Influence of Cattaneo-Christov model on Darcy-Forchheimer flow of Micropolar Ferrofluid over a stretching/shrinking sheet

doi:10.1016/j.icheatmasstransfer.2019.104385

International Communications in Heat and Mass Transfer

Volume 110, January 2020, 104385

https://doi.org/10.1016/j.icheatmasstransfer.2019.104385 Get rights and content

Abstract

This work deals with the study of Darcy-Forchheimer flow of micropolar ferrofluid on a porous and dynamic (stretching/shrinking) sheet under the influence of thermal radiations subjected to both suction and injection. The effects of the external electric and magnetic fields are considered as well. Water is used as a base fluid and Fe₃O₄ (iron oxide) as electro-magnetite nanoparticles. The mathematical equations developed in this study are based on the Cattaneo-Christov model consisting of coupled nonlinear partial differential equations. These equations are transformed into a set of coupled ordinary differential equations (ODEs) by using similarity transformations. These ODEs are solved by applying the standard mathematical technique of homotopy analysis (HAM). The effects produced by different parameters on the velocity, micro-rotational velocity and temperature profiles are shown graphically for positive as well as negative mass transfer flow and for both stretching and shrinking cases. It has been observed that the velocity profile increases (decreases) with the increasing electric field strength and microrotation parameter during the stretching (shrinking) of the surface in both suction (S > 0) and injection (S < 0) cases. Furthermore, similar results has been observed for the velocity profile with the increasing inertial coefficient, porosity, magnetic and boundary parameters during flow over the stretching (shrinking) surface for both S > 0 and S < 0. The micro-rotational velocity increases with higher values of microrotation parameter for stretching, while decreases for the shrinking of the surface. The temperature profile displays an increasing trend with the increasing values of heat energy source and sink terms and thermal radiation parameter for stretching as well as shrinking of the sheet for S > 0 as well as S < 0. The temperature profile also changes with the variation in thermal relaxation parameter and Prandlt number.

Introduction

There are numerous scientific challenges for effective heat transfer in various fields like medicine, food, electronics, chemical sensors, solar cells, fuel cells, batteries, etc. These challenges can be handled by applying the recently developed field of nanotechnology. One of the major achievements of nanotechnology is the introduction of nanofluids. Nanofluid is a wonderful addition of Modern science and Technology. Due to nanometer dimensions, the nanofluid can flow through micro-size channels. Because of the enhanced convection between the base liquid surfaces and the nanoparticles, the nano-suspensions exhibit large thermal conduction. The suspension of the nanoparticles in the base fluid increases the heat conduction capacity, thermal conductivity, effective surface area, interparticles collisions, and the various interactions between the constituent particles. The nanofluid utilization for the first time was introduced by Choi in 1995 [1] for enhancing the thermal conductivity of the classical fluid. After its introduction, the nanofluid research has attracted many investigators throughout the world due its fascinating thermal properties and possible applications in different fields like medicine, electronics, petrochemical, refining, heating, air-conditioning and transportation. The experimental investigation of two different kinds of nanoparticles γ-alumina (Al₂O₃) and titaniumdioxide (TiO₂) on the turbulent heat transfer flow was given by Pak and Cho [2]. It has been observed that the addition of the suspended nanoparticles caused an increase in the convective heat transfer coefficient of water. Eastman et al. [3] has claimed that the heat transfer coefficient in Cu-water suspended nanoparticles is higher than as compared with for pure water by analyzing some initial suspended nanoparticles experiments. Qiang and Yimin [4] have found better results than Eastman et al. [3] by experimentally studying the convective heat transfer characteristics of Cu- water based nanofluid flow. The coefficient of convective heat transfer of carbon nanotubes nanofluid in the presence of constant heat flux was experimentally studied by Rashidi and Nezamabad [5]. They found a considerable increase in the convective heat transfer coefficient of the carbon nanotubes nanofluid and also found it as function of axial distance from the inlet. Mahanta and Abramson [6] investigated the thermal conductivities of graphene and graphene oxide and obtained that the higher thermal conductivity of multilayer graphene platelets can be associated with the interlayer coupling due to covalent interactions provided by oxygen atoms. Sun et al. [7] have examined the convective heat transfer properties of water based on ferrofluid nanofluid inside copper tubes and showed that the addition of dispersants resulted a significant enhancement of the stability of the nanofluids. Walvekar et al. [8] have experimentally investigated the heat transfer enhancement during the turbulent flow of carbon nanotube (CNT) nanofluids by varying the CNT concentration and found that heat transfer enhances as a function of CNT concentration and temperature. Dawar et al. [9] have recently examined the radiative heat transfer and the MHD carbon nanotubes Casson nanofluid flow between two rotating plates and investigated the effects produced by various parameters of interest on the velocity and temperature profiles.

The mathematical models which researchers employ for studying different behaviours of nanofluids are Buongiorno and Tiwari-Das (TD) models [10,11]. Inertia, Magnus effect, diffusiophoresis, fluid drainage, gravity, thermophoresis, and Brownian diffusion, are the seven different mechanisms in which slip take place among the solid and fluid phases in the Buongiorno model. Buongiorno studied all these seven mechanisms and obtained that when turbulent effect is absent, the theromphoresis and the Brownian diffusion are the significant and leading mechanisms in the nanofluid dynamics. The Buongiorno model has been used in various studies [[12], [13], [14], [15], [16], [17]]. Later on, the Buongiorno model has been further extended by different researchers to investigate variety of problems related to the nanofluid flow on stretching as well as shrinking surfaces. For example, Zaimi et al. [18] studied numerically the boundary layer and heat transfer flow of a nanofluid over a permeable stretching (shrinking) surface. By analyzing the effects of various parameters, they found that the suction widens the range of stretching (shrinking) parameter for which the solution exists. Hayat et al. [19] has analytically studied the three dimensional MHD nanofluid flow on a stretching surface in the presence of heat generation (absorption) and convective heat transfer by considering the effects of Brownian motion and thermophoresis. They found that the temperature distribution varies directly with Biot number and magnetic parameter. Muhammad et al. [20] has investigated the MHD boundary layer flow of Maxwell nanofluid on a porous stretching surface based on Darcy-Forchheimer model and studied the temperature and concentrations profiles by varying the different relevant parameters. Muhammad et al. [21] studied the three dimensional Darcy-Forchheimer nanofluid flow on a bidirectional stretching surface by including the thermophoretic and Brownian diffusion effects in the energy and volume fraction of nanoparticles expressions. They explained the results of their work through different graphs and tables. Hayat et al. [22] examined the MHD flow of electrically conducting nanofluid over a porous stretching surface by taking into account the Brownian motion and thermophoretic diffusion of nanoparticles and studied graphically the impacts produced by different physical parameters on the velocity, temperature and nanoparticle concentration. Hayat et al. [23] undertook the three dimensional MHD flow of a couple stress electrically conducting nanofluid on a bidirectional stretching surface in the presence of thermophoresis and Brownian motion effects. They have graphically shown the effects produced by different emerging parameters on the temperature and nanoparticles concentration and concluded that the thickness of the thermal boundary layer is an increasing function of radiative effect. Further relevant study can be found in Hayat et al. [24,25]. In contrast to the Buongiorno model which mainly concentrates on the the thermophoresis effects and Brownian motion, the TD model considers the nanoparticles volume fraction. Under different physical conditions, many researchers use TD model to study the different heat transfer characteristics of the nanofluids. The convective heat transfer of Ag and Cu-water base nanofluids on a stretching (shrinking) surface during steady flow of a boundary layer has been studied by Yacob et al. [26]. They found that the heat energy transfer rate decreases with increasing convective parameter while increases with higher values of nanoparticle volume fraction. It was also observed that heat transfer rate at the Cu-water nanofluid surface is larger than at the surface of Ag-water nanofluid, although Ag is a good thermal conductor as compared to Cu. Furthermore, Hussanan et al. [27] has examined the magnetite ferrofluid flow on a stretching (shrinking) surface subjected to suction and injection in order to investigate the micro-structure and inertial properties of the substructure of iron oxide (Fe₃O₄) nanoparticles mixed with water (base fluid). During this study, they observed that micro-rotation velocity changes remarkably with positive and negative values of mass transfer parameter and also concluded that micropolar ferrofluid has higher velocity as compared to classical nanofluid. The TD model has further been applied to other relevant research and can be found in the references [[28], [29], [30], [31], [32], [33], [34]].

The most important model for the transfer of heat due to conduction is the Fourier's law [35]. Cattaneo [36] in 1948 recommended a thermal relaxation time factor to the Fourier's model to improve the heat transfer. But the main difficulty is the different thermal relaxation times of the different materials. Keeping this fact in mind, a very useful time derivative model was developed for the effective heat transfer mechanism by Christov [37], which is called the Cattaneo-Christov heat flux model. Meraj Mustafa [38] applied it to study the viscoelastic rotating fluid flow bounded by a stretching surface. He found an inverse relation between velocity and viscoelastic fluid parameter and also observed that fluid temperature varies inversely with the relaxation time and with the Prandtl number. The heat flux model of Cattaneo-Christov was also applied by Hayat et al. [39] in order to study the boundary layer flow on a dynamic surface of variable thickness and to explore the various aspects of heat transformation with varying thermal conductivity. They analyzed and explain the dependence of temperature and velocity distributions on different pertinent parameters. Ali and Sandeep [40] have used Cattaneo-Christov model and numerically studied the heat transfer due to radiations of the MHD Casson-ferrofluid. The different studies where Cattaneo-Christov model has been used in [[41], [42], [43]].

In this work, we examine the Cattaneo-Christov ferrofluid model on a porous and stretching (shrinking) surface by taking into account the effects of thermal radiations subjected to suction and injection. The effects of electric and magnetic fields are considered as well. In this brief study, water is taken as base fluid mixed with Fe₃O₄ (iron oxide) electro-magnetite nanoparticles. The mathematical equations are developed by using the Cattaneo-Christov model of nanofluid. We solve these model equations through the standard homtopy analysis procedure by using the similarity variables. In Section 2, the mathematical model consisting of different equations is outlined. Section 3 is devoted to the solution of these models equations by homotopy analysis. The results and discussion are presented in Section 4. We have concluded our work in Section 5.

Section snippets

Problem statement

We consider the boundary layer two-dimensional flow of a micropolar ferrofluid on a linear porous and stretching (shrinking) sheet. The x-axis is taken parallel while the y-axis is taken perpendicular to the sheet. Both magnetic and electric fields are assumed along the y-axis. The stretching (shrinking) velocity of the sheet is taken as U_w(x) = αx, where α > 0 (<0) for the stretching (shrinking) of the sheet. Due to the assumed geometry and conditions, the system of equations which govern the

Homotopy analysis method

In order to solve Eqs. (11), (12), (13) by applying the boundary (14), (15), we use the standard mathematical technique (HAM) [49,50]. With the auxiliary parameter ℏ, we adjust and control the solution convergence.

The initial assumptions are taken in the following form [51]: $f_{0} (ξ) = - α e^{- ξ} + S + α, g_{0} (ξ) = αδ e^{- ξ}, θ_{0} (ξ) = \frac{1}{e^{ξ}}$

The operators L_f, L_g and L_θ, which are linear in nature are defined as: $L_{f} (f) = \frac{d^{3} f}{d ξ^{3}} - \frac{df}{dξ}, L_{g} (g) = \frac{d^{2} g}{d ξ^{2}} - g, L_{θ} (θ) = \frac{d^{2} θ}{d ξ^{2}} - θ$ having the following properties $L_{f} (ε_{1} + ε_{2} e^{- ξ} + ε_{3} e^{ξ}) = 0, L_{g} (ε_{4} + ε_{5} e^{ξ}) = 0, L_{θ} (ε_{6} e^{- ξ} + ε_{7} e^{ξ}) = 0$ where ε

Results and discussions

We present graphically here the impact of the various parameters on f^'(ξ) (velocity profile), g(ξ) (micro-rotational velocity profile), and θ(ξ) (temperature profile). To see the variations produced by different embedded parameters on these three profiles for the stretching and shrinking (α > 0, α < 0) cases, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12, Fig. 13, Fig. 14, Fig. 15, Fig. 16, Fig. 17, Fig. 18, Fig. 19, Fig. 20, Fig. 21, Fig. 22, Fig. 23, Fig. 24

Conclusions

In the current study, we have investigated the Cattaneo-Christov model of electromagnetite ferrofluid on a porous stretching (shrinking) sheet under the influence of the thermal radiation subjected to both suction and injection. The developed mathematical equations are tackled with the help of similarity variables through HAM. We have investigated the effects produced by different parameters on the velocity, micro-rotation velcoity and temperature functions. The most important findings of the

Declaration of Competing Interest

The author declares that they have no competing interests.

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (DF-866-130-1441). The authors, therefore, gratefully acknowledge DSR for technical and financial support.

References (53)

B. Sun et al.
Flow and convective heat transfer characteristics of fe2o3–water nanofluids inside copper tubes
Int. Commun. Heat Mass Transf.
(2015)
R.K. Tiwari et al.
Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids
Int. J. Heat Mass Transf.
(2007)
A. Kuznetsov et al.
Natural convective boundary-layer flow of a nanofluid past a vertical plate
Int. J. Therm. Sci.
(2010)
W.N. Mutuku et al.
Hydromagnetic bioconvection of nanofluid over a permeable vertical plate due to gyrotactic microorganisms
Comput. Fluids
(2014)
H. Xu et al.
Fully developed mixed convection flow in a horizontal channel filled by a nanofluid containing both nanoparticles and gyrotactic microorganisms
Eur. J. Mech. B Fluids
(2014)
W. Khan et al.
MHD nanofluid bioconvection due to gyrotactic microorganisms over a convectively heat stretching sheet
Int. J. Therm. Sci.
(2014)
T. Hayat et al.
An analytical solution for magnetohydrodynamic oldroyd-b nanofluid flow induced by a stretching sheet with heat generation/absorption
Int. J. Therm. Sci.
(2017)
T. Muhammad et al.
A revised model for darcy-forchheimer flow of Maxwell nanofluid subject to convective boundary condition
Chin. J. Phys.
(2017)
T. Muhammad et al.
A revised model for darcy-forchheimer three-dimensional flow of nanofluid subject to convective boundary condition
Results Phys.
(2017)
T. Hayat et al.
On squeezing flow of nanofluid in the presence of magnetic field effects
J. Mol. Liq.
(2016)

T. Hayat et al.

On darcy-forchheimer flow of carbon nanotubes due to a rotating disk

Int. J. Heat Mass Transf.

(2017)

M. Sheikholeslami et al.

MHD forced convection flow of nanofluid in a porous cavity with hot elliptic obstacle by means of lattice boltzmann method

Int. J. Mech. Sci.

(2018)

A. Hussanan et al.

Microstructure and inertial characteristics of a magnetite ferrofluid over a stretching/shrinking sheet using effective thermal conductivity model

J. Mol. Liq.

(2018)

K. Vajravelu et al.

Convective heat transfer in the flow of viscous ag–water and cu–water nanofluids over a stretching surface

Int. J. Therm. Sci.

(2011)

M. Hamad

Analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field

Int. Commun. Heat Mass Tran.

(2011)

M. Hamad et al.

Magnetic field effects on free convection flow of a nanofluid past a vertical semi-infinite flat plate

Nonlinear Anal.

(2011)

M. Sheikholeslami et al.

Analytical investigation of NHD nanofluid flow in a semi-porous channel

Powder Technol.

(2013)

M. Sheikholeslami et al.

Heat transfer of Cu-water nanofluid flow between parallel plates

Powder Technol.

(2013)

R. Kandasamy et al.

Impact of chemical reaction on cu, al2o3 and swcnts–nanofluid flow under slip conditions

Eng. Sci. Technol. Int. J.

(2016)

C. Christov

On frame indifferent formulation of the Maxwell–cattaneo model of finite-speed heat conduction

Mech. Res. Commun.

(2009)

M. Ali et al.

Cattaneo-christov model for radiative heat transfer of magnetohydrodynamic casson-ferrofluid: a numerical study

Results Phys.

(2017)

M. Waqas et al.

Cattaneo-christov heat flux model for flow of variable thermal conductivity generalized burgers fluid

J. Mol. Liq.

(2016)

T. Hayat et al.

Three-dimensional flow of nanofluid with cattaneo–christov double diffusion

Results Phys.

(2016)

M. Turkyilmazoglu

Flow of a micropolar fluid due to a porous stretching sheet and heat transfer

Int. J. Non-Linear Mech.

(2016)

S.U. Choi et al.

Enhancing Thermal Conductivity of Fluids With Nanoparticles

(1995)

B.C. Pak et al.

Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles

Exp. Heat Transf. Int. J.

(1998)

Cited by (66)

Recent advances on entropy analysis of composite nanofluids-A critical review
2024, Results in Engineering
The present study analyzes on the reviews carried out by the previous researchers on the entropy generation caused by the flow and heat transfer of several composite nanofluids subject to varieties of geometries under the influence of several constraints of motions. The composite nanofluids include binary hybrid nanofluids and ternary hybrid nanofluids involving disparate base fluids with different suitable nanoparticles. The geometries considered are deformed surfaces (stretched/shrunk sheets, cylinders, and disks), channels and cavities. Several numerical techniques such as Runge-Kutta method and finite element/difference/volume methods are implemented in the investigations. The major and significant outcome of the review analysis is the entropy comparison for different types of nanofluids. Growth of the volume percentage of nanoparticles at a fixed mass flow rate enhances entropy formation. When porosity strength increases, the irreversible entropy production caused by fluid friction results in increased thermal efficiency of certain systems than magnetic entropies. Compared to single nanofluids, hybrid nanofluids display the maximum entropy. With increasing Rayleigh numbers, the frictional and magnetic entropies in composite nanofluids ameliorate significantly than thermal entropy.
Understanding the impact of magnetic dipole and variable viscosity on nanofluid flow characteristics over a stretching surface
2024, Journal of Magnetism and Magnetic Materials
The purpose of this research article is to investigate the impact of magnetic dipole on the flow of nano-fluids over the stretching surface with variable viscosity and thermal conductivity. Fluid fills the porous space. Heat transfer analysis is carried out in the presence of Newtonian heating. Water and ethylene glycol are used as base fluid while gamma alumina acts as a nanoparticle. System of ordinary differential equations is obtained by appropriate transformations. Numerical solutions are computed for the resulting nonlinear differential system by applying shooting method with RK-4 scheme. Impact of different parameters on the velocity and temperature profiles is examined. Computations for surface drag force and heat transfer rates are presented and examined for the influences of pertinent parameters. It is observed that fluid velocity reduces for larger viscosity parameter while opposite behavior is observed for temperature. Increasing values of ferro hydrodynamic interaction parameter and Curie temperature correspond to the enhancement in temperature. Surface drag force increases for larger values of nanoparticles volume fraction. Gamma alumina-ethylene glycol nano-fluid produce less surface drag force than gamma alumina-water nano-fluid. Heat transfer rate is increasing function of Prandtl number.
Hydromagnetic flow of two immiscible nanofluids under the combined effects of Ohmic and viscous dissipation between two parallel moving plates
2023, Journal of Magnetism and Magnetic Materials
The hydromagnetic flow of immiscible nanofluids between two parallel plates is investigated. Two regions, namely, (i) Region 1 is filled with base fluid kerosene oil with silver $(Ag)$ nanoparticles, while (ii) Region 2 is filled with base fluid water with titanium dioxide $(Ti O_{2})$ nanoparticles. Analytic solutions of nonlinear coupled differential equations have been obtained. The velocity and temperature profile results are discovered to flatten out when the magnetic field's inclination angle decreases. Impact of key parameters like inclination of the magnetic field, Hartmann number and load parameter with thermal conductivity ration have been illustrated graphically.
Pulsating magnetohydrodynamic flow of Fe<inf>3</inf>O<inf>4</inf>-blood based micropolar nanofluid between two vertical porous walls with Cattaneo–Christov heat flux and entropy generation
2023, Journal of Magnetism and Magnetic Materials
The present article is aimed to investigate the impact of entropy generation on pulsating hydromagnetic flow of a micropolar nanofluid between two vertical porous walls using the Cattaneo–Christov heat flux model. Here Fe $_{3}$ O $_{4}$ (magnetite) is taken as a nanoparticle and blood as micropolar fluid (base fluid). The significance of viscous dissipation, Ohmic heating, and thermal radiation are considered. This model is noteworthy in the field of magnetic bioseparation, pressure surges, magnetofection agent, biomedical engineering, cancer therapeutic, artificial kidney, brain tumors, and nano-drug delivery in the arteries. The governing partial differential equations are transformed into the system of ordinary differential equations by deploying the perturbation process and then solved numerically by employing the fourth-order Runge–Kutta scheme with the support of the shooting technique. The flow variables like velocity, microrotation, temperature, entropy generation, and Bejan number are depicted graphically and discussed in detail. The heat transfer rate is displayed through a table. The results depict that the velocity is diminishing with the enhancement of coupling parameter, Hartmann number, and nanoparticle volume fraction. The temperature of micropolar nanofluid is increasing with an increment of viscous dissipation, thermal radiation, and heat source while it is reducing with the enhancement of magnetic field, thermal relaxation time, and heat sink parameter. The entropy generation is diminished by increasing the values of Hartmann number, thermal relaxation, and coupling parameter. Further, the Bejan number is enhanced by varying thermal radiation and nanoparticle volume fraction.
Rigid plate submerged in a Newtonian fluid and fractional differential equation problems via Caputo fractional derivative
2022, Partial Differential Equations in Applied Mathematics
The study of fractional variational problems with derivatives in the sense of Caputo is a recent subject in Newtonian fluids. In this article, the homotopy perturbation method (HPM), the variational iteration method (VIM), and the Akbari–Ganji method (AGM) were used to solve linear fractional differential equations order in fluid mechanics and fractional optimal control problems in the sense of Caputo. Achieving an exact solution to these equations has been one of the researchers’ goals; according to the research results, the proposed methods provide an exact solution. Comparing the results obtained by the proposed methods with those obtained by the Shehu method reveals that the present methods are very effective and convenient in applied fields. Also, the VIM and AGM methods are more proper than HPM in solving these equations. The solutions obtained by the proposed methods agree with the solutions available in the literature.
Dynamics of magneto-electric hybrid nanoparticles with chemically reacting and radiated moving plate: Entropy analysis
2022, International Communications in Heat and Mass Transfer
Citation Excerpt :
They emphasized that the understanding the basics of how energy moves through nanofluids is important for making nanofluids that are very energy efficient for a variety of heat transfer applications. Subsequently, other researchers [2–6] considered stretching surface as a geometry and studied various nanofluids under varying boundary conditions. Maleki et al. [7–9] discussed different nanofluid flows over porous surface in the presence/absence of slip parameters.
In this new era of fluid field, researchers are interested in hybrid nanofluid because of its thermal properties and potential, which are better than those of nanofluid when it comes to increasing the rate at which heat is transferred. In comparison to the dynamics of chemically reactive Ethylene Glycol - Zinc Oxide (nanofluid) and Ethylene Glycol - Zinc Oxide - Titanium dioxide (hybrid nanofluid) over a moving surface, nothing is known in terms of thermal radiation, entropy formation, viscous dissipation, and the Lorentz force. The thermo-physical characteristics of Ethylene Glycol, Zinc Oxide nanoparticles, and Titanium dioxide nanoparticles are used in this study to derive the governing equations for the transport of both dynamics. Flow-driven equations are transformed into nonlinear ODEs and solved using MATLAB bvp4c solver. It is noticed that, when magnetic field parameter (M) takes input in the range 0 ≤ M ≤ 3, skin friction coefficient decreases at a rate of 0.22052 (in case of hybrid nanofluid) and 0.19203 (in case of nanofluid) per unit value of magnetic field parameter. It is detected that the rise in Brinkmann number causes a rise in entropy generation and Bejan number decreases as Brinkmann number increases. Furthermore, when Eckert number rises, the heat transmission rate decreases. The heat transmission rate drops at a rate of 0.0869 (in case of hybrid Nanofluid) and 0.07429 (in case of nanofluid) when Eckert number (Eck) takes input in the range 0 ≤ Eck ≤ 0.7. The mass transmission rate increases at a rate of 1.013383 (in case of hybrid nanofluid) and 1.013125 (in case of nanofluid) when Schmidt number (Sc) takes input in the range 0 ≤ Sc ≤ 1. A comparison of the current results to previous results indicated a satisfactory agreement under certain conditions.

View all citing articles on Scopus

View full text

Influence of Cattaneo-Christov model on Darcy-Forchheimer flow of Micropolar Ferrofluid over a stretching/shrinking sheet

Abstract

Introduction

Section snippets

Problem statement

Homotopy analysis method

Results and discussions

Conclusions

Declaration of Competing Interest

Acknowledgments

Int. Commun. Heat Mass Transf.

Int. J. Heat Mass Transf.

Int. J. Therm. Sci.

Comput. Fluids

Eur. J. Mech. B Fluids

Int. J. Therm. Sci.

Int. J. Therm. Sci.

Chin. J. Phys.

Results Phys.

J. Mol. Liq.

Int. J. Heat Mass Transf.

Int. J. Mech. Sci.

J. Mol. Liq.

Int. J. Therm. Sci.

Int. Commun. Heat Mass Tran.

Nonlinear Anal.

Powder Technol.

Powder Technol.

Eng. Sci. Technol. Int. J.

Mech. Res. Commun.

Results Phys.

J. Mol. Liq.

Results Phys.

Int. J. Non-Linear Mech.

Enhancing Thermal Conductivity of Fluids With Nanoparticles

Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles

Exp. Heat Transf. Int. J.