MAGNETIC EFFECTS OF FERROFLUID FLOW IN A HEATED MICROCHANNEL WITH PERMEABLE WALLS

Z.M. KEFENE^†, O.D. MAKINDE^‡ and G.L. ENYADENE^‡

† Department of Applied Mathematics, Adama Science and Technology University, Adama, Ethiopia.

mesfink2005@gmail.com, senalemi2007@gmail.com

‡ Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa.

makinded@gmail.com

Cite this article as: 

Kefene, Z.M., Makinde, O.D., Enyadene, G.L. (2022) “Magnetic effects of ferrofluid flow in a heated mi-crochannel with permeable walls”, Latin American Applied Research 52(3), pp 247-257.

Abstract--In this paper, we analyze the combined effects of magnetic field, nanoparticles volume fractions, variable viscosity of unsteady ferrofluid flow in a microchannel with parallel permeable walls oriented horizontally whose temperatures are held asymmetrically. Using non-dimensionalization process, the governing momentum, energy equations and the associated initial boundary conditions (IBCs) are reduced to a set of dimensionless nonlinear partial differential equations with IBCs. The numerical method used for solving these equations is semi-discretization via centered finite difference with Runge-Kutta Fehlberg integration techniques. The effects of pertinent parameters on dimensionless fluid velocity, temperature, skin friction and heat transfer coefficient are analyzed through graphical depiction using MAPLE software. It is disclosed that the ferrofluid velocity decreased with an increase in both magnetic field and nanoparticles volume fraction, whereas it increased with pressure gradient and variable viscosity parameter. It is also shown that an increase in nanoparticles volume fraction and Eckert number rise the ferrofluid temperature. Moreover, the Nusselt number augmented at both microchannel walls as suction parameter increases, while a decreasing behavior is observed with injection parameter.

Keywords--Ferrofluid; Permeable microchannel; Variable viscosity; Suction/injection; Modified Buongiorno.

I. INTRODUCTION

Fluid flow and heat transfer through microchannel have become an active area of research and have been studied by many scholars in the analysis of microscale phenomena in science and engineering fields. Microchannels are delineated in terms of hydraulic diameter. Microchannels are channels whose largest polygonal dimension is less than 1mm and its hydraulic diameter is between 10 micro-meters and 200 micro-meters (Tuckerman and Pease, 1981). An exhaustive study in this field has gained importance in practical applications including drug delivery for medicine, cooling for microelectronics (Wie et al., 2007), and pumping technology for medical engineering (Salari et al., 2015). In this perspective, the use of microchannels provide enhancement in heat transfer rates and reduced temperature gradients across the channel width (Harms et al., 1999; Obot, 2002). Due to smaller diameter of the microchannels, there is high frictional resistance of fluid/coolant flow in the microchannel and rapid transformation of kinetic energy of the flow to internal energy which leads to generation of high temperature within the flowing fluid. This has a significant effect on fluid property like viscosity and thermal conductivity. The most susceptible property of fluid that may be affected by an increase in temperature is viscosity (Namburu et al., 2007). For instance, when the temperature mounts from 10^oC (viscosity, m=0.0131 gm/(cm s)) to 50^oC (viscosity, m=0.00548 gm/(cm s)) the viscosity of water dwindles by 270 percent (Nguyen et al., 2007).

One of the techniques that help to rise up heat transfer of fluid flow is to enhance the working fluid’s thermal conductivity. Nanofluid improves the heat transfer capability by adding nanoparticles of higher thermal conductivity than the carrier fluid. The nanoparticles inherently has high surface area to volume ratio which provides a larger contact surface area for conductive heat transfer between the base fluid and the solid particles, which also helps to enhance the effective conductivity of the nanofluid (Taylor et al., 2013). Nanofluid fluid flow through microchannel has been investigated (Malvandi and Ganji, 2016; Hosseini et al., 2018; Belhadj et al., 2018). Makinde (2018) has analyzed numerically the flow and heat transfer of a variable viscosity nanofluid containing etheylene glycol(EG)-water mixture with silver(Ag) nanoparticles in a microchannel with symmetric convective cooling at slip wall. Snoussi et al. (2018) analyzed heat transfer and nanofluid flow in a microchannel heat sink (MCHS) numerically by FLUENT software. The effects of thermal radiation, nanoparticles shape and valume fraction, and temperature dependent viscosity on microchannel single phase flow of ethylene glycol/silver (EG/Ag) nanofluid with convective cooling studied numerically by Monaledi and Makinde (2018). Kefene et al., (2020) studied MHD variable viscosity mixed convection flow of nanofluid through microchannel in the presence of suction and injection. The thermal and hydrodynamic behavior of unsteady flow of variable viscosity nanofluid flow through a microchannel filled with porous medium is investigated by Hindebu et al., (2021). Ferahani et al., (2021) presented numerical study on the effect of microchannel-porous media and nanofluid on temperature and performance of solar concentrator photovoltaic (CPV) system. Ferrofluids are a special class of nanofluid made up of superparamagnetic nanoparticles and a non-magnetic carrier fluid which is manipulated by

Figure 1: Flow geometry with coordinate system

an applied external magnetic field (Rosensweig, 1997). The most common functions of ferrofluids found in a loud speaker to improve its audio response, and sound amplitude amplification while reducing sound distortion. Not only these applications but also they used in fluid seals for devices like rotating shaft seal to alleviate frictional losses in better way than the traditional/conventional mechanical seals or high-speed computer disk drives to prevent ingress of dust particles and impurities (Clark, 2013). The problem of microchannel flow using ferrofluids under the influence of applied external magnetic field has been studied by several researchers (Kurtoglu et al., 2012; Lee and Seo, 2013; Asfer et al., 2016; Sheikholeslami and Ganji, 2016; Shashikumar et al., 2018). Weng and Lo (2015) studied experimentally and numerically the convective heat transfer of ferrofluid in an isothermally heated microtube. Their investigation revealed the influence of nanoparticles volume fraction and magnetic field on velocity, temperature, flow drag and heat transfer rate. Gui et al. (2018) analyzed theoretically and experimentally ferrofluid characterization and measurement of heat transfer rate for single-phase forced convective ferrofluidic flow in microchannel in the presence of external magnetic field. They found that the heat transfer rate increased with an increase in nanoparticles volume fraction from 0.2 to 0.4%, but decreased with magnetic field. Rosengarten et al. (2019) demonstrated experimentally the heat transfer enhancement rate in microchannel flow using single-phase and two-phase ferrofluid with externally imposed magnetic field. They have shown that the flow and heat transfer rate appreciably influenced by the interaction of ferrofluid and the magnetic field both in single-phase and two-phase flow. Nguyen et al. (2020) numerically studied the heat transfer and entropy generation of ferrofluid flow through porous ribbed microchannel using single-phase and two-phase model without magnetic field. They obtained that the maximum value of heat transfer coefficient is observed at the higher values of nanoparticles volume fractions, Reynolds number and porosity. Also, it is shown that the entropy generation increases by increasing porosity value.

The newness in this study is the investigation of the combined effects of unsteadiness, pressure gradient, suction/injection, nanoparticles volume fractions, magnetic field, variable viscosity of ferrofluid flow through parallel plate’s microchannel. The main objective is to solve this problem numerically by semi-discretization finite difference method and Runge-Kutta Fehlberg integration technique (Na, 1979). The influence of many parameters

Table 1 Base fluid and nanoparticles thermophysical properties

content	water	ferro-nanoparticles
density	997.1	5180
specific heat	4179	670
thermal conductivity	0.613	9.7
electrical conductivity	0.05	0.0000025

on steady state flow velocity, temperature field, skin friction and Nusselt number along with transient analysis on fluid velocity and temperature are illustrated graphically and explained qualitatively. The study proceeds by modeling the problem, tackling the governing equations numerically and discussing the results.

II. MATHEMATICAL FORMULATION

We consider unsteady flow of an incompressible, temperature dependent variable viscosity Magneto-hydrodynamic (MHD) ferrofluid in a horizontal parallel-plates microchannel of width units with permeable walls, where the bottom plate is kept at temperature of and the top plate at  such that . A uniform transverse magnetic field is applied in the direction orthogonal to -axis. The schematic diagram is depicted in Fig. 1. It is assumed that a 2D-coordinate system is used, wherein the bottom plate is fixed at , the top plate is fixed at  (the -axis is aligned vertically) and the -axis is parallel to the plates.

The ferrofluid flow through the microchannel is set in motion due to the effect of pressure gradient only. It is also assumed that the magnetic Reynolds number is so small that the induced magnetic field becomes diminutive to be ignored as compared to the imposed magnetic field. Using the result (Brinkman, 1952), temperature dependent dynamic viscosity of nanofluid is given by

,

where  is the base fluid dynamic viscosity at the ambient temperature,  is the nanoparticles volume fraction,  is the viscosity variation parameter and  is the nanofluid temperature. The values of thermophysical properties of water and ferro-nanoparticles (Fe₃O₄) nanoparticles used in the model are tabulated in Table 1 (Sheikholeslami and Ganji, 2016).

Under the above assumptions and applying MHD model equation, the governing equations of the problem for continuity, momentum and energy are given as:

,                                   (1)

,                              (2)

,                          (3)

with the initial and boundary conditions

, , (4)

, ,

, ,  (5)

where  is the nanofluid velocity in - direction,  is the temperature of nanofluid,  is the suction/injection velocity,  is the nanofluid pressure,  is the time,  is the temperature of the bottom plate,  is the temperature of the top plate,  is the microchannel width,  is the effective density of the nanofluid,  is the effective dynamic viscosity of the nanofluid,  is the electrical conductivity of the nanofluid,  is the thermal conductivity of the nanofluid,  is the heat capacitance of the nanofluid. Following the results (Brinkman, 1952), thermophysical properties of nanofluid in terms of its base fluid and particles volume fractions are given as

,

,

, ,

,

where  is the density of the base fluid,  is the nanoparticles density,  is the heat capacitance of the base fluid,  is the heat capacitance of the nanoparticles,  is the thermal conductivity of the base fluid,  is the thermal conductivity of the nanoparticle,  is the electrical conductivity of the base fluid,  is the electrical conductivity of the nanoparticles,  is dimensionless electrical conductivity.

Equations (2)-(5) turn in to non-dimensional equations by using and incorporating the following variables and parameters:

, , , , , ,

, , , ,

, ,                           (6)

Then the non-dimensional forms of partial differential Eqs. (2)-(5) are:

                               (7)

     (8)

with initial and boundary conditions

, ,  (9)

, ,

, ,  (10)

where is the Reynolds suction/injection parameter,  is the pressure gradient parameter, is the viscosity variation parameter,  is the magnetic field parameter,  is the Prandtl number and  is the Eckert number.

The important physical quantities of interest in the study of the problem are the local skin friction , and the local Nusselt number , where

   and                                      (11)

where  is the shear stress,  is the heat flux; and  and .

Equation (11) in non-dimensional form:

 and      (12)

but .

III. NUMERICAL METHOD

The model Eqs. (7)-(10) are a system of nonlinear IBVPs and their solutions can be tackled numerically using semi-discretization via finite difference scheme along with Runge-Kutta Fehlberg integration technique. The space interval [0,1] is  subdivided into  equal subintervals. The nodal spacing and points are defined respectively as  and , for  where  is the number of interior nodal points in [0, 1]. The first and second spatial derivatives are replaced by a central finite difference approximation of order  accuracy. Let  and  be represented by  and , respectively. Then the semi-discretization method with centered finite difference scheme for Eqs. (7)-(10) becomes:

                           (13)

                                        (14)

with initial conditions

                                   (15)

The boundary conditions at  are transformed to incorporate as follows:

.                                             (16)

Equations (13)-(16) is a system of first order nonlinear ordinary differential equations with well known initial conditions and can be solved iteratively by Runge-Kutta Fehlberg integration techniques (Na, 1979).

IV. RESULT AND DISCUSSION

The effects of magnetic field and temperature dependent variable viscosity on unsteady ferrofluid flow in asymmetrically heated horizontal parallel permeable plates microchannel has been studied. Solutions to nonlinear initial boundary value problems (IBVPs) have been tackled by semi-discretization via centered finite difference

Figure 2: Fluid velocity and temperature profiles across the channel with increasing time.

Figure 3: Unsteady fluid velocity profiles with increasing distance and time.

Figure 4: Unsteady fluid temperature profiles with increasing distance and time.

scheme along with Runge-Kutta Fehlberg integration technique. Numerical results of velocity profile, temperature profile, skin friction and Nusselt number are analyzed by using table 1 and considering governing parameter values of the flow , , , , , ,  if not mentioned. The obtained results have been highlighted through Figs. (2)-(18).

A. Transient Analysis

Outlines of evolution of velocity and temperature profiles are depicted in Figs. (2)-(4). Figure 2(a) shows the rise of fluid velocity from a value of zero in time and space at both top and bottom walls to its value of ceiling within the microchannel. The fluid temperature augments from its zero value at the bottom wall to its highest value at the top wall as illustrated in Fig. 2(b). From Figs. 3(a) and (b), it is observed that the fluid highest velocity value is reached near the center of the microchannel within short period of time and a steady state flow velocity value is obtained with increasing time. Also, a rise and fall of fluid flow velocity value is seen close to the bottom wall of the microchannel. It can be noted from Figs. 4(a) and (b) that the fluid temperature boosts in the whole micro-

Figure 5: Velocity profiles with increasing .

Figure 6: Velocity profiles with increasing , .

Figure 7: Velocity profiles with increasing .

channel width before a steady state temperature is achieved and its highest value is observed at the top wall of the microchannel. In addition, a vacillation of fluid temperature value is obtained very near to the top wall of the microchannel for . This is due to the fact that magnetic-nanoparticles in the ferrouid near the magnetic field move to the hot wall under the action of Kelvin force, as a result a dynamic fluctuating temperature distribution is produced for a short while.

B. Steady State Velocity Profiles

Figures (5)-(7) point up the effects of various model parameters on steady state nanofluid flow velocity profiles. The influence of suction/injection parameter  on nanofluid velocity is displayed in Figs. 5 (a) and (b). In

Figure 8: Temperature profiles with increasing , .

Figure 9: Temperature profiles with increasing .

Fig. 5 (a), it is observed that increasing suction parameter >0 dwindles the nanofluid velocity near the bottom permeable wall and the center of the microchannel but there is a wavering of flow velocity before an opposite flow velocity trends to happen in the vicinity of the top permeable wall. Increasing injection parameter <0, increases the fluid velocity at the bottom permeable wall which is followed by fluctuation of fluid velocity before a reverse phenomenon happened at the centerline of the channel and top permeable wall as shown in Fig. 5 (b). Figure 6 (a) demonstrates fluid flow velocity for various values of variable viscosity parameter . It is observed that as  increases, the flow velocity increases in the microchannel. This is because increasing will decrease the thickness of the fluid, which in turn allows increasing the flow velocity. The influence of magnetic field  on velocity is highlighted in Fig. 6 (b). It is seen that increasing in  reduced the fluid velocity, but a strong reducing effect is seen at the centerline and weak reducing effect is seen at both top and bottom permeable walls. In Figs. 7 (a) and (b), the effects of pressure gradient  and nanoparticles volume fraction  on velocity profile are illustrated. It is seen that the rise of nanofluid velocity with an increase in pressure gradient  in Fig. 7 (a), but a decrease flow velocity with an increase in nanoparticle volume fraction  in Fig. 7 (b).

C. Steady State Temperature Profiles

The temperature variation profiles for different values of physical parameters are graphically presented in Figs. (8) and (9). Figure 8(a) displays the impact of Prandtl number  on temperature profile. From the figure, it is observed that the nanofluid temperature profile decreases with an increase in . Nanofluid temperature variation profile with Eckert number  is highlighted in Fig. 8 (b). A positive  is directly proportional to heating (heat is

Figure 10: Skin friction versus L with increasing .

Figure 11: Skin friction versus L with increasing Re < 0.

Figure 12: Skin friction versus M with increasing A.

Figure 13: Skin friction versus  with increasing .

being provided across the wall into the fluid). Therefore, an increase in  rises the temperature of the nanofluid in the microchannel. Nanofluid temperature profile enhanced with introduction of nanoparticles  is displayed in Fig. 9 (a).

D. Skin friction and Nusselt number

The system profile has been attained its steady state within short period of time as shown in Figs. 3 and 4. Therefore, it is sensible to consider quantities of engineering interest skin friction  and Nusselt number  at steady state. Figures (10)-(18) demonstrate the skin friction and Nusselt number with variation in numerical values of various pertinent parameters. Skin frictions are plotted graphically in Figs. (10)-(14). The influence of suction/injection parameter  on skin friction at the bottom and top permeable walls of the microchannel is seen in Figs. (10) and (11). From these figures, skin friction lessens with an increase in suction parameter  and enlarges with an increase in injection parameter  at both permeable walls of the microchannel. It is also

Figure 14: Skin friction versus  with increasing .

Figure 15: Nusselt number versus  with increasing .

Figure 16: Nusselt number versus  with increasing .

found that increasing variable viscosity variation parameter  escalates and lightens the skin friction at bottom and top microchannel walls, respectively. The combined effects of magnetic field parameter  and pressure gradient  on skin friction at bottom and top walls of the micro-channel are revealed in Figs. 12 (a) and (b), respectively. In Fig. 12 (a), magnetic field  has a reducing effect on skin friction  but pressure gradient parameter  has a magnifying effect on  at the bottom wall. In fig. 12 (b), increasing  amplify, but  diminishes  at the top wall. Figures 13 (a) and (b) depicts the simultaneous impact of Eckert number  and pressure gradient parameter  on skin friction at the bottom and top walls of the microchannel. It is observed from the figures that skin friction  increases with an increase in both  and  at bottom wall but it decreases at the top wall of the microchannel. Skin friction versus pressure gradient  with increasing values of nanoparticles volume

Figure 17: Nusselt number versus  with increasing .

Figure 18:  Nusselt number versus  with increasing .

fraction  at both bottom and top walls are illustrated, respectively, in Figs. 14 (a) and (b). Figure 14 (a) entail that increasing  is obtained at bottom wall as both  and  augment. A similar trend of dwindled  is observed at top wall in Fig. 14 (b) with increasing both  and .

The Nusselt number characterizes the heat flux from a solid surface to a fluid and it is known as dimensionless heat transfer coefficient. The variation of Nusselt number  with viscosity variation parameter  for varying values of suction/injection parameters  at bottom and top permeable walls are shown in Figs. (15) and (16). From these figures, Nusselt number increase at both bottom and top walls as suction parameter  increases, while a reverse trend is found at both walls as injection parameter  increases. Also, increasing viscosity variation parameter lightens the Nusselt number at the bottom wall and escalates the Nusselt number at the top wall in the presence of both suction and injection parameter. Figures 17 (a) and (b) represent the Nusselt number  versus suction parameter  with increasing magnetic field parameter  at bottom and top walls, respectively. In Figs. 17 (a) and (b), the Nusselt number varies directly with the magnetic field parameter  at the bottom wall, but it varies inversely at the top wall. Nusselt number dependence on simultaneous increment of magnetic field parameter  and nanoparticle particles volume fraction  at bottom and top walls respectively are depicted in Figs. 18 (a) and (b). It can be seen that both magnetic field parameterand nanoparticle particles volume fraction  help in increasing Nusselt number  at the bottom wall but they have a decreasing effect on  at the top wall of the microchannel.

V. SUMMARY AND CONCLUSIONS

In this study, the combined effects of magnetic field, nanoparticles concentration, temperature dependent viscosity of unsteady ferrofluid flow through microchannel with parallel permeable walls and asymmetrically heated walls were thoroughly investigated. The effects of relevant parameters on ferrofluid velocity, temperature, skin friction and Nusselt number were scrutinized and analyzed with the help of their graphical representations. From graphical results, the following pertinent conclusions are drawn:

(i) The vacillation of nanofluid velocity and temperature is observed, respectively, very close to the bottom and top permeable walls of the microchannel.

(ii) Increase the effects of ,  leads to speed up the nanofluid velocity while it retards with increase of  and . In addition, a significant speed up/reduction of nanofluid velocity is seen at/around the centerline of the microchannel with rising of influential parameters on velocity.

(iii) Temperature of the nanofluid across the microchannel width from bottom to top wall increases with increase in  and , but decreased with .

(iv) Skin friction  decreases at both permeable walls of the microchannel with an increase in suction parameter  but  increases at both walls with an increase in injection parameter.

(v)  Skin friction  increases with the increase of the value of , ,  at the bottom wall while the reverse phenomenon occurs at the top wall of the microchannel.

(vi)  Increasing suction parameter enhances  at both walls. Butdecreases with increase.

(vii)  Increase in  and , augments  at the bottom wall and dwindle  at the top wall of the microchannel.

REFERENCES

Asfer, M., Mehta, B., Kumar, A., Khandekar, S. and Panigrahi, P. K. (2016) Effect of magnetic field on laminar convective heat transfer characteristics of ferrofluid flowing through a circular stainless steel tube. Int. J. Heat and Fluid Flow. 59, 74-86.

Belhadj, A., Bouchenafa R. and Saim, R. (2018) Numerical investigation of forced convection of nanofluid in microchannels heat sinks. J. Therm. Eng. 4, 2263-2273.

Brinkman, H.C. (1952) The viscosity of concentrated suspensions and solutions. J. Chem. Phys. 20, 571-571.

Clark, N. A. (2013) Ferromagnetic ferrofluids. Soft-Matter Phy. 504, 229-230.

Farahani, S.D., Alibeigi, M., Zakinia, A. and Goodarzi, M. (2021) The effect of microchannel‑porous media and nanofluid on temperature and performance of CPV system. J. Thermal Analysis and Calorimetry. https://doi.org/10.1007/s10973-021-11087-5.

Gui, N.G., Stanley, C., Nguyen, N.-T. and Rosengarten, G.  (2018) Ferrofluids for heat transfer enhancement under an external magnetic field. Int. J. Heat and Mass Trans. 123, 110-121.

Harms, T.M., Kazmierczak, M.J. and Gerner, F.M. (1999) Developing convective heat transfer in deep rectangular microchannels. Int. J. Heat Fluid Flow. 20, 149-157.

Hindebu, B., Makinde, O.D. and Guta, L. (2021) Unsteady mixed convection flow of variable viscosity nanofluid in a micro-channel filled with a porous medium. Indian J. Phys. https://doi.org/10.1007/ s12648-021-02116-y.

Hosseini, S.R., Sheikholeslami, M., Ghasemian, M. and Ganji, D.D. (2018) Nanofluid heat transfer analysis in a microchannel heat sink (MCHS) under the effect of magnetic field by means of KKL model. Powder Technol. 324, 36-47.

Kefene, M.Z., Makinde, O.D. and Enyadene, L.G. (2020) MHD variable viscosity mixed convection of nanofluid in a microchannel with permeable walls. Indian J. Pure & Appl. Phys. 58, 892-908.

Kurtoglu, E., Bilgin, A., Sesen, M., Msrloglu, B., Yildiz, M., Acar, H.F.Y. and Kosar, A. (2012) Ferrofluid actuation with varying magnetic fields for micropumping applications. Micro. Nano. 13, 683-694.

Lee, M.-Y. and Seo, J.-H. (2013) Numerical investigation on heat and flow characteristics of temperature-sensitive ferrofluid in a square cavity. Adv. in Mech. Eng. 2013, 1-9.

Makinde, O.D. (2018) Heat transfer in variable viscosity micro-channel flow of EG-Water/Ag nanofluids with convective cooling. Defect and Diffusion Forum. 387, 182-193.

Malvandi, A. and Ganji, D.D. (2016) Mixed convection of Alumina/Water nanofluid in microchannels using modified Buongiornos model in presence of heat source/sink. J. Appl. Fluid Mech. 9, 2277-2289.

Monaledi, R.L. and Makinde, O.D. (2018) Entropy analysis of a radiating variable viscosity EG/Ag nanofluid flow in microchannels with buoyancy force and convective cooling. Defect and Diffusion Forum. 387, 273-285.

Na, T.Y. (1979) Computational methods in engineering boundary value problems. Math. Sci. Eng. 145, Academic Press, Michigan.

Namburu, P.K. , Kulkarni, D.P., Dandekar, A. and Das, D.K. (2007) Experimental investigation of viscosity and specific heat of silicon dioxide nanofluids. Micro. Nano. Lett. 2, 67-71.

Nguyen, C.T., Desgranges, F., Roy, G., Galanis, N., Mare, T., Boucher, S. and Minsta, H.A. (2007) Temperature and particle-size dependent viscosity data for water based nanofluids hystresis phenomenon. Int. J. Heat and Fluid Flow. 28, 1492-1506.

Nguyen, Q., Sedeh, S.N., Toghraie, D. and Karimipour,  A. (2020) Numerical simulation of the ferrofluid flow in a porous ribbed microchannel heat sink: investigation of first and second laws of thermodynamics with single-phase and two-phase approaches. J. of the Brazilian Society of Mech. Sci. and Eng. 42, 492.

Obot, N.T. (2002) Toward a better understanding of friction and heat/mass transfer in microchannel - A literatures review. Microscale Thermophys. Eng. 6 155-173.

Rosensweig, R.E. (1997) Ferrohydrodynamics, Dover edition ed., Dover Publications Inc., Mineola, New York.

Rosengarten, G., Gan, N. and Stanley, C. (2019) Heat transfer enhancement using ferrofluids. Proc. 6th Int. Conf. Fluid Flow, Heat and Mass Transf. Ottawa, Canada. 152.

Salari, A., Navi, M. and Dalton, C. (2015) A novel alternating current multiple array electrothermal micropump for lab-on-a-chip applications. Biomicrofluidics. 9, 014113.

Shashikumar, N.S., Gireesha, B.J., Mahanthesh, B. and Prasannakumara, B.C. (2018) Brinkman-Forchheimer flow of SWCNT and MWCNT magneto-nanoliquids in a microchannel with multiple slips and Joule heating aspects. Multidiscipline Modeling in Materials and Structures Emerald Publishing Limited, 1573-6105.

Sheikholeslami, M. and Ganji, D.D. (2016) Ferrofluid convective heat transfer under the influence of external magnetic source. Alexandria Eng. J. 57, 49-60.

Snoussi, L., Ouerfelli, N., Sharma, K.V., Vrinceanu, N., Chamkhaf, A.J. and Guizani, A. (2018) Numerical simulation of nanofluids for improved cooling efficiency in a 3D copper microchannel heat sink (MCHS), Phys. and Chem. Liquids. 56, 311-331.

Taylor, R., Coulombe, S., Otanicar, T., Phelan, P., Gunawan, A. and Lv, W. (2013) Small particles, big impacts: a review of the diverse applications of nanofluids. J. Appl. Phys. 113, 011301.

Tuckerman, D.B. and Pease, R.F.W. (1981) High-performance heat sinking for VLSI. IEEE Electron Device Lett. 2, 126-129.

Weng, H.C. and Lo, K.J. (2015) Convective heat transfer of magnetic nanofluids in a microtube. Smart Sci. 3, 56-64.

Wie, X., Joshi, Y. and Patterson, M. K. (2007) Experimental and numerical study of a stacked microchannel heat sink for liquid cooling of microelectronic devices. J. of Heat Transf. 129, 1432-1444.

Received: October 15, 2020

Sent to Subject Editor: January 13, 2021

Accepted: January 17, 2022

Recommended by Subject Editor German Prieto