Mass Flux Boundary Conditions Research Articles

GFEM analysis of MHD nanofluid flow toward a power‐law stretching sheet in the presence of thermodiffusive effect along with regression investigation

AbstractIn the current study, we use Galerkin finite‐element simulation to analyze the concept of triple diffusive flow with magnetic field effect toward a power law stretching sheet. The fluid comprises dissolved solutal particles and nanoparticles in the base fluid. The three important mechanisms that are responsible for rise in phenomenon of convective transportation are diffusophoresis, thermophoresis along with Brownian motion have been considered. Recently, the proposed nanoparticles' mass flux and heat flux boundary conditions have been imposed. Nanoparticle mass transportation, solutal mass transportation with heat transportation for prominent physical parameters, such as stretching parameter, magnetic influence parameter, thermophoresis parameter, and Brownian motion parameter are calculated. To further verify and understand the strength of the relationship between heat transportation rate and controlling parameters, the multiple regression process is used. The finite difference approach was adopted to numerically solve the nonlinear governing equations and the linked boundary conditions. In the present study, we used MATLAB software for finding the final outcomes and relating the concluding results for with extant outcomes in the literature as a limiting case in the absence of the magnetic intensity parameter and an excellent agreement was noted. It was observed that the magnetic field has a positive effect on heat and mass transfer. This study also helps in understanding and thus controlling the velocity of the flow along with solutal depositions, which has a significant engineering application in the process of extrusion. The findings of the present study help to control the rate of heat and mass transfer, aiding manufacturing companies in obtaining the desired quality of product.

Theoretical and analytical analysis of convective heat transport of radiated micropolar fluid over a vertical plate under nonlinear Boussinesq approximation

PurposeIn heat transfer problems, if the temperature difference is not sufficiently so small then the linear Boussinesq approximation is not adequate to describe thermal analysis. Also, nonlinear density variation with respect to temperature/concentration has a significant impact on heat and fluid flow characteristics. Because of this reason, the impact of nonlinear density variation in the buoyancy force term cannot be neglected. Therefore in this paper, the unsteady flow and heat transfer of radiating magneto-micropolar fluid by considering nonlinear Boussinesq approximation is investigated analytically.Design/methodology/approachThe flow is fully developed and time-dependent. Heat and mass flux boundary conditions are also accounted in the analysis. The governing equations of transport phenomena are treated analytically using regular perturbation method. To analyze the tendency of the obtained solutions, a parametric study is performed.FindingsIt is established that the velocity field is directly proportional to the nonlinear convection parameter and the same trend is observed with the increase of the value of Grashof number. The micro-rotational velocity profile decreases with increase in the nonlinear convection parameter. Further, the temperature profile increases due to the presence of radiative heat aspect.Originality/valueThe effectiveness of nonlinear Boussinesq approximation in the flow of micropolar fluid past a vertical plate in the presence of thermal radiation and magnetic dipole is investigated for the first time.

Heat Transfer Analysis of Three-Dimensional Mixed Convective Flow of an Oldroyd-B Nanoliquid over a Slippery Stretching Surface

The present article examines Soret and Dufour effects on the three-dimensional mixed convective flow of an Oldroyd-B nanoliquid. The flow is caused due to bidirectional stretching of the surface in the presence of an induced magnetic field and heat generation/absorption. Besides, concentration and thermal buoyancy impacts are inspected. The velocity slip, convective and zero nanoparticle mass flux boundary condition at the surface are taken into account. Nonlinear system of equations which are highly coupled is solved via optimal homotopy algorithm. The influence of pertinent parameters on velocity, temperature, and concentration are analyzed graphically. The impact of Dufour number is quite substantial on temperature whereas Soret number increases the concentration. To see the legitimacy of the present work, the present results are compared with the results available in the literature and noted an excellent agreement for the limiting cases.

A Numerical Exploration of Modified Second-Grade Nanofluid with Motile Microorganisms, Thermal Radiation, and Wu’s Slip

This study is carried out to scrutinize the gyrotactic bioconvection effects on modified second-grade nanofluid with motile microorganisms and Wu’s slip (second-order slip) features. The activation energy and thermal radiation are also incorporated. The suspended nanoparticles in a host fluid are practically utilized in numerous technological and industrial products such as metallic strips, energy enhancement, production processes, automobile engines, laptops, and accessories. Nanoparticles with high thermal characteristics and low volume fraction may improve the thermal performance of the base fluid. By employing the appropriate self-similar transformations, the governing set of partial differential equations (PDEs) are reduced into the ordinary differential equations (ODEs). A zero mass flux boundary condition is proposed for nanoparticle diffusion. Then, the transmuted set of ODEs is solved numerically with the help of the well-known shooting technique. The numerical and graphical illustrations are developed by using a collocation finite difference scheme and three-stage Lobatto III as the built-in function of the bvp4c solver via MATLAB. Behaviors of the different proficient physical parameters on the velocity field, temperature distribution, volumetric nanoparticles concentration profile, and the density of motile microorganism field are deliberated numerically as well as graphically.

Entropy generation analysis for peristaltic motion of Carreau–Yasuda nanomaterial

Peristaltic transport of magneto nanofluid in a symmetric channel is numerically discussed. Carreau–Yasuda model is used to explore the shear thickening and shear thinning characteristics. Joule heating and viscous dissipation effects are included in the energy equation. Effects of slip velocity, temperature jump and zero mass flux boundary conditions for channel walls are further considered. Entropy generation and Bejan number are studied. This research has been carried out employing lubrication approximation. Solutions are numerically developed and described.

Non-Similar Solution of G-jitter Induced Unsteady Magnetohydrodynamic Radiative Slip Flow of Nanofluid

We present a mathematical model and numerical simulation of the unsteady 2-D g-jitter-free and forced the convective flow of water-based nanofluid from a flat plate, considering both the velocity slip and thermal slip conditions imposed on the wall of the plate. The Darcian model is used, and both cases of a calm and moving free stream are considered. In place of the extensively used linearly varying radiative heat flux, the nonlinearly varying heat flux calculation is applied to produce practically useful results. Further, we incorporate the “zero mass flux boundary condition” which is believed to be more realistic than the earlier extensively used “actively” controlled model. The parameter influences the non-dimensional velocity, temperature, nanoparticle volume fraction, skin friction and heat transfer rates are visualized graphically and discussed in detail. Special cases of the results are benchmarked with those existing in the literature, and a good arrangement is obtained. It is found that the rate of heat transfer is lower for the calm free stream rather than the moving free stream.

Time-dependent 3D flow of viscoelastic nanofluid over an unsteady stretching surface

This article addresses the unsteady three-dimensional flow of second grade nanofluid due to thermophoresis and Brownian motion effects. The flow is generated due to unsteady stretching in two lateral directions. The zero nanoparticles mass flux boundary conditions are imposed to find the desired solutions. By introducing the dimensionless variables the governing system of partial differential equations are transformed into the system of ordinary differential equations. Series solutions of the transformed ordinary differential equations are computed. Solutions are carried out for dimensionless temperature and nanoparticles concentration. The behavior of physical parameters on the temperature and nanoparticles concentrations is examined through graphs and tabular data. The temperature and concentration distribution at the surface are also computed and analyzed. It can be noted that both the temperature and nanomaterial are reduced for higher values of unsteadiness parameter. It is also observed that the friction factor of second grade fluid parameter is less compared with viscous nanofluid.

A novel model to analyze Darcy Forchheimer nanofluid flow in a permeable medium with Entropy generation analysis

A novel mathematical model is envisaged to scrutinize the Darcy Forchheimer 3D Powell Eyring nanofluid flow in a porous medium. Flow is taken under the influence of zero mass flux and convective boundary conditions at the surface and a chemical reaction in the mass equation. The heat transfer flow is scrutinized with non-linear thermal radiation. Entropy generation analysis of the envisioned model is also conducted. The Homotopy Analysis method to yield the series solutions for the envisioned model. The graphs are plotted to witness the characteristics of several parameters versus velocity, heat, and mass distributions and are well cogitated accordingly. The findings show that the velocity is decreasing the function of Darcy-Forchheimer number. Further, the Biot number large values boost the fluid temperature. The outcomes obtained in the analysis are substantiated when compared with a published result in the literature. An outstanding matching is achieved in this regard.

Scrutinization of 2D and mixed convection flow of generalized Newtonian fluid with nanoparticles and magnetic field

In this research paper we focus on presenting the local non-similar solutions for two-dimensional steady and mixed convective flow of electrically conducting Carreau nanofluid under the influence of Brownian motion and thermophoresis effects. This research paper also presents the new mass flux boundary condition of nanoparticles. The governing partial differential equations are converted into ordinary differential equations by using the local non-similar transformations called the Sparrow–Quack–Boerner local non-similar method. A numerical approach is used to investigate the local non-similar solutions of the entire problem. Outcomes for the skin friction and rate of heat and mass transfer have been obtained and discussed for parametric variations of the buoyancy parameter ξ, magnetic parameter M, Weissenberg number We, viscosity ratio parameter β*, Brownian motion parameter Nb, thermophoresis parameter Nt, Prandtl number Pr, and Schmidt number Sc. From the obtained results a considerable increase in the skin friction coefficient is observed with an increase in buoyancy parameter ξ. Also, ξ shows relative growth for the momentum and concentration profiles.

Entropy generation analysis in flow of thixotropic nanofluid

Purpose The purpose of this paper is to address entropy generation in flow of thixotropic nonlinear radiative nanoliquid over a variable stretching surface with impacts of inclined magnetic field, Joule heating, viscous dissipation, heat source/sink and chemical reaction. Characteristics of nanofluid are described by Brownian motion and thermophoresis effect. At surface of the sheet zero mass flux and convective boundary condition are considered. Design/methodology/approach Considered flow problem is mathematically modeled and the governing system of partial differential equations is transformed into ordinary ones by using suitable transformation. The transformed ordinary differential equations system is figure out by homotopy algorithm. Outcomes of pertinent flow variables on entropy generation, skin friction, concentration, temperature, velocity, Bejan, Sherwood and Nusselts numbers are examined in graphs. Major outcomes are concluded in final section. Findings Velocity profile increased versus higher estimation of material and wall thickness parameter while it decays through larger Hartmann number. Furthermore, skin friction coefficient upsurges subject to higher values of Hartmann number and magnitude of skin friction coefficient decays via materials parameters. Thermal field is an increasing function of Hartmann number, radiation parameter, thermophoresis parameter and Eckert number. Originality/value The authors have discussed entropy generation in flow of thixotropic nanofluid over a variable thicked surface. No such consideration is yet published in the literature.

NANOFLUID SLIP FLOW THROUGH POROUS MEDIUM WITH ELASTIC DEFORMATION AND UNIFORM HEAT SOURCE/SINK EFFECTS

The uniform heat source/sink effect on second-grade nanofluid flow over a stretching sheet embedded in Darcian porous medium is studied with elastic deformation. The partial slip, heat flux, and mass flux boundary conditions are considered. The magnetic field is applied in various directions. The nanofluid model is considered with viscoelasticity, Brownian motion and theromophosis mechanisms. Mathematical equations governing the problem are solved numerically using the fourth-order Runge-Kutta method with shooting iteration technique. The flow and heat transfer phenomena are analyzed through plots for various sets of physical parameters. It is found that the presence of elastic deformation and the uniform heat source increase the thickness of the nanofluid thermal and concentration boundary layers.

Unsteady three-dimensional stagnation point magnetohydrodynamic flow of bionanofluid with variable properties

Unsteady three-dimensional laminar stagnation point forced convective boundary layer magnetohydrodynamic flow of a bionanofluid with variable transport properties is studied theoretically and numerically. Thermal convective and zero mass flux boundary conditions are incorporated in this study. The transport properties are assumed to be a function of nanoparticle volume fraction to get physically realistic results. The dimensional boundary layer equations along with the coupled boundary conditions are transformed via similarity transformations into a system of ordinary differential equations. The transformed equations are solved numerically using the Runge–Kutta–Fehlberg fourth-, fifth-order numerical method. The effect of selected governing parameters, namely, viscosity, thermal conductive, mass diffusivity, microorganism diffusivity, magnetic field and bioconvection Schmidt number, on the dimensionless velocity, temperature, nanoparticle volume fraction, microorganism, skin friction coefficient, heat transfer rate, mass transfer rate and microorganism transfer rate, is illustrated graphically and interpreted in detail. Comparisons with previous works are carried out for some limiting cases and found to be in good agreement.

On Model for Three-Dimensional Flow of Nanofluid With Heat and Mass Flux Boundary Conditions

The present paper examines magnetohydrodynamic (MHD) three-dimensional (3D) flow of viscous nanoliquid in the presence of heat and mass flux conditions. A bidirectional nonlinearly stretching surface has been employed to create the flow. Heat and mass transfer attribute analyzed via thermophoresis and Brownian diffusion aspects. Viscous liquid is electrically conducted subject to applied magnetic field. Problem formulation is made through the boundary layer approximation under small magnetic Reynolds number. Appropriate transformations yield the strong nonlinear ordinary differential system. The obtained nonlinear system has been solved for the convergent homotopic solutions. Effects of different pertinent parameters with respect to temperature and concentration are sketched and discussed. The coefficients of skin friction and heat and mass transfer rates are computed numerically.

Double dispersion effects on non-Darcy free convective boundary layer flow of a nanofluid over vertical frustum of a cone with convective boundary condition

AbstractIn this paper, a numerical analysis is performed to investigate the effects of double dispersion and convective boundary condition on natural convection flow over vertical frustum of a cone in a nanofluid saturated non-Darcy porous medium. In addition, Brownian motion and thermophoresis effects have taken into consideration, and the uniform wall nanoparticle condition is replaced with the zero nanoparticle mass flux boundary condition to execute physically applicable results. For this complex problem, the similarity solution does not exist and hence suitable non-similarity transformations are used to transform the governing equations along with the boundary conditions into non-dimensional form. The Bivariate Pseudo-Spectral Local Linearisation Method (BPSLLM) is used to solve the reduced non-similar, coupled partial differential equations. To test the accuracy of proposed method, the error analysis and convergence tests are conducted. The effect of flow influenced parameters on non-dimensional velocity, temperature, nanoparticle volume fraction, regular concentration field as well as on the surface drag, heat transfer, nanoparticle and regular mass transfer rates are analyzed.

Bioconvection nanofluid slip flow past a wavy surface with applications in nano-biofuel cells

A theoretical study is presented to examine free convective boundary layer flow of water-based bio-nanofluid containing gyrotactic microorganisms past a wavy surface. Buongiorno's nanofluid model with passively controlled boundary condition is applied to investigate the effects of the emerging parameters on the physical quantities namely, skin friction, Nusselt numbers and density number of motile microorganisms. The effects of the both hydrodynamic and thermal slips are also incorporated. Local similarity and non-similarity solutions are obtained using the seventh-order Runge–Kutta–Fehlberg method (RKF7) coupled with shooting quadrature. In order to compare our numerical results with the existing data, the active mass flux boundary condition is also used to benchmark MAPLE numerical solutions with earlier similar and non-similar solutions for a smooth stationary surface. It is found that the passive boundary condition reduces the skin friction and enhances local Nusselt numbers. Also the wavy surface is found to result in higher skin friction and higher local Nusselt numbers compared with a stationary surface. It is found that motile micro-organism density number is elevated with increasing bioconvection Péclet number whereas the motile micro-organism species boundary layer thickness is reduced with increasing bioconvection Lewis number. The work finds applications in heat transfer enhancement in bio-inspired nanoparticle-doped fuel cells.

Three-dimensional flow of nanofluid with heat and mass flux boundary conditions

This article presents magnetohydrodynamic three-dimensional flow of nanofluid over an exponentially stretching surface with prescribed heat flux (PHF) and prescribed concentration flux (PCF) conditions. An exponentially stretching surface is used to generate the flow. Effects of Brownian motion, thermophoresis, viscous dissipation and Joule heating are also taken into account. Boundary layer and small magnetic Reynolds number assumptions are considered in the mathematical modeling. The conversion from nonlinear partial differential equations to nonlinear ordinary differential equations is taken into place by using suitable transformations. The resulting nonlinear system is solved via convergent homotopic approach. Graphs are sketched in order to analyze that how the temperature and concentration profiles are affected by distinct physical flow parameters. Skin-friction coefficients and local Nusselt and Sherwood numbers are also computed and analyzed.

Hydromagnetic nanofluid flow past a stretching cylinder embedded in non-Darcian Forchheimer porous media

The present article presents the hydromagnetic nanofluid flow past a stretching cylinder embedded in non-Darcian Forchheimer porous media by using Buongiorno’s mathematical model (Buongiorno in J Heat Transf 128:240–250, 2006; Nadeem et al. in J Taiwan Inst Chem Eng 45:121, 2014, Nadeem et al. Appl Nanosci 4:625–631, 2014). Thermal radiation via Roseland’s approximation (Akbar et al. in Chin J Aeronaut 26:1389–1397, 2013; Nadeem and Haq in J Aerosp Eng 28:04014061, 2012), Brownian motion, thermophoresis and Joule heating effects are also considered. To explore thermal characteristics, prescribed heat flux and prescribed mass flux boundary conditions are deployed. Governing flow problem consists of PDEs in the cylindrical form, which are converted into system of nonlinear ODEs by applying applicable similarity transforms. ODEs are tackled by RK–Fehlberg fourth–fifth-order numerical integration scheme with shooting algorithm. Impact of numerous involving physical parameters on flow features like temperature distribution, velocity distribution, Sherwood number, local Nusselt number and skin friction coefficient is shown through graphs and tables.

Unsteady Magnetoconvective Flow of Bionanofluid with Zero Mass Flux Boundary Condition

Induced magnetic field stagnation point flow for unsteady two-dimensional laminar forced convection of water based nanofluid containing microorganisms along a vertical plate has been investigated. We have incorporated zero mass flux boundary condition to get physically realistic results. The boundary layer equations with three independent variables are transformed into a system of ordinary differential equations by using appropriate similarity transformations. The derived equations are then solved numerically by using Maple which use the fourth-fifth order Runge-Kutta-Fehlberg algorithm to solve the system of similarity differential equations. The effects of the governing parameters on the dimensionless velocity, induced magnetic field, temperature, nanoparticle volume fraction, density of motile microorganisms, skin friction coefficient, local Nusselt number and motile density of microorganisms transfer rate are illustrated graphically and tabular form. It is found that the controlling parameters strongly affect the fluid flow and heat transfer characteristics. We compare our numerical results with published results for some limiting cases and found excellent agreement.

Numerical Study of Variable Fluid Properties and Magnetic Field on Convectively Heated Inclined Plate Utilizing Nanofluids

The present paper consists of numerical study of the heat transfer process in a two-dimensional steady natural convective flow of a nanofluid over an inclined plate. The considered transport model includes the effect of Brownian motion and thermophoresis and the zero nanoparticles mass flux boundary condition is also employed. The analysis accounts for temperature dependent viscosity, thermal conductivity and magnetic field. The nonlinear governing differential equations are solved numerically using the Finite Element Method. Results for the dimensionless velocity and temperature and nanoparticle volume fraction profiles are displayed graphically delineating the effect of various nanofluid flow characterizing parameters. It is observed that with the increase in thermal conductivity parameter, velocity and temperature in the respective boundary layers is increased. Whereas, magnetic filed have negative effect on fluid velocity and positive effect on temperature profile. Application of such kind of problem is prevalent in electronic packing and glassblowing.

Unsteady slip flow of Carreau nanofluid over a wedge with nonlinear radiation and new mass flux condition

This article addresses a numerical investigation for the unsteady 2D slip flow of Carreau nanofluid past a static and/or moving wedge with the nonlinear radiation. A zero nanoparticle mass flux and convective boundary conditions are implemented. Further, the most recently devised model for nanofluid is adopted that incorporates the effects of Brownian motion and thermophoresis. A set of suitable transformation is demonstrated to alter the nonlinear partial differential equations into nonlinear ordinary differential equations and then tackled numerically by employing bvp4c in Matlab package. The numerical computations for the wall heat flux (Nusselt number) and wall mass flux (Sherwood number) are also performed. Effects of several controlling parameters on the velocity, temperature and nanoparticles concentration are explored and discussed in detail. Our study reveals that the temperature and the associated thermal boundary layer thickness are enhancing function of the temperature ratio parameter for both shear thickening and shear thinning fluids. Moreover, it is noticed that the velocity in case of moving wedge is higher than static wedge.