Influence of homogeneous‐heterogeneous reactions and induced magnetic field on the nonlinear thermal convective radiative Jeffrey liquid flow with heat source/sink

An analysis has been carried out to investigate the effect of homogeneous‐heterogeneous reactions and induced magnetic field on the unsteady two‐dimensional incompressible nonlinear thermal convective velocity slip flow of a Jeffrey fluid in the presence of nonlinear thermal radiation and heat source/sink. We assumed that the flow is generated due to injection at the lower plate and suction at the upper plate. We obtained a numerical solution for the reduced nonlinear governing system of equations via the shooting technique with fourth‐order Runge‐Kutta integration. We plotted the graphs for various nondimensional parameters, like Deborah number, heat source/sink parameter, nonlinear convection parameter, nonlinear radiation parameter, magnetic Reynolds number, Strommer's number, velocity slip parameter, strengths of homogeneous, heterogeneous reaction parameters and skin friction over the nondimensional flow, temperature, concentration profiles and magnetic diffusivity fields. Also, we calculated the numerical values of boundary properties, such as the skin friction and heat transfer rate. We noticed that the temperature of the fluid is enhanced with the radiation parameter, whereas the concentration decreases with increase of the magnetic Reynolds number. The present results have good agreement with published work for the Newtonian case.     

Numerical study of activation energy and thermal radiation effects on Oldroyd-B nanofluid flow using the Cattaneo–Christov double diffusion model over a convectively heated stretching sheet

This paper investigates a theoretical model of a mixed convective Oldroyd-B nanofluid with thermal radiation and activation energy effects. A thorough analysis is done by employing the nonhomogeneous Buongiorno model in the presence of velocity slip and suction. The surface is porous in nature, and nanoparticle mass flux is maintained passively at the surface. The thermal and concentration equations are modeled with the Cattaneo–Christov theory of heat and mass flux, respectively. Proper transformations are utilized for the conversion of transport equations and boundary conditions. The similarity solution is obtained through a numerical approach by utilizing the Runge–Kutta–Fehlberg method and shooting technique. The vital outcomes of this study and the influence of controlling parameters on the flow field, temperature, and concentration profiles are discussed graphically and in a tabular manner. Furthermore, a detailed discussion is provided to explain the results physically. The velocity of the nanofluid increases when the porosity parameter is increased, and temperature decreases with increasing thermal relaxation parameter. The outcomes elucidate that the suction parameter, thermal radiation parameter, and thermal relaxation parameter are positively correlated with the heat transfer coefficient. The result of passive control of nanoparticles at the surface is that the Brownian motion parameter has no influence on the temperature of the Oldroyd-B nanofluid flow and rate of heat transfer at the surface.     

Approximate Analytical Solution of Stagnation Point Flow and Heat Transfer over an Exponential Stretching Sheet with Convective Boundary Condition

This study is concerned with the stagnation point flow and heat transfer over an exponential stretching sheet via an approximate analytical method known as optimal homotopy asymptotic method (OHAM). The governing partial differential equations are converted into ordinary nonlinear differential equations using similarity transformations available in the literature. The heat transfer problem is modeled using two‐point convective boundary condition. These equations are then solved using the OHAM approach. The effects of controlling parameters on the dimensionless velocity, temperature, friction factor, and heat transfer rate are analyzed and discussed through graphs and tables. It is found that the OHAM results match well with numerical results obtained by Runge–Kutta Fehlberg fourth‐fifth order method for different assigned values of parameters. The rate of heat transfer increases with the stretching parameter. It is also found that the stretching parameter reduces the hydrodynamic boundary layer thickness whereas the Prandtl number reduces the thermal boundary layer thickness.     

The effect of viscous dissipation and rarefaction on rectangular microchannel convective heat transfer

The effect of viscous dissipation and rarefaction on rectangular microchannel convective heat transfer rates, as given by the Nusselt number, is numerically evaluated subject to constant wall heat flux (H2) and constant wall temperature (T) thermal boundary conditions. Numerical results are obtained using a continuum based, three-dimensional, compressible, unsteady computational fluid dynamics algorithm with slip velocity and temperature jump boundary conditions applied to the momentum and energy equations, respectively. For the limiting case of parallel plate channels, analytic solutions for the thermally and hydrodynamically fully developed momentum and energy equations are derived, subject to both first- and second-order slip velocity and temperature jump boundary conditions, from which analytic Nusselt number solutions are then obtained. Excellent agreement between the analytical and numerical results verifies the accuracy of the numerical algorithm, which is then employed to obtain three-dimensional rectangular channel and thermally/hydrodynamically developing Nusselt numbers. Nusselt number data are presented as functions of Knudsen number, Brinkman number, Peclet number, momentum and thermal accommodation coefficients, and aspect ratio. Rarefaction and viscous dissipation effects are shown to significantly affect the convective heat transfer rate in the slip flow regime.     

Upshot of radiative rotating Prandtl fluid flow over a slippery surface embedded with variable species diffusivity and multiple convective boundary conditions

Here, modeling and computations are performed to explore the impact of variable molecular diffusivity, nonlinear thermal radiation, convective boundary conditions, momentum slip, and variable molecular diffusivity on Prandtl fluid past a stretching sheet. By using the compatible transformation, the partial differential equations regarding momentum, energy, and concentration are reformed into ordinary differential equations and furthermore, these equations are handled numerically via the shooting method. The behavior of intricate parameters that emerge during numerical simulation is displayed in the form of tables and graphs. These outcomes are supplemented with the information for the heat transfer rate and surface drag coefficients. It is perceived that an uplift in the temperature profile occurs by virtue of augmentation in the temperature convection parameter, and furthermore, mass fraction field escalates owing to an amplification in the chemical reaction coefficient.     

Second law analysis for chemically reacting natural convective second‐grade fluid flow between porous parallel plates with Hall and Ion slip

In this article, we performed the entropy generation of free convective chemically reacting second‐grade fluid confined between parallel plates in the influence of the Hall and Ion slip with heat and mass fluxes. Let there be a periodic suction/injection along with the plates, the governing flow field equations are reduced as a set of coupled nonlinear ordinary differential equations by using appropriate similarity transformations then solved numerically with shooting method based on Runge‐Kutta 4th order scheme. The results are analyzed for velocity in axial and radial directions, temperature distribution, concentration distribution, entropy generation number, Bejan number, mass and heat transfer rates with respect to distinct geometric, and fluid parameters and shown graphically and tables. It is observed that the entropy generation is enhanced with Prandtl number, whereas decreases with a second‐grade parameter, the effects of Hall and Ion slip parameters on velocity components, temperature and entropy generation number are the same. The entropy generation number the fluid is enhanced with the suction‐injection parameter whereas, the concentration of the fluid decreases with the increasing of chemical reaction parameter.     

Finite element analysis of couple stress micropolar nanofluid flow by non‐Fourier's law heat flux model past stretching surface

Numerical analysis has been done to investigate magnetohydrodynamics nonlinear convective flow of couple stress micropolar nanofluid with Catteneo‐Christov heat flux model past stretching surface with the effects of heat generation/absorption term, chemical reaction rate, first‐order slip, and convective boundary conditions. The coupled highly nonlinear differential equation governing the steady incompressible laminar flow has been solved by a powerful numerical technique called finite element method. The impacts of diverse parameters on linear velocity, angular velocity (microrotation), temperature, concentration profile, local skin friction coefficient, local wall couple stress, local Nusselt number, and Sherwood number are presented in graphical and tabular form. The result pointed out that the enhancement in material parameter β increases the velocity of the fluid while the couple stress parameter K has quite opposite effect. Heat and mass transfer rate of the fluid are enhanced by increasing material parameter while couple stress parameter shows the opposite influence. Moreover, heat and mass transfer rate are higher with the Catteneo‐Christov heat flux model than Fourier's law of heat conduction. The accuracy of the present method has been confirmed by comparing with previously published works.     

Heat transfer and buoyancy‐driven convective MHD flow of nanofluids impinging over a thin needle moving in a parallel stream influenced by Prandtl number

The current study focuses on investigating the influence of transverse magnetic field, variable viscosity, buoyancy, variable Prandtl number, viscous dissipation, Joulian dissipation, and heat generation on the flow of nanofluids over thin needle moving in parallel stream. The theory of nanofluids that includes the Buongiorno model featured by slip mechanism, such as Brownian motion and thermophoresis, has been implemented. Further, convective boundary condition and zero mass flux condition are considered. The nondimensionally developed boundary layer equations have been solved by Runge–Kutta–Fehlberg method with shooting technique for different values of parameters. The most relevant outcomes of the present study are that the augmented magnetic field strength, viscosity parameter, buoyancy ratio parameter, and the size of the needle undermine the flow velocity, establishing thicker velocity boundary layer while Richardson number and Brownian motion show opposite trend. Another most important outcome is that increase in the size of the needle, viscous dissipation, convective heating, and heat generation upsurges the fluid temperature, leading to improvement in thermal boundary layer. The effects of different natural parameters on wall shear stress and heat and mass transfer rates have been discussed.     

Second law analysis of pseudo-plastic nanofluid stream past a stretching sheet with a sliding consequence

The entropy generation (second law of thermodynamics) analysis of gyrotactic microorganism flow of power-law nanofluid with slip effects and combined effect of heat and mass transfer past a stretching sheet has been studied. The flow is maintained with Lorentz force and thermal radiation. The governing nonlinear partial differential equations are transformed into ordinary differential equations using similarity transformations. The impact of different physical parameters, such as convective bouncy parameter, power-law parameter, Brownian motion parameter, thermophoresis parameter, and slip parameter for velocity and temperature on the entropy generation number (Ns) are plotted graphically with the help of MATLAB built in bvp4c solver technique. Further, the uniqueness of this study is to find out the ratios of various irreversibilities due to thermal and mass diffusions, momentum diffusion, and microorganism over the total entropy generation rate. Our results showed that the power-law parameter and Brownian motion parameter influenced entropy generation positively. The slip parameter for velocity and temperature and the thermophoresis parameter helps to reduce the entropy production.     

Double-diffusive natural convective stream due to moving vertical plate with nonlinear thermal radiation and Newton's boundary constraint

This article investigates the heat and mass transmission of the double-diffusive convective stream over a moving vertical plate with nonlinear thermal radiation and newton boundary conditions. The governing partial differential equations of the stream, heat, and concentration profiles were transformed into a system of nonlinear ordinary differential equation by utilizing resemblance transformation. This system was then resolved numerically by applying the fourth order Runge-Kutta method with most efficient shooting technique. The effect of convection, buoyancy ratio, nonlinear thermal radiation, Prandtl number, Rayleigh number and Schmidt number are graphically scrutinized. The numerical results are obtained for velocity, temperature, and concentration profiles. It is found that when the velocity profile increases, heat and mass transfer rate decreases with an increase in the parametric value of buoyancy ratio parameter. It is found that the effect of nonlinear thermal radiation stabilizes the thermal boundary layer growth. The skin friction coefficient decreases with an increase in Prandtl number. However, the Nusselt number increases with an increase in the local convective heat transfer rate. The present results are very much promising, and further, there is a very good agreement of results when compared with earlier published results for some limiting conditions.     

Cattaneo–Christov double-diffusion model with Stefan blowing effect on copper–water nanofluid flow over a stretching surface

We often encounter many processes where the cooling rate is a key factor in deciding the features of a desired product. Due to increasing demands of controlled cooling systems, an effort is made to theoretically study the effect of volume fraction on mixed convective Cu–water nanofluid flow over a stretching surface with activation energy and thermal radiation. The nonlinear dynamical system is simplified using apt similarity variables and the obtained ordinary differential equations are dealt numerically using Runge–Kutta–Fehlberg method and shooting scheme. The thermal and solutal equations are modeled considering Cattaneo–Christov double-diffusion model. The flow problem is studied considering velocity slip and zero mass flux state at the surface. As a novelty, the present case considers the blowing effect at the surface to study massive species transport during nanofluid flow with Cattaneo–Christov double-diffusion model. The results show that an increase in strength of thermal radiation increases temperature and buoyancy ratio parameter, thereby escalating the skin friction coefficient. When thermal relaxation parameter changes from 0.001 to 0.005, heat transfer coefficient improves by 24.36%. Furthermore, with the change in value of the blowing parameter from 0.1 to 0.1015, the maximum value concentration of nanoparticles that is attained during the flow is increased by 7.15%.     

Heat Transfer Phenomena of Assemblages of Smooth Slip Spheres in Newtonian Fluids

Combined effects of slip velocity and volume fraction of slip spheres on the heat transfer characteristics of multiple slip spheres are numerically investigated within the framework of a free surface cell model combined with a linear slip velocity along the surface of the slip spheres. The governing conservation equations of the mass, momentum, and energy are solved by a segregated approach using a simplified marker and cell algorithm implemented on a staggered grid arrangement in spherical coordinates. The convection and diffusion terms of conservation equations are discretized using quadratic upstream interpolation for convective kinematics and second‐order central differencing schemes, respectively. Prior to obtaining new results, this numerical solver is validated by comparison of present results with the existing literature values. Further new results are obtained for a range of conditions as; Reynolds number, Re: 0.1–200; Prandtl number, Pr: 1–100; volume fraction of slip spheres, Φ: 0.1–0.5 and slip parameter, λ: 0.01–100. The effects of these dimensionless parameters on isotherm contours and local and average Nusselt numbers are thoroughly delineated. Finally, a new empirical correlation for the average Nusselt number of multiple smooth slip spheres is proposed on the basis of present numerical results.     

Radiation and slip effects on MHD point flow of nanofluid towards a stretching sheet with melting heat transfer

In the present paper, the melting heat transfer of a nanofluid over a stretching sheet is investigated. Magnetohydrodynamic stagnation point flow with thermal radiation and slip effects is considered for this study. The governing model of the flow is solved by Runge–Kutta fourth-order method using appropriate similarity transformations. Temperature and velocity fields are presented for various flow pertinent parameters. Nondimensional physical parameters such as Prandtl number, radiation parameter, Brownian motion parameter, Lewis number, thermophoresis parameter, magnetic parameter, and melting parameter on fluid velocity, heat, concentration, skin friction, Sherwood number, and Nusselt number are presented graphically and discussed numerically. Heat transfer rate can be increased by increasing slip, melting, or radiation parameter. Mass transfer increases for greater values of melting parameter or slip parameter while radiation parameter shows the opposite impact on mass transfer.     

Spectral relaxation method analysis of Casson nanofluid flow over stretching cylinder with variable thermal conductivity and Cattaneo–Christov heat flux model

This article deals with non‐Newtonian Casson nanofluid flow and heat transfer over stretching cylinder in a porous medium. The mode of heat transfer is presented considering temperature‐dependent thermal conductivity by integrating the Cattaneo–Christov heat flux and mass flux models. Boundary layer theory is applied to develop the governing partial differential equations from the physical problem. Employing proper similarity transformation, the governing boundary layer equations are transformed into dimensionless system of nonlinear ordinary differential equations. Then, the resulting problem is numerically solved by means of spectral relaxation method. The convergence analysis of the proposed numerical scheme is presented via a table, which confirms almost the 10th order of approximation is enough for the convergence of the skin friction coefficient, local heat transfer, and mass transfer rates. The effects of various embedded parameters on velocity, temperature, and concentration profiles as well as skin friction coefficient, surface heat and mass transfer rates are examined through graphs and tables. The findings reveal that the growth of permeability and velocity slip parameters appears to decelerate the velocity distributions of fluid. Thermal boundary layer thickness tends to develop with greater values of permeability and Brownian motion parameters. Also, the local heat transfer rate is less with Fourier's law of heat conduction than Cattaneo–Christov heat flux model. Furthermore, the validity and accuracy of the present result is checked with the available literature, and very sound agreement has been obtained.     

Nonlinear convective boundary layer flow of micropolar‐couple stress nanofluids past permeable stretching sheet using Cattaneo‐Christov heat and mass flux model

The present work aims to examine the effects of viscous dissipation and unsteadiness parameters on nonlinear convective laminar boundary layer flow of micropolar‐couple stress nanofluid past a permeable stretching sheet with non‐Fourier heat flux model in the presence of suction/injection variable. The unsteadiness in the flow, temperature, and concentration profile is caused by the time‐dependence of the stretching velocity, surface temperature, and surface concentration of the boundary layer flow. Similarity transformation is applied to transform the time‐dependent boundary layer flow equations into the corresponding highly nonlinear coupled ordinary differential equations with appropriate boundary conditions. The robust numerical technique called Galerkin finite element method is used to solve the obtained dimensionless governing equations of the flow. The effects of Eckert number, unsteadiness parameter, suction/injection parameter, mixed convection parameter, material parameter, Schmidt number, and couple stress parameter on linear velocity, angular velocity, temperature, concentration, local skin friction coefficient, local wall couple stress, local Nusselt number, and local Sherwood number is analyzed with the help of graphical and tabular form. Under special conditions, the present result is compared with the existing literature and revealed good agreement. Our result shows that as unsteadiness parameter boost, both heat and mass transfer rate rises. The present study has a significant application in material processing technology.     

Hydromagnetic non‐Newtonian nanofluid transport phenomena from an isothermal vertical cone with partial slip: Aerospace nanomaterial enrobing simulation

In this article, the combined magneto‐hydrodynamic heat, momentum, and mass (species) transfer in external boundary layer flow of Casson nanofluid from a vertical cone surface with convective conditions under an applied magnetic field is studied theoretically. The effects of Brownian motion and thermophoresis are incorporated in the model in the presence of both heat and nanoparticle mass transfer convective conditions. The governing partial differential equations (PDEs) are transformed into highly nonlinear, coupled, multidegree, nonsimilar PDEs consisting of the momentum, energy, and concentration equations via appropriate nonsimilarity transformations. These transformed conservation equations are solved subject to appropriate boundary conditions with a second‐order, accurate finite difference method of the implicit type. The influences of the emerging parameters, that is, magnetic parameter (M), Casson fluid parameter (β), Brownian motion parameter (Nb), thermophoresis parameter (Nt), Lewis number (Le), Prandtl number (Pr), velocity slip (S_f) and thermal slip (S_T) on velocity, temperature, and nanoparticle concentration distributions is illustrated graphically and interpreted at length. Validation of solutions with a Nakamura tridiagonal method has been included. The study is relevant to enrobing processes for electrically conductive nanomaterials, of potential use in aerospace and other industries.     

Impact of Soret and Dufour on MHD mixed convective flow of non-Newtonian fluid over a coagulated surface

In the present study, we investigated the steady, two-dimensional mixed convective stagnation point flow of an electrically conducting micropolar fluid due to stretching of a variable thicked surface in the attendance of viscous dissipation. The flow is incompressible and laminar. The combined heat and mass transfer features are investigated. Convective and diffusion conditions are considered. The nonlinear thermal radiation, thermo-diffusion, and diffusion thermal effects are considered. The governing partial differential equations are converted to ordinary differential equations by using the appropriate similarity transformations. The obtained nonlinear and coupled ordinary differential equations are elucidated numerically using the fourth-order Runge–Kutta based shooting technique. The influence of various nondimensional parameters on the flow field like velocity, microrotation, temperature, and concentration is examined with the assistance of graphs. Results indicate that the Dufour number has a proclivity to increase the distributions of concentration and temperature correspondingly. Also, fluid temperature and concentration enhance for increasing values of the wall thickness parameter.     

Natural convection MHD mass transfer through an infinite vertical porous plate in the existence of radiation embedded in porous medium

This research focuses on studying the effects of heat and mass transfer convective flow passing through an infinite vertical plate embedded in porous media under radiation and chemical reaction with constant heat and mass flux. A magnetic field of strength is functional throughout the fluid region. The novelty of the present work is to examine the heat and mass transfer magnetohydrodynamics flow in the presence of thermal radiation. The equations governing the flow, heat and mass transfer are solved analytically using the perturbation technique. Expressions for velocity, temperature, concentration, skin-friction, Nusselt, and Sherwood numbers are obtained. The influence of physical parameters on the flow domain is described graphically and in tabular form. It is found that increase in radiation parameter reduces the velocity and temperature. Moreover, internal friction of the plate decreased with increasing values of radiation parameter.     

Lie Group Analysis and Numerical Solutions for Magnetoconvective Slip Flow along a Moving Chemically Reacting Radiating Plate in Porous Media with Variable Mass Diffusivity

A numerical investigation of magnetoconvective boundary layer slip flow along a nonisothermal continuously moving permeable nonlinear radiating plate in Darcian porous media is reported. The concentration dependent mass diffusivity, viscous dissipation, Joule heating, and chemical reaction are taken into account. A Lie group of transformation is applied to the governing transport equations and boundary condition to find the corresponding similarity equations. Furthermore, the similarity equations with the relevant boundary conditions are solved numerically using the Runge‐Kutta‐Fehlberg fourth‐fifth order numerical method. Numerical results for the dimensionless velocity, temperature, and concentration distributions as well as friction factor, local Nusselt, and local Sherwood numbers are discussed for various controlling parameters. It is found that that the dimensionless concentration increases whilst the rate of mass transfer decreases with the mass diffusivity parameter. An excellent correlation is found between the present results and published results. The study finds applications in the polymer industry and metallurgy.     

Numerical analysis of MHD nanofluid flow over a wedge,including effects of viscous dissipation and heat generation/absorption,using Buongiorno model

The key purpose of this article is to examine magnetohydrodynamics flow, generative/absorptive heat, and mass transfer of nanofluid flow past a wedge in the presence of viscous dissipation through a porous medium. The investigation is completely theoretical, and the present model expresses the influence of Brownian motion and thermophoresis using the nanofluid Buongiorno model. The fundamental model of partial differential equations is reframed into the structure of ordinary differential equations implementing the nondimensional similarity transformation, which are tackled through the fourth–fifth-order Runge–Kutta–Fehlberg algorithm together with the shooting scheme. The analysis of sundry nondimensional controlling parameters, such as magnetic parameter, Eckert number, heat generation/absorption parameter, porosity parameter, Brownian motion parameter, and thermophoresis parameter on velocity, temperature, and concentration profiles are discussed graphically. The effects of the physical factors on the rate of momentum and heat and mass transfer are also determined with appropriate analysis in terms of skin friction, Nusselt number, and Sherwood number. The outcomes illustrate that the local Nusselt number and local Sherwood number are reduced for higher values of the thermophoresis parameter. Besides, it is found that higher estimations of heat generation/absorption and viscous dissipation parameters increase temperature. Moreover, it is found that the temperature profile increases with the involvement of the Brownian motion parameter, while an opposite trend is observed in the concentration profile. A comparison is also provided for limiting cases to authenticate our obtained results.