow, magnetic parameter, rotation parameter, and different types of nanoparticles are studied graphically.