Numerical study of heat transfer and viscous flow in a dual rotating extendable disk system with a non‐Fourier heat flux model

Nonlinear, steady‐state, viscous flow, and heat transfer between two stretchable rotating disks spinning at dissimilar velocities are studied with a non‐Fourier heat flux model. A nondeformable porous medium is intercalated between the disks and the Darcy model is used to simulate matrix impedance. The conservation equations are formulated in a cylindrical coordinate system and via the von Karman transformations are rendered into a system of coupled, nonlinear ordinary differential equations. The emerging boundary value problem is controlled by number of dimensionless parameters, that is, Prandtl number, upper disk stretching, lower disk stretching, permeability, non‐Fourier thermal relaxation, and relative rotation rate parameters. A perturbation solution is developed and the impact of selected parameters on radial and tangential velocity components, temperature, pressure, lower disk radial, and tangential skin friction components and surface heat transfer rate are visualized graphically. Validation of solutions with the homotopy analysis method is included. Extensive interpretation of the results is presented which are relevant to rotating disk bioreactors in chemical engineering.     

Heat and mass transfer analysis of MWCNT‐kerosene nanofluid flow over a wedge with thermal radiation

A comparison between the unsteady and steady magnetohydrodynamics Tiwari‐Das model Williamson nanofluid flow through a wedge occupied by carbon nanotubes of multiwalled type nanoparticles and kerosene as base fluid is presented in this analysis. A suitable similarity variable technique is adopted to transmute the governing partial differential equations into a set of nonlinear ordinary differential equations (ODEs). To solve these ODEs together along with boundary conditions, we have utilized finite element analysis. The behavior of concentration, temperature, and velocity sketches for diverse values of the pertinent parameters is plotted through graphs. The impact on the above parameters on the rates of velocity, heat, and concentration is also evaluated and depicted through tables. It is noted that as the values of nanoparticle volume fraction parameter rises, the rates of temperature increase in both the unsteady and steady cases.     

Influence of carbon nanotubes on heat transfer in MHD nanofluid flow over a stretchable rotating disk: A numerical study

The transfer of heat is an important phenomenon in the several areas due to its numerous applications in industries. Several fluids like water, ethylene glycol and oil, and so on have very‐low thermal conductivities due to which the transfer of heat in these fluids become very low. To enhance heat transfer rate, carbon nanotubes (CNTs) including single‐walled CNTs and multi‐walled CNTs are suspended into base fluids, this mixture is known as nanofluid. The aim of this study is to examine the heat transfer rate of nanofluid in the presence of CNTs over a stretchable rotating disk. The mathematical model, developed by Tiwari and Das, is used and solved numerically by using the shooting method. The impacts of governing constraints on the dimensionless velocities, temperature, skin friction, and Nusselt number are investigated. It is noted that heat transfer rate increases by enhancing the concentration of CNTs into base fluids. The numerical results show that the solid volume fraction of the CNTs augment heat transfer rate more in ethylene glycol as compared with water.     

On the mixed convective carbon nanotube flow over a convectively heated curved surface

In this article, mixed convective boundary layer stream of nanofluid flow with carbon nanotube as nanoparticles and transmission of heat over a coiled stretched surface are studied. The influence of magnetic orientation and velocity slip is also encountered in this problem. Two classes of carbon nanotubes, SWCNT and MWCNT, are considered as nanoparticles and water as a pure liquid. The foremost leading partial differential equations (PDEs) are formulated through curvilinear coordinate system subjected to proper boundary conditions. To simplify this nonlinear PDE‐associated model, we have employed a compatible similarity conversion and acquired the nonlinear dimensionless ordinary differential equations (ODEs). To determine the requisite numerical solution of the transformed problem, a shooting procedure embedded with RK‐4 technique has been applied. Various pictorial attempts have been initiated against different parametric inputs to reveal the hydrothermal scenario. Some physical quantities like skin friction and Nusselt numbers are calculated to investigate flow distribution inside the preferred system. A comparison with earlier research depicts parallel outcomes. Results assured that velocity is a cumulative function with positive increment of curvature parameter, but an opposite scenario is shown for temperature for both type of nanofluids. The amount of heat transition has been declined against the improvement of the magnetic parameter.     

Unsteady flow and heat transfer of a ferrofluid between two rotating,vertical moving and stretching disks

Unsteady flow and heat transfer of a magnetic fluid between two rotating disks is investigated. Both the disks are stretchable and the lower disk moves in the vertical direction. A new approach of similarity transformation is adopted to transform the equation of continuity, momentum, and the energy equation into ordinary nonlinear coupled differential equations. The numerical solution of the converted nonlinear differential equations is obtained using the finite element method. The effects of magnetization force, rotational viscosity, Prandtl number, and Eckert number on the velocity and temperature distributions are studied. The impact of stretching, movement, and rotation of the disk is also considered in this computational study. The skin friction coefficients and heat transfer rate on the lower disk for different physical parameters are calculated. Different types of motion of the disks and the magnetization force are crucial aspects in the stress distribution and heat transfer rate near the lower disk.     

Thermal transportation analysis of nanoliquid squeezed flow past a sensor surface with MCWCNT and SWCNT

This article considers the flow and heat transfer of a single and multi‐walled carbon nanotube over a sensor surface. For this persistence, a mathematical forming is established with the aspects of thermal radiation. In addition, the stimuli of magnetic properties and variable thermal conductivity are presented. By means of noteworthy conversions, nonlinear PDEs are altered into nonlinear ODEs and elucidated via a numerical approach in virtue of the Runge‐Kutta fourth order method scheme. The repercussion of countless variables of flow and energy transfer characteristics are portrayed and conferred. These upshots portray that the enhancement of heat is bounteous in a single‐wall nanotube when compared with multiwall nanotubes. Further, the velocity field is contracted for enhancing the values of .     

Computational study on heat transfer and MHD-electrified flow of fractional Maxwell nanofluids suspended with SWCNT and MWCNT

Recent developments in fluid dynamics have been focusing on nanofluids, which preserve significant thermal conductivity properties and magnify heat transport in fluids. Classical nanofluid studies are generally confined to models described by partial differential equations of an integer order, where the memory effect and hereditary properties of materials are neglected. To overcome these downsides, the present work focuses on studying nanofluids with fractional derivatives formed by differential equations with Caputo time derivatives that provide memory effect on nanofluid characteristics. Further, heat transfer enhancement and boundary layer flow of fractional Maxwell nanofluid with single-wall and multiple walls carbon nanotubes are investigated. The Maxwell nanofluid saturates the porous medium. Also, buoyancy, magnetic, electric, and heating effects are considered. Governing continuity, momentum, and energy equations involving Caputo time-fractional derivatives reduced nondimensional forms using suitable dimensionless quantities. Numerical solutions for arising nonlinear problems are developed using finite difference approximation combined with L1 algorithm. The influence of involved physical parameters on flow and heat transfer characteristics is analyzed and depicted graphically. Our simulations found out that surface drag of Maxwell nanofluid with single-walled carbon nanotubes dominates nanofluids with multiple walls carbon nanotubes, but the reverse trend is noticed for larger Grashof number values.     

Thin film flow of the water‐based carbon nanotubes hybrid nanofluid under the magnetic effects

In this article, the effects of magnetic field versus the thin liquid film water‐based ferrum oxide (Fe₃O₄) and carbon nanotubes (CNTs) nanofluids have been studied through stretching cylinder. The iron oxide and CNTs (single‐wall [SWCNTs] or multi‐wall [MWCNTs]) have been used as nanoparticles in carrier fluid water (H₂O). To the flow field, magnetic effects are applied vertically. The modeled system of partial differential equations are transformed to nonlinear ordinary differential equations by selecting variables. The analytic solution has been obtained through homotopy analysis method. The obtained results are further compared with the numerical ND‐solve method. The embedded constraints impacts are focused on pressure distribution, velocity profile, heat transfer, Nusselt number, and Skin friction through graphical illustration and tables. The dispersion of Fe₃O₄ and CNTs in base fluid significantly enhanced the mechanism of heat transfer. Moreover, from the results, it has been observed that the MWCNTs have a greater impact on heat transfer, velocity, and pressure profile.     

Analysis of thermal behavior of moving longitudinal porous fin wetted with water‐based SWCNTs and MWCNTs

In this study, the consequences of natural convection and radiation on longitudinal porous fin are scrutinized numerically. Here, the fin is moving and is wetted with nanoliquid containing single‐walled carbon nanotubes and multiwalled carbon nanotubes. Darcy's model is employed in modeling the present physical phenomenon. The derived steady‐state energy equation has been solved numerically via the Runge‐Kutta‐Fehlberg fourth‐fifth‐order method. The impact of significant parameters on the thermal performance of the wet fin has been illustrated through graphs for both types of carbon nanotubes. The comprehensive study of carbon nanotubes explores that better heat transfer from a wet fin can be achieved in the case of multiwalled carbon nanotubes.     

Exact solutions of micropolar SWCNTs nanofluid with heat transfer

The present study is aimed to analyze the unsteady micropolar nanofluid flow passing over an oscillating infinite vertical plate. The flow is affected by thermal radiation and Newtonian heating. Single‐walled carbon nanotubes (SWCNTs) are added to enrich the thermal properties of the micropolar fluid. Kerosene is taken as the base liquid to enhance heat transfer. By using dimensional analysis, the governing equations for temperature, velocity, and microrotation are reduced to dimensionless form and after that, these equations have been solved by applying Laplace transform method to get the exact solutions. Finally, we have presented the effects of material and flow parameters and illustrated graphically by the Mathcad software. We found that microrotation, temperature, and velocity are decreasing functions of Prandtl number but have shown increasing behavior for Grashof number. Furthermore, we found that SWCNTs‐water‐based nanofluid has a comparatively higher heat transfer rate than SWCNTs‐kerosene and SWCNTs‐engine oil‐based nanofluids.     

Entropy optimized CNTs based Darcy-Forchheimer nanomaterial flow between two stretchable rotating disks

Here entropy optimized CNTs based flow of viscous liquid is addressed between two stretchable surfaces of disks. Both upper and lower disks stretch and rotate with different rates and angular frequencies. Carbon nanotubes (single and multi-walls) are considered as a nanoparticles and water as continuous phase liquid. Xue model is utilized in the mathematical modeling for the transport of nanoparticles. Energy expression is developed through first law of thermodynamics and discussed in the presence of viscous dissipation. Main attention is given to the modeling of entropy generation subject to CNTs nanoparticles. Total entropy rate is calculated. Average residual error is calculated through implementation of optimal homotopy analysis method. Flow parameters are graphically discussed for both single and multi-walls carbon nanotubes. Furthermore, engineering quantities like skin friction and Nusselt number are numerical calculated and discussed through Tables. From obtained outcomes it is examined that entropy rate boosts up versus larger Brinkman number and nanoparticles volume friction. No such attempt is yet done by the researchers on entropy optimized Darcy-Forchheimer CNTs based nanomaterial flow between two rotating disks. The obtained outcomes are compared with published literature and found good agreement.     

MHD boundary layer flow of SWCNT‐water and MWCNT‐water nanofluid over a vertical cone with heat generation/absorption

The analysis is carried out to investigate the magneto hydro dynamics (MHD) boundary layer flow, heat and mass transfer characteristics of two carbon nanotubes, namely, single‐wall carbon nanotubes (SWCNTs) and multiwall carbon nanotubes (MWCNTs), with water as the base fluid by taking thermal radiation and chemical reaction into consideration. Suitable similarity conversions are employed to reduce nonlinear partial differential equations into the system of ordinary differential equations, and these equations together with boundary conditions are solved numerically using the finite element method. Velocity, temperature, and concentration distributions as well as skin friction coefficient, Nusselt number, and Sherwood number for diverse values of influencing parameters are examined in detail, and the results are displayed graphically and in tabular form. It is found that the rate of heat transfer is remarkably higher in water‐based MWCNTs than the SWCNTs as the value of the nanoparticle volume fraction parameter rises in the boundary layer regime.     

Second law analysis in radiative mixed convective squeezing flow of Casson fluid between parallel disks with Soret and Dufour effects

The present communication deals the entropy generation by cause of heat and mass transform in an unsteady mixed convective radiative squeezing flow of a Casson fluid confined between two parallel disks in the presence of diffusion‐thermo and thermal‐diffusion effects and temperature jump. The lower disk is taken to be porous and the upper one is impermeable. The governing PDE is converted as nonlinear ordinary differential equations (ODE) by using well‐established similarity transformations; then, the reduced nonlinear ODE are solved by shooting method with Runge‐Kutta fourth‐order approach. The influence of distinct nondimensional fluid and geometric‐related parameters on the velocity profiles, temperature, concentration, entropy generation number, and Bejan number are studied in detail and represented in the form of graphs. The entropy of the Casson fluid is increased with the Eckert number, whereas the concentration profile is decreased by squeezing Reynolds number. The current results are correlated with existing results for the viscous case and found to be in better agreement.     

Molecular dynamics of the generation process of double‐walled carbon nanotubes from peapods

The generation process of a double‐walled carbon nanotube (DWNT) from a “peapod” was studied by classical molecular dynamics simulation. Starting from a peapod structure, defined by five C₆₀ molecules inside a (10,10) single‐walled carbon nanotube (SWNT), polymerized fullerenes, a peanut‐like structure and an almost nanotube‐like structure were obtained under suitable conditions of temperature control. The mean distance between the two layers of the DWNT agreed with an experimental report that it is larger than the interlayer spacing found in multi‐walled carbon nanotubes (MWNTs). In addition, the chirality dependence of the potential energy of a DWNT on the relative chirality of its constituent tubes was examined using a 6‐12 Lennard‐Jones potential. It was found that the potential energy depends only on the distance between the two layers, not on the relative chiralities. © 2006 Wiley Periodicals, Inc. Heat Trans Asian Res, 35(4): 254–264, 2006; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20115     

An investigation on the forced convection heat transfer in the gap of two rotating disks with laminar inflow

This study presents forced convection in the gap between two rotating disks with the laminar radial inward flow. The disk surfaces are held at a constant temperature different from the temperature of the fluid flowing. The disks' surfaces may also receive a heat flux. The temperature of the fluid flowing in the gap is predicted by solving the coupled equations of momentum, energy, and continuity in cylindrical coordinate numerically. The finite difference method is used to discretize the energy equation into nonlinear algebraic equations. The tridiagonal matrix algorithm is employed to solve the resulting algebraic equations. Predominantly, throughflow Reynolds number, rotational Reynolds number, gap ratio, speed ratio, and Peclet number are the parameters that affect the temperature distribution for the fixed disk temperature and for the heat flux boundary conditions. The Nusselt number compares reasonably well with the numerical results of other investigators. The heat flow into the fluid is higher for corotating disks than for contrarotating disks for both constant temperatures as well as heat flux boundary conditions. This is the first investigation that predicts temperature distribution due to forced convection in the gap of two rotating disks with laminar inflow.     

Radiative CNT-based hybrid magneto-nanoliquid flow over an extending curved surface with slippage and convective heating

This study concentrates on the hydrothermal prominence of a mixed convective flow of a hybrid nanoliquid over a convectively heated extending curved surface under the influence of a uniform transverse magnetic field. Two types of carbon nanotubes (CNTs), namely single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs), and magnetite nanoparticles are dispersed in the host liquid (water) to simulate the hybrid nanoliquid flow model. First- and second-order velocity slip conditions and nonlinear radiative heat flux are incorporated in this model. First, the system of governing partial differential equations is changed into nonlinear ordinary differential equations through the utilization of appropriate transformations and computed numerically via MATLAB built-in function bvp4c based on the three-stage Lobatto IIIA technique. The consequences of physical and geometrical parameters pertinent to this analysis on the dimensionless physical quantities of interest are deliberated using requisite graphs and tables. Our simulation communicates that the first-order velocity slip parameter decreases the velocity profile, whereas the second-order velocity slip parameter is found to be augmented. The suspension of CNTs in the magnetite nanoliquid improves the local surface drag force but diminishes the local heat flux. Moreover, it is examined that SWCNTs have greater impacts than MWCNTs.     

Silver–ethylene glycol and copper–ethylene glycol based thermally radiative nanofluid characteristics between two rotating stretchable disks with modified Fourier heat flux

In the present analysis, our aim is to investigate the mass and heat transport of silver (Ag)–ethylene glycol (EG) and copper (Cu)–EG-based nanofluids between two rotating stretchable disks under the convective boundary conditions. We have also incorporated Cattaneo–Christov heat flux, thermal radiation, and chemical reaction in the fluid flow. The system of coupled partial differential equations is transformed into ordinary differential equations by using similarity transformations. The finite element method has been accomplished to find numerical solutions to transformed equations. The behavior of radial and tangential velocity, temperature fields, and concentration fields influenced by the various parameters are sketched through graphs. The local skin friction coefficient, Nusselt number, and Sherwood number are also calculated for the pertinent parameters and displayed the results through tables. It is perceived that velocity sketches of both nanoliquids degenerate with larger values of thermal relaxation parameters. Also, the values local Nusselt number of both Ag–EG, and Cu–EG based Cattaneo–Christov nanofluid intensifies with improving values of stretching parameter at the lower disk, whereas, it impedes at the upper disk.     

Mixed Convection Over a Vertical Plate in a Doubly Stratified Fluid‐Saturated Non‐Darcy Porous Medium with Cross‐Diffusion Effects

The present article investigates the influence of Dufour and Soret effects on mixed convection heat and mass transfer over a vertical plate in a doubly stratified fluid‐saturated porous medium. The plate is maintained at a uniform and constant wall heat and mass fluxes. The Darcy–Forchheimer model is employed to describe the flow in porous medium. The nonlinear governing equations and their associated boundary conditions are initially transformed into dimensionless forms. The resulting system of nonlinear partial differential equations is then solved numerically by the Keller‐box method. The variation of the dimensionless velocity, temperature, concentration, heat, and mass transfer rates for different values of governing parameters involved in the problem are analyzed and presented graphically. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21114     

Heat transfer analysis of CNTs-water nanofluid flow between nonparallel plates: Approximate solutions

Approximate solutions are given to investigate the problem of heat transfer and flow of a nanofluid between two nonparallel plates. Two different types of carbon nanotube (CNT) nanoparticles, namely functionalized single-walled carbon nanotubes and functionalized multi-walled carbon nanotubes are considered with water as the base fluid. The chemical functionalization is attributed to the attachment of the covalent bond of functional groups onto carbon nanotubes. Using the appropriate similarity variables, the model partial differential equations are transformed into ordinary differential equations and tackled analytically via the homotopy perturbation method. Comparison of the obtained results is carried out with those obtained by the previously published in the literature as well as the Runge–Kutta–Fehlberg method and an excellent agreement is achieved. The effects of various emerging parameters on nanofluid velocity, temperature, skin friction coefficient, and Nusselt number are illustrated graphically and comprehensively discussed.     

Effect of heat source/sink on MHD flow due to porous rotating disk with variable thickness in the presence of cross diffusion

An analysis of steady magnetohydrodynamic axisymmetric flow of a viscous incompressible electrically conducting fluid due to porous rotating disk with variable thickness in the presence of heat source/sink is presented. Soret and Dufour effects (cross‐diffusion) are also considered. The governing partial differential equations are transformed into a system of nonlinear ordinary differential equations. The homotopy analysis method is used to solve the resulting coupled nonlinear equations under appropriate transformed boundary conditions. A parametric study of the physical parameters is made and results are presented through graphs and tables. The results indicate that the thermal boundary layer is thicker for the flow problems having a heat source when compared with that of the problems without a heat source. It is also found that thickness of the disk is having a high impact on fluid velocity, temperature, and concentration.