Analysis of fractional derivatives in thermal and solutal transfer equations of second-grade non-Newtonian fluids: A numerical study

A numerical analysis is carried out using Atangana–Baleanu and Caputo–Fabrizio time-fractional derivatives to study the mixed convective unsteady flow of a second-grade fluid past an infinite vertical porous plate under the influence of a uniform transverse magnetic field. As finding the exact solutions of fluid equations presents huge difficulties due to the vagueness or uncertainty associated with the fluid parameters, fuzzy theoretic concepts are used rather than the classical crisp theoretic ones. Governing partial differential equations are made dimensionless and are then subject to fuzzification. The finite-difference scheme is used to discretize the equations, and hence suitable programming codes are developed in PYTHON for AB and CF fractional derivatives. The results are obtained and plotted graphically. Interpretations based on these physical parameters imply that both AB and CF methods agreed well.     

Fundamental solution for a line source of heat in the fractional order theory of thermoelasticity using the new Caputo definition

Abstract

The new Caputo Fabrizio fractional differential operator is used to investigate a problem in the fractional order theory of thermoelasticity. The problem concerns an infinite elastic space under the effect of a continuous line source of heat. The problem is solved using asymptotic expansions valid for short times. Laplace and Hankel transforms are used to solve the problem. A brief study to the nature of propagation of waves is introduced. Graphical results are presented and discussed.     

A Ginzburg–Landau model for material aging depending on temperature

ABSTRACT

We consider a model describing the behavior of a body subject to aging and fatigue. These phenomena are supposed to be affected by both mechanical and thermal effects. The material is assumed to be viscoelastic where the stress–strain relation is based on a new fractional derivative proposed in Caputo and Fabrizio. The order of derivative is regarded as a new variable whose evolution is ruled by a Ginzburg–Landau equation. The model also includes an evolutive equation for the temperature deducing from the first law of thermodynamics. In this article, thermodynamic compatibility is shown and some numerical simulations are performed.     

Fractional order simulation for unsteady MHD nanofluid flow in porous medium with Soret and heat generation effects

In this paper, unsteady magnetohydrodynamics nanofluid flow with thermo-diffusion and heat generation effects is studied. The fluid flow at the plate is considered exponentially accelerated through a porous medium. The governing system of equations is made dimensionless with the help of similarity transformation. A Caputo–Fabrizio fractional-order derivative is employed to generalize the momentum, energy, and concentration equations, and the exact expression is obtained using Laplace transformation techniques. To realize the physics of the problem, numerical results of velocity, temperature, and concentration profiles are obtained and presented through graphs. Also, the numerical values of the Nusselt number and Sherwood number are obtained and compared which strongly agree with the previous studies. From the results, it is concluded that velocity distribution decline by improving the value of the chemical reaction and magnetic field while the reverse trend is observed for volume fraction and micropolar parameter. It is also seen that the heat transfer process improves with heat generation and thermal radiation whereas, mass transfer declines with the chemical reaction parameter.     

Thermo-electrokinetic rotating non-Newtonian hybrid nanofluid flow from an accelerating vertical surface

This paper explores the combined effects of Coriolis force and electric force on the rotating boundary layer flow and heat transfer in a viscoplastic hybrid nanofluid from a vertical exponentially accelerated plate. The hybrid nanofluid comprises two different types of metallic nanoparticles, namely silver (Ag) and magnesium oxide (MgO) suspended in an aqueous base fluid. The Casson model is deployed for non-Newtonian effects. An empirical model is implemented to determine the thermal conductivity of the hybrid nanofluid. Rosseland's radiative diffusion flux model is also utilized. An axial electrical field is considered and the Poisson–Boltzmann equation is linearized via the Debye–Hückel approach. The resulting coupled differential equations subject to prescribed boundary conditions are solved with Laplace transforms. Numerical evaluation of solutions is achieved via MATLAB symbolic software. A parametric study of the impact of key parameters on axial velocity, transverse velocity, nanoparticle temperature and Nusselt number is conducted for both the hybrid (Ag–MgO)–water nanofluid and also unitary (Ag)–water nanofluid. With increasing volume fraction of silver nanoparticles, there is a reduction in both axial velocity and temperatures, whereas there is a distinct elevation in transverse velocity for both unitary and hybrid nanofluids. Elevation in the heat absorption parameter strongly decreases axial velocity, whereas it enhances transverse velocity. Increasing the radiation parameter strongly boosts temperatures. Increasing the heat absorption parameter significantly accelerates the transverse flow. Negative values of Helmholtz–Smoluchowski velocity decelerate the axial flow whereas positive values accelerate it; the opposite behavior is observed for transverse velocity. Increasing Taylor number significantly damps both the axial (primary) and transversal (secondary) flow. Increasing thermal Grashof number strongly enhances the axial flow but damps the transverse flow. The unitary nanofluid achieves higher Nusselt numbers than the hybrid nanofluid but these are decreased with greater radiative effect (due to greater heat transport away from the plate surface), Prandtl number and heat absorption. Nusselt number is significantly reduced with greater time progression and values are consistently higher for the unitary nanofluid compared with hybrid nanofluid. The computations provide insight into more complex electrokinetic rheological nanoscale flows of relevance to biomedical rotary electro-osmotic separation devices.     

Zero‐mass flux and Cattaneo–Christov heat flux through a Prandtl non‐Newtonian nanofluid in Darcy–Forchheimer porous space

The Darcy–Forchheimer Prandtl fluid flow due to moving sheet is described here. The familiar energy transfer model, namely, the Cattaneo–Christov model of heat transportation, is adopted under thermal radiation phenomenon. The Prandtl non‐Newtonian nanofluid is accounted as a functioning fluid. The functioning fluid flows in Darcy–Forchheimer porosity space. The boundary‐layer and similarity variables are executed to reframe the mathematical expressions into simplified and single independent variable. Numerical solutions of nonlinear dimensionless expressions are calculated. The variations of distinct constraints on important quantities are demonstrated through tabular and pictorial forms. It is visualized that the velocity of non‐Newtonian nanofluid is enhanced significantly by incrementing the elastic parameter. Improving the thermophoretic and Brownian movement parametric values leads to higher profile of Prandtl nanofluid temperature.     

Numerical investigation of non‐Newtonian nanofluid mixed convection in a two‐opposite direction lid‐driven cavity with variable properties

The mixed convection fluid flow in a square cavity filled with AL₂O₃‐water non‐Newtonian nanofluid is numerically analyzed. The left and right vertical boundaries of the enclosure have been kept in the constant temperature. Remaining walls of the cavity have been considered to have adiabatic boundary condition. Two different cases have been considered. In the first case, left and right side walls have been moved vertically with constant speed V_b in opposite directions. In the second case, the directions of their motions have been reversed. The transport equations, written in terms of the primitive variables for the non‐Newtonian nanofluid, have been solved numerically using the finite volume method. The shear stresses were calculated using the Ostwald‐de Waele model for the shear‐thinning nanofluid. The model introduced by Patel et al was used to obtain the thermal conductivity of the nanofluid. The variation of the fluid flow with respect to the Richardson number and volume fraction of the nanoparticles was investigated through a parametric study. Even though increasing the volume fraction of nanoparticles leads to heat transfer enhancement, for the second case of this study, for Ri = 1, the average Nusselt number initially drops sharply by increasing the volume fraction of nanoparticles, then remains constant.     

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.     

Fractional analysis of radiative blood transport through a porous channel containing multishaped cobalt nanoparticles: An application to hemodynamics

In this work, impacts of dispersing nonspherical shaped cobalt nanoparticles in the blood are analyzed for magnetohydrodynamic radiative transport of blood inside a vertical porous channel. An Oldroyd-B model is used to feature flow characteristics of blood along with Fourier's principle of heat transmission for the mathematical modeling of the problem. A fractional system is constructed by employing the idea of the Caputo–Fabrizio derivative on subsequent differential equations. The Laplace transform method is adopted to solve the fractional flow and energy equations subject to generalized boundary conditions, which involve time-dependent functions $h\left(\tau \right)$ and $g\left(\tau \right)$, respectively. Instead of promoting the analytic velocity and energy expressions, Zakian's numerical algorithm is operated to achieve the reverse transformation purpose of Laplace domain functions. To certify the obtained solutions, two additional numerical algorithms named Stehfest's algorithm and Durbin's algorithm are inculcated in this study, and comparative illustrations are drawn. For the extensive investigation of shear stress and heat transfer phenomenon, numerical simulations for the coefficient of skin friction and Nusselt number are performed, and outcomes are communicated through various tables. The impacts of shape-dependent viscosity and other significant parameters on flow patterns are investigated through graphs for multiple motion types of the left channel wall. Meanwhile, the thermal performance of nanofluid is examined for platelet, brick, cylinder, and blade shape nanoparticles, along with other thermal parameters. In addition, some recently reported results and flow profiles for Maxwell, second-grade, and viscous fluids are deduced graphically as special cases of the current study.     

Modeling of a Solar Water Collector with Water‐Based Nanofluid Using Nanoparticles

The pressure‐velocity form of the Navier–Stokes equations, energy equation, and concentration equation are used to represent the mass, momentum, energy, and concentration conservations of the nanofluid medium in the solar collector. The governing equations and corresponding boundary conditions are converted to dimensionless form and solved numerically by the finite element method. The physical domain is discretized by triangular mesh elements with six nodes. The working fluid is water‐based nanofluid with two nanoparticles, namely, silver (Ag) and copper oxide (CuO). The study includes computations for different values of buoyancy ratio (Nr) and Schmidt number (Sc). Flow, heat, and mass transfer characteristics are presented in the forms of streamlines, isotherms, and iso‐concentrations. In addition, results for the average radiative, convective heat and mass transfer, mean temperature and concentration of nanofluid, mid‐height horizontal‐vertical velocities, and subdomain average velocity field are offered and discussed for the above‐mentioned parametric conditions. Results show that the effects of Nr and Sc on the convective‐radiative heat and mass transfer phenomenon inside the collector are significant for all values of Nr and Sc studied. Comparison and validation with the standard experimental/numerical data is given in brief. The variation of the obtained result is presented as 34% with the result of experimental data. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(3): 270–287, 2014; Published online 30 September 2013 in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21080     

Nanofluid flow over three different geometries under viscous dissipation and thermal radiation using the local linearization method

This study deals with the transfer of mass and heat of nanofluid flow over three different geometries of the non‐Darcy permeable vertical cone/wedge/vertical plate. Influence of the Brownian motion and thermophoresis takes place due to the nanofluid. Boundary condition on the temperature is introduced at the surface where the thermal conductivity of the fluid obeys a linear relation with the temperature. The local linearization method is introduced for solving the governing equations, and is based on spectral discretization. To verify the numerical scheme, we compared our results with those in the existing literature. The impact of the governing parameters on the fluid velocity, temperature distribution, and concentration distribution of nanoparticles along with the Nusselt number and Sherwood number is discussed. Some important outcomes of the present study are that the Nusselt number is higher for the plane plate than that for the vertical cone and it significantly decreases with introduction of the radiation parameter. The nanofluid Lewis number decreases the diffusivity of mass of the nanofluid, and as a result it helps enhance the Sherwood number.     

Comparative analysis of a solar‐driven novel salt‐based absorption chiller with the implementation of nanoparticles

The present study exemplifies the comprehensive thermal analysis to compare and contrast ammonia‐lithium nitrate (NH₃‐LiNO₃) and ammonia‐sodiumthiocynate (NH₃‐NaSCN) absorption systems with and without incorporation of nanoparticles. A well‐mixed solution of copper oxide/water (CuO/H₂O) nanofluid is considered inside a flat‐plate collector linked to an absorption chiller to produce 15‐kW refrigeration at ?5°C evaporator temperature. Enhancements in heat transfer coefficient, thermal efficiency, and useful heat gain of the collector are evaluated, and the effect of these achievements on the performance of both absorption chillers have been determined for different source temperatures. A maximum 121.7% enhancement is found in the heat transfer coefficient with the application of the nanofluid at 2% nanoparticle concentration. The maximum coefficient of performance observed for the NH₃‐NaSCN chiller is 0.12% higher than that for the NH₃‐LiNO₃ chiller at 0°C evaporator temperature. Contradictory to this, the average system coefficient of performance of the NH₃‐LiNO₃ absorption system has been found 5.51% higher than that of the NH₃‐NaSCN system at the same evaporator temperature. Moreover, the application of the nanofluid enhanced the performance of the NH₃‐NaSCN and NH₃‐LiNO₃ systems by 2.70% and 1.50%, respectively, for lower generator temperature and becomes almost the same at higher temperatures, which altogether recommends the flat‐plate collector–coupled NH₃‐LiNO₃ absorption system be integrated with a nanofluid.     

Memory response for thermal distributions moving over a magneto-thermoelastic rod under Eringen’s nonlocal theory

Abstract

Enlightened by the Caputo fractional derivative, this study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of magnetic field and moving heat source in a rod in the context of Lord–Shulman (LS) theory of thermoelasticity based on Eringen’s nonlocal elasticity. Both ends of the rod are fixed and heat insulated. Employing Laplace transform as a tool, the problem has been transformed into the space domain and solved analytically. Finally, solutions in the real-time domain are obtained by applying the inverse Laplace transform. Numerical calculation for stress, displacement, and temperature within the rod is carried out and displayed graphically. The effects of moving heat source speed, time instance, memory-dependent derivative, magnetic field and nonlocality on temperature, stress, and temperature are studied.     

Influence of Viscous Dissipation on non‐Darcy Mixed Convection Flow of Nanofluid

In this paper, the steady fully developed non‐Darcy mixed convection flow of a nanofluid in a vertical channel filled with a porous medium with different viscous dissipation models is analyzed. The Brinkman‐Forchheimer extended Darcy model is used to describe the fluid flow pattern in the channel. The transport equations for a nanofluid are solved analytically using the seminumerical‐analytical method known as differential transformation method, and numerically with the Runge‐Kutta shooting method. Finally, the influence of pertinent parameters, such as solid volume fraction, different nanoparticles, mixed convection parameter, Brinkman number, Darcy number, and inertial parameter on the velocity and temperature fields are shown graphically. The results show that velocity and temperature are enhanced when the mixed convection parameter, Brinkman number, and Darcy number increases whereas solid volume fraction and inertial parameter decreases the velocity and temperature fields. The obtained results show that the nanofluid enhances the heat transfer process significantly.     

Cu‐Al2O3/H2O hybrid nanofluid flow past a porous stretching sheet due to temperatue‐dependent viscosity and viscous dissipation

The resent development of research in the field of nano technology introduced hybrid nanofluids which are advanced classes of fluids with augmented thermal properties and it gives better results comparing to regular nanofluid. The aim of the present work is to study the significant effects of variable viscosity and viscous dissipation on a porous stretching sheet in the presence of hybrid nanofluid and radiative heating. In this model, two types of nanoparticles, namely copper (Cu) and alumina oxide (Al₂O₃), are suspended in the base fluid H₂O to form a hybrid nanoliquid. The novelty of this study is to introduce variable viscosity along with natural convection in the momentum equation and viscous dissipation in the energy equation. Mathematical modeling is employed in this study, whereby partial differential equations for the fluid flow are constructed and transformed to a set of ordinary differential equations, and hence resolved computationally by Runge‐Kutta‐Fehlberg method along with shooting scheme. The most important results for relevant parameters concerning the flow heat measure, surface drag, and heat transfer coefficients are thoroughly examined and presented graphically for both Cu‐Al₂O₃/water hybrid nanofluids. There is an increase in hybrid nanofluid velocity profile with mounting values of $\lambda$, and the Cu‐water nanofluid converges to the boundary more quickly than the hybrid nanofluid due to the occurrence of variable viscosity. The results concluded that the Nusselt number of the viscous fluid is lower than that of the nanofluid and hence the hybrid nanofluid (ie, heat transfer rate: normal fluid < nanofluid < hybrid nanofluid). The outcomes of present investigations are in close agreement with the viscous fluid as a particular case.     

Numerical computation on Marangoni convective flow of two‐phase MHD dusty nanofluids under Brownian motion and thermophoresis effects

The current theoretical study describes the Marangoni thermal convective flow of magnetohydrodynamic dusty nanofluids along a wavy vertical surface. The two‐phase mathematical model is developed under the influence of thermal radiation and exponentially varying space‐dependent heat source. Pure and hybrid nanoparticles together with dust particle suspension in the base fluid are taken into consideration to characterize the behavior of the flow. Brownian motion and thermophoresis mechanisms are considered, since it enhances the convection features of dusty nanofluid. Appropriate transformations are adopted to modify the flow governing equations and boundary conditions to dimensionless form. The forward finite difference scheme is implemented to illustrate the resultant coupled partial differential equations. The Newton quasi‐linearization technique is utilized to reduce the nonlinear system into a linear form, which is solved thereafter by Thomas algorithm. The responses of velocity, temperature, concentration, friction factor, and heat and mass transfer rate profiles with various governing parameters are discussed and portrayed graphically. The study evidences that the radiation and space‐dependent heat generating parameters strengthen the temperature distribution. Also, the heat transfer rate appreciably rises with the increment in Marangoni convection. The solution methodology and accuracy of the model is validated by generating the earlier outcomes for nonradiating nanofluid flow without heat source/sink.     

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.     

Marangoni convective nanofluid flow over an electromagnetic actuator in the presence of first‐order chemical reaction

The present study aims to investigate Marangoni‐forced convective nanofluid flow over an electromagnetic actuator (Riga plate). A first‐order homogeneous chemical reaction is considered. The thermocapillary and solutocapillary Marangoni effect developed by the surface tension is considered as a driving force for the nanofluid. In addition, Grinberg‐term is accounted to involve the impact of Lorentz force impinged by the actuator in the model. A set of nonlinear ordinary differential equations is obtained via suitable transformations for a nonsimilar analysis. Series solutions are achieved through homotopy to discuss the behavior of the velocity field, thermal distribution, and concentration of the nanoparticles graphically. The variation in Nusselt and Sherwood numbers is discussed. The outcomes declared that the flow parallel to the surface of the plate is assisted by the Lorentz forces generated by electromagnetic bars of the actuator resulting in an enhancement in the fluid motion. Furthermore, the stronger Marangoni effect resulted in the declining trend of the temperature profile. The concentration of nanoparticles near the surface reduced intensive chemical reaction inside the nanofluid.     

Modeling of double diffusive buoyant flow in a solar collector with water‐CuO nanofluid

The behavior of a prism‐shaped solar collector with a right triangular cross sectional area is investigated numerically. The water‐CuO nanofluid is taken as the functioning liquid through the solar collector. The leading differential equations with boundary conditions are solved by the penalty finite element method using Galerkin's weighted residual scheme. The performance of parameters in terms of temperature, mass, velocity distributions, radiative, convective heat and mass transfer, mean temperature and concentration of nanofluid, mid height horizontal‐vertical velocities, and sub‐domain average velocity field are investigated systematically. These parameters include the Rayleigh number Ra and the solid volume fraction φ. The outcome explains that the performance of the solar collector can be enhanced with the largest Ra and φ. The code validation shows excellent concurrence with the hypothetical outcome obtainable in the literature. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21039     

Surveying the hybrid of radiation and magnetic parameters on Maxwell liquid with TiO2 nanotube influence of different blades

In this paper, the impacts of Maxwell nanoliquid transmission, rectangular with titanium oxide nanoparticles are explored over the triangular, chamfer blades. The innovation of this paper is the use of the number of chamfers, rectangular, and triangular blades at the top and bottom of a stretched plate to study physical nanofluid parameters such as temperature and the effects of magnetism. Also, by determining the appropriate height and length for the blades, we achieve the best optimization of temperature and velocity of nanofluid between the plate and the blades, which improves heat transfer and with a more and better effect of magnetic effects. The finite element method is utilized for the calculated differential equations. In this paper, by utilizing the reaction surface strategy, we optimized the titanium oxide nanofluid velocity and temperature, and magnetic parameter passing from the extending sheet. On average, the titanium oxide nanoparticle velocity around the two rectangular blades at the beginning of the sheet is 73.09% higher than triangular blades and 66.98% higher than chamfer blades. Based on the outcomes got from the titanium oxide nanofluid speed charts and the warm exchange cantors and magnetic impacts within the Design-Expert computer program, the most excellent optimization occurred for TiO₂ nanofluid speed and TiO₂ nanofluid temperature and TiO₂ magnetic parameter with u = 0.523, T = 3.25, and H = 2.671.