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Research Papers

# Assisting and Opposing Combined Convective Heat Transfer and Nanofluids Flows Over a Vertical Forward Facing StepOPEN ACCESS

[+] Author and Article Information
H. A. Mohammed

Mem. ASME
Department of Thermofluids,
Faculty of Mechanical Engineering,
Universiti Teknologi Malaysia,
UTM Skudai,
Johor Bahru 81310, Malaysia
e-mail: Hussein.dash@yahoo.com

Omar A. Hussein

Department of Mechanical Engineering,
College of Engineering,
Tikrit University,
PO Box 42,
Tikrit, Salahuddin, Iraq

1Corresponding author.

Manuscript received February 9, 2014; final manuscript received July 4, 2014; published online August 6, 2014. Assoc. Editor: Calvin Li.

J. Nanotechnol. Eng. Med 5(1), 010903 (Aug 06, 2014) (13 pages) Paper No: NANO-14-1010; doi: 10.1115/1.4028009 History: Received February 09, 2014; Revised July 04, 2014

## Abstract

Numerical simulations of two-dimensional (2D) laminar mixed convection heat transfer and nanofluids flows over forward facing step (FFS) in a vertical channel are numerically carried out. The continuity, momentum, and energy equations were solved by means of a finite volume method (FVM). The wall downstream of the step was maintained at a uniform wall heat flux, while the straight wall that forms the other side of the channel was maintained at constant temperature equivalent to the inlet fluid temperature. The upstream walls for the FFS were considered as adiabatic surfaces. The buoyancy assisting and buoyancy opposing flow conditions are investigated. Four different types of nanoparticles, Al2O3, CuO, SiO2, and ZnO with different volumes' fractions in the range of 1–4% and different nanoparticle diameters in the range of 25–80 nm, are dispersed in the base fluid (water) are used. In this study, several parameters, such as different Reynolds numbers in the range of 100 < Re < 900, and different heat fluxes in the range of 500 ≤ qw ≤ 4500 W/m2, and different step heights in the range of 3 ≤ S ≤ 5.8 mm, are investigated to identify their effects on the heat transfer and fluid flow characteristics. The numerical results indicate that the nanofluid with SiO2 has the highest Nusselt number compared with other nanofluids. The recirculation region and the Nusselt number increase as the step height, Reynolds number, and the volume fraction increase, and it decreases as the nanoparticle diameter increases. This study has revealed that the assisting flow has higher Nusselt number than opposing flow.

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## Introduction

The flow over a FFS and the heat transfer associated with it are considered as fundamental problems. The interest in these problems is expected to grow in the near future, as more sophisticated flow fields and heat transfer processes become involved in modern engineering design. Experimental data for the flow field are available but not from many sources. Most of the research regarding this problem was done in the late fifties and early sixties. Only one source provides partial heat transfer information for the turbulent case, Luzhanskiy and Solntsev [1]. Only one experimental research was reported later by Baker [2]. The problem has not yet been given the analytic attention it deserves. Hence, one of the major objectives of the current study is the evaluation and prediction of the flow field and the heat transfer associated with laminar flow over a FFS.

One of the ways to enhance heat transfer in the separated regions is to employ nanofluids. Nanofluids are fluids that contain suspended nanoparticles such as metals and oxides. These nanoscale particles keep suspended in the base fluid. Thus, it does not cause an increase in pressure drop in the flow field. Past studies showed that nanofluids exhibit enhanced thermal properties, such as higher thermal conductivity and convective heat transfer coefficients, compared to the base fluid. Several authors studied have been witnessed extensive in the literature on the convective heat transfer in nanofluids; see, for example, Daungthongsuk and Wongwises [3], Wang and Mujumdar [4], Al-aswadi et al. [5], and Mohammed et al. [6-7].

The problem of laminar flow over backward facing step (BFS) geometry in natural, forced, and mixed convection has been investigated extensively in the past, both numerically and experimentally by Lin et al. [8], Hong et al. [9], and Abu-Mulaweh et al. [10,11], and the references cited therein. On the other hand, the problem of laminar flow over a FFS has received very little attention.

The most studied case of the flow over a FFS available for laminar and the turbulent situations has been reported both numerically and experimentally, in rectangular and cylindrical co-ordinates, by Wilhelm and Kleiser [12], Largeau and Moriniere [13], Gandjalikhan Nassab et al. [14], and Barbosa-Saldana et al. [15,16].

A general review of the recent literature on laminar mixed convective flow over a FFS including 2D and three-dimensional studies, vertical, horizontal, and inclined ducts orientation were reported by Abu-Mulaweh [17]. Several numerical and experimental studies have been conducted to analyze the effects of the buoyancy forces on the velocity field and on the physical parameters defining the flow over the FFS. However, a very limited number of studies have been conducted to analyze the three-dimensional mixed convective flow over a FFS, and even fewer studies have been conducted to analyze the horizontal case where the buoyancy forces and the mainstream flow are perpendicular to each other.

On the other hand, the configuration of a FFS has been investigated much less than the BFS. Stuer et al. [18] mentioned in their publication that very little has been published referring to the laminar separation over a FFS, and neither its topology nor its recirculation zones are known in a predictable form. Abu-Mulaweh [17] reported that the phenomena of convection over the FFS have not been studied due to its complexity. He concluded that depending on the magnitude of the flow Reynolds number, one or two flow-separation regions may develop adjacent to the step.

Others authors like Asseban et al. [19] had conducted their studies for the FFS geometry to analyze the mixed convective flow in vertical plates or for studying the mixed convective flow in 2D channel for assisting and opposing flow as presented by Abu-Mulaweh et al. [20-22]. Although an important effort to analyze the flow passing a three-dimensional FFS has been made, most of the studies are limited to the 2D case.

Mixed convective flow over a FFS in inclined channel has been investigated by several researchers in the past decades. Abu-Mulaweh [23,24] investigated experimentally mixed convective flow over a 2D vertical FFS. They examined the effect of FFS height on turbulent mixed convection along a vertical flat plate. The upstream and downstream walls and the FFS itself were heated to a uniform and constant temperature. Results of interests, such as time–mean velocity and temperature, intensities of velocity and temperature fluctuation, reattachment lengths and local Nusselt number distributions, are reported to illustrate the effect of FFS heights on turbulent mixed convection along a vertical flat plate.

The first numerical study to investigate the flow and heat transfer over a BFS using nanofluids was done by Abu-Nada [25]. The Reynolds number and nanoparticles volume fraction used were in the range of 200 ≤ Re ≤ 600 and 0 ≤ Ø ≤ 0.2, respectively, for five types of nanoparticles which are Cu, Ag, Al2O3, CuO, and TiO2. He reported that the high Nusselt number inside the recirculation zone mainly depended on the thermophysical properties of the nanoparticles and it is independent of Reynolds number. Numerical analysis of forced and mixed convection over horizontal and vertical BFS in a duct using different nanofluids was conducted by Mohammed et al. [6-7]. The effects of Reynolds number (in the range of 75 ≤ Re ≤ 225), temperature difference (in the range of 0 ≤ ΔT ≤ 30 °C), and nanofluid type (such as Au, Ag, Al2O3, Cu, CuO, diamond, SiO2, and TiO2) were investigated on the fluid flow and heat transfer characteristics. It is found that a recirculation region developed straight behind the BFS which appeared between the edge of the step and few millimeters before the corner which connects the step and the downstream wall. In the few millimeters zone between the recirculation region and the downstream wall, a U-turn flow was developed opposite to the recirculation flow which is mixed with the unrecirculated flow and travels along the channel. It is inferred that Au nanofluid has the highest maximum peak of Nusselt number, while diamond nanofluid has the highest minimum peak in the recirculation region.

The study of steady laminar mixed convection flow over a FFS utilizing nanofluids in a 2D vertical configuration under uniform heat flux (UHF) boundary conditions seems not to have been investigated in the past and this has motivated the present study. Thus, the present study deals with different types of nanofluids, such as Al2O3, CuO, SiO2, and ZnO, with different volume fractions and different nanoparticle diameters. The effects of heat flux and Reynolds number on the velocity distribution, skin friction coefficient, and Nusselt number are studied and reported to illustrate the effect nanofluids on these parameters for buoyancy assisting and opposing flows.

## Numerical Model

###### Physical Model.

Considering the FFS placed in channel as shown schematically in Fig. 1. The step height and expansion ratio (ER) are fixed at 4.8 mm and 2, respectively. The upstream and downstream walls are 24 mm and 144 mm, respectively. The wall downstream of the step (Xe) is maintained at a uniform wall heat flux (qw), while the straight wall that forms the other side of the channel is maintained at constant temperature equivalent to the inlet fluid temperature (To). The wall upstream of the step (Xi) and the step itself (S) are considered as adiabatic surfaces.

Nanofluids flow at the channel entrance is considered to be hydrodynamically steady and the fully developed flow is attained at the edge of the step, and the streamwise gradients of all quantities at the channel exit where set to be zero.

The nanoparticles and the base fluid (i.e., water) are assumed to be in a thermal equilibrium and no slip condition occurs. The fluid flow is assumed to be Newtonian and incompressible. The nanofluid is treated as a single phase and it is adopted in this work because of the similarity between the single and two-phase models results as outlined by Bianco et al. [26]. Radiation heat transfer and viscous dissipation terms are neglected. The internal heat generation is not conducted in this study. The thermophysical properties of the nanofluids are assumed to be constant and it is only affected by the buoyancy force, which means that the body force acting on the fluid is the gravity; the density is varied and can be adequately modeled by the Boussinesq approximation.

###### Governing Equations.

To complete the computational fluid dynamics (CFD) analysis of FFS, it is important to setup the governing equations (continuity, momentum, and energy). Using the Boussinesq approximation and neglecting the viscous dissipation effect and compressibility effect, the dimensionless governing equations for 2D laminar incompressible flows can be written as follows [25]:

The continuity equationDisplay Formula

(1)$∂U∂X+∂V∂Y=0$

The X-momentum equationDisplay Formula

(2)$U∂U∂X+V∂U∂Y=-∂P∂X+μnfρnυnf1Re(∂2U∂X2+∂2U∂Y2)$

The Y-momentum equationDisplay Formula

(3)$U∂V∂X+V∂V∂Y=-∂P∂Y+μnfρnυnf1Re(∂2V∂X2+∂2V∂Y2)±(ρβ)nfρnfβnfGrRe2θ$

The energy equationDisplay Formula

(4)$U∂θ∂X+V∂θ∂Y=αnfαfRePr(∂2θ∂X2+∂2θ∂Y2)$

The dimensionless variables are as follows:

$X=XS, Y=YS, θ=T-ToTw-To, U=uUo,V=υUo, and Re=ρuoDhμ$

###### Boundary Conditions.

The boundary conditions for the present problem imposed at the solid walls are mainly no-slip boundary condition in addition to the specified wall temperature. The condition assumed at the inlet section are those of ambient conditions (ambient temperature and velocity), while the only condition imposed at the exit is uniform ambient pressure. The boundary conditions for the above set of governing equations are

1. (1)Upstream inlet conditions at
$X=-XiDh, SDh≤Y≤HDh, and, Ui=uiuo, V=0, θ=0$
2. (2)Downstream exit conditions at
$X=XeDh, 0≤Y≤HDh, and, ∂2U∂X2=0, ∂2V∂X2=0, ∂2θ∂X2=0$
All the quantities at the channel exit are set to zero.
3. (3)Top wall conditions at
$X=-XiDh≤X≤XeDh, Y=HDh, and, U=0, V=0, θ = 0$
4. (4)Sidewalls conditions at
$X=-XiDh≤X≤XeDh, 0≤Y≤HDh, and, U=0, V=0, θ=0$
5. (5)Stepped wall condition
1. (a)Upstream of the step at
$-XiDh≤X≤0, Y=SDh, and, U=0,V=0, ∂θ∂Y=0$
2. (b)At the step
$X=0, 0≤Y≤SDh, and, U=0, V=0, ∂θ∂Y=0$
3. (c)Downstream of the step at
$0

###### Grid Independence Test.

Numerical studies are used to conduct the grid independence test for the effect of heat transfer of different nanofluids over BFS geometry. The grid tests are conducted with nanofluid as a working fluid at Re = 50 and 175, the temperature difference between the downstream wall and the inlet flow is ΔT = 15 °C. Grid densities of 140 × 50, 128 × 50, and 140 × 60, were selected to perform a grid independence test. A grid size of 128 × 50 confirms the grid independent solution. It shows less than 4% difference in velocity and temperature distribution compared with other grid sizes. The distance between the nodes has a last/first ratio of 14 in the x-direction and 3.5 in the y-direction. In this study, quadrilateral elements and a non-uniform grid system are employed. The grid is highly concentrated near the step edge, close to the step, and walls, to ensure the accuracy of the numerical simulations and for saving both the computational time and grid size. The expansion successive ratio is the mathematical relation used for setting up the grid. The second grid tests are carried out for Reynold number (Re = 100) of air flow where the downstream wall is subjected to a UHF of qw = 200 W/m2. The grid nodes are performed with different grid densities and grid numbers in x-direction (x = 100, 120, and 140) and in y-direction (y = 50, 60, and 70) to ensure the grid independence solution. A grid of 100 × 50 confirms the grid independence solution. It shows less than 3% difference in Nusselt number compared with other grid sizes as shown in Table 1. In addition, a fine grid is used in the x-direction nearer to the step wall section, while it increases with a constant ER (first/last) of 0.1 in x-direction away from the step wall. Other fine grid of the expansion ratios (first/last) of 0.285 in y-direction are utilized nearer to the top wall, bottom wall, and at the step edge to ensure the accuracy of the grid independent solution.

###### Code Validation.

The process of numerical validation method consists of running the numerical code under specific conditions for benchmark problems and then comparing the obtained results with the available experimental or theoretical numerical data in the literature. The first test case was simulating forced convective nanofluids flow over a horizontal BFS placed in a duct [5]. For this validation, the Reynolds numbers chosen are in the laminar regime (Re = 50 and 175). Figures 2–4 show the results of the comparison with Al-aswadi et al. [5] for different nanofluids. The velocity distributions for various X/s in the recirculation region along the duct are shown in Figs. 2 and 3. The recirculation region clearly appears and the size of the recirculation region decreases as the distance between the step and the stepped wall increases until the flow reaches the reattachment point where the flow exhibits zero velocity, as shown in Figs. 2(a)2(d) and 3(a)3(d). Figures 2(a)2(c) and 3(a)3(c) show that a recirculation region developed downstream the step. Downstream of this point, the flow starts to become fully developed flow as the fluid flows toward the duct exit as shown in Figs. 2(d) and 3(d). Figure 4 shows the skin friction coefficient at the bottom wall downstream the step. The skin friction coefficient increases and reaches as the distance downstream from the step increases. It is then decreases monotonically until it reaches its maximum peak and the minimum peak. It increases until it reaches a point where the skin friction coefficient remains constant along the rest of the bottom wall. This shows that the skin friction coefficient is approaching asymptotically the fully developed channel flow. The second test case was simulating mixed convective airflow over BFS placed in a duct [9]. The Reynolds number (Re = 100) of air flow is maintained constant. The downstream wall is subjected to a UHF of qw = 200 W/m2. The velocity distribution and Nusselt number at 0 deg angle are validated and compared as shown in Figs. 5(a) and 5(b). The comparison of the present results with the results of Refs. [5] and [9] shows a good agreement.

###### Numerical Parameters and Procedures.

The numerical computation was carried out by solving the governing conservation equations along with the boundary conditions, Eqs. (1)–(4). Equations for solid and fluid phase were simultaneously solved as a single domain. The discretization of governing equations in the fluid and solid regions was done using the FVM. The diffusion term in the momentum and energy equations is approximated by second-order central difference which gives a stable solution. In addition, a second-order upwind differencing scheme is adopted for the convective terms. The flow field was solved using the SIMPLE algorithm [27]. This is an iterative solution procedure where the computation is initialized by guessing the pressure field. Then, the momentum equation is solved to determine the velocity components. The pressure is updated using the continuity equation. Even though the continuity equation does not contain any pressure, it can be transformed easily into a pressure correction equation [27].

###### Thermophysical Properties of Nanofluids.

In order to carry out simulations for nanofluids, the effective thermophysical properties of nanofluids must be calculated first. In this case, the nanoparticles being used are A12O3, CuO, SiO2, and ZnO. Basically, the required properties for the simulations are effective thermal conductivity (keff), effective dynamic viscosity (μeff), effective mass density (ρeff), effective coefficient of thermal expansion (βeff), and effective specific heat ($Cpeff$). The effective properties of mass density, specific heat, and coefficient of thermal expansion are actually calculated according to the mixing theory.

By using Brownian motion of nanoparticles for flow over BFS, the effective thermal conductivity can be obtained using the following mean empirical correlation [28]:Display Formula

(5)$keff=kStatic+kBrownian$

Static thermal conductivityDisplay Formula

(5.1)$kStatic=kbf[knp+2kbf-2(kbf-knp)∅knp+2kbf+(kbf-knp)∅]$

Brownian thermal conductivityDisplay Formula

(5.2)$kBrownian=5×104β∅ρbfcp,bfkT2ρnpRnp·f(T,∅)$

where Boltzmann constant $k=1.3807×10-23J/K$; Modeling function, β [29]: $βAl2O3=8.4407(100∅)-1.07304$; $βCuO=9.881(100∅)-0.9446$; $βSiO2=1.9526(100∅)-1.4594$; $βZnO=8.4407(100∅)-1.07304$

Modeling function, f(T, Ø)

$f(T,∅)=(2.8217×10-2∅)+3.917×10-3)(TTo) +(-3.0699×10-2∅-3.91123×10-3)$

The effective viscosity can be obtained by using the following empirical correlation [30]:Display Formula

(6)$Viscosity: μeffμf=11-34.87(dpdf)-0.3∅1.03$
Display Formula
(7)$Equivalent diameter of base fluid molecule:df= [6MNπρbf]1/3$

The density of the nanofluid ($ρnf$) can be calculated using [28]Display Formula

(8)$ρnf=(1-∅)ρf+∅ρnp$

The effective heat capacity at constant pressure of the nanofluid $(ρcp)nf$ can be calculated using [28]Display Formula

(9)$(ρcp)nf=(1-∅)(ρcp)f+∅(ρcp)np$

The effective coefficient of thermal expansion of nanofluid, (ρβ)nf can be calculated using [28]Display Formula

(10)$(ρβ)nf=(1-∅)(ρβ)f+∅(ρβ)np$

Table 2 shows the thermophysical properties for pure water and different nanofluids at T = 300 K.

## Results and Discussion

This section is designated to present the numerical results for the study of mixed convective flow over 2D vertical FFS. The results of velocity distribution, skin frication coefficient, and Nusselt number at different nanofluids, nanoparticles volume fractions, nanoparticles diameters; Reynolds numbers, heat fluxes, and step heights are predicted and discussed in this section. The Reynolds numbers are maintained in the range of 100 ≤ Re ≤ 900. The downstream wall of the step is maintained at a uniform wall heat flux in the range of 500 ≤ qw ≤ 4500 W/m2. The flow over a vertical FFS is characterized by two different recirculating regions. After that, the velocity profile is reattached and redeveloped approaching a fully developed flow as the fluid flows towards the channel exit.

###### The Effect of Nanofluids Parameters.

Different types of nanoparticles which are Al2O3, CuO, SiO2, and ZnO and pure water as a base fluid are used. In order to see the effects of different nanofluids on the heat transfer enhancement, all other parameters should be fixed, $∅$ = 0.04 and dp = 25 nm at Re = 300 and qw = 500 W/m2 along the downstream wall. Figure 6(a) shows that the Nusselt number at the heated wall increases with increasing the distance from the downstream wall to a maximum value near the step wall at some distance downstream wall of the reattachment point, and then it decreases slowly as the distance continues to increase in the streamwise direction. It is found that the base fluid with SiO2 nanofluids has the highest maximum peak in Nusselt number followed by Al2O3, CuO, and ZnO. This is because inside the recirculation zone the nanofluid needs to have lower density and thermal conductivity. This result was also obtained by Kherbeet et al. [31,32].

In this study, the effect of the nanoparticle volume fraction $∅$ in the range of 1–4% on the heat transfer characteristics is studied. Figure 6(b) shows the effect of different nanoparticle volume fractions on the Nusselt number for constant nanoparticle diameters dp = 25 nm at Re = 300 and qw = 500 W/m2. As $∅$ increased, the Nusselt number of SiO2 nanofluids becomes higher than that of pure fluids. This is because increasing the volume fraction leads to enhance the heat transfer as shown in Fig. 6(b). This happens because the temperature difference decreases as the volume fraction increases and therefore the heat transfer enhancement increases.

To study the effect of the nanoparticle diameter on the heat transfer, it is evident that the Nusselt number varies significantly with the mean nanoparticle diameter between 25 and 80 nm, while Re, qw, and $∅$ are fixed at 300, 500 W/m2 and 4%, respectively. The result of Nusselt number for SiO2 nanofluids is shown in Fig. 6(c). As the mean nanoparticle diameter increases, the Nusselt number decreases. This is because the nanofluids properties increase with decreasing nanoparticles diameter. The properties of nanofluid come from relatively high surface area to the volume ratio. Thus, it is concluded that using smaller diameter of nanoparticles will lead to get better heat transfer enhancement. It can be observed that Nusselt number increases with decreasing the nanoparticles diameter.

###### Velocity Distribution.

The velocity distributions of SiO2 nanofluid for different types of Reynolds numbers in range of 100 ≤ Re ≤ 900 with qw = 500 W/m2, $∅$ = 4%, and dp = 25 nm at different x/S sections along the stepped wall are shown in Fig. 7. The velocity profile increases as Reynolds number increases. It is noticed that the higher Reynolds number has a higher parabolic velocity profile for SiO2 nanofluid as shown in Fig. 7(a). It is found that the velocity increases as Reynolds number increases, which enhance the recirculation size and the flow reattached farther from the step. The distance between the step and the stepped wall increases until the flow reaches the reattachment point and the size of the recirculation region decreases as shown in Figs. 7(b) and 7(c). Downstream of this point, the flow starts to redevelop and then approach fully developed flow as the fluid flows towards the exit as shown in Fig. 7(d).

###### Skin Friction Coefficient.

The skin friction coefficient of SiO2 nanofluid for different Reynolds numbers in range of 100 ≤ Re ≤ 900 with qw = 500 W/m2, $∅$ = 4%, and dp = 25 nm along the downstream wall is shown in Fig. 8(a). The trend of skin friction coefficient along the heated stepped wall is more sensitive to vortices and recirculation flows. It is noticed to increase from the corner joining the step wall and the downstream wall until it reaches a maximum point in upstream the exit. It is indicated that the skin friction coefficient decreases as the Reynolds number increases. This is because the skin friction coefficient is inversely proportional to the velocity.

###### Nusselt Number.

The Nusselt number of SiO2 nanofluid for different Reynolds numbers in range of 100 ≤ Re ≤ 500 with qw = 500 W/m2, $∅$ = 4%, and dp = 25 nm along the downstream wall is shown in Fig. 8(b). It is found that the Nusselt number increases from downstream the step wall until it reaches step wall. After that, it decreases gradually along the heated downstream wall and becomes steeper in the vicinity of the exit. The results show that the flow with high Re number is found to have the lowest minimum peak of Nu number at the mixing region. Along this region, it is found that the SiO2 nanofluid has higher Nusselt number at higher Reynolds number.

###### Velocity Distribution.

The velocity distribution of SiO2 nanofluid for different heat fluxes in the range of 500 ≤ qw ≤ 4500 W/m2 with Re = 100, $∅$ = 4%, and dp = 25 nm at different x/S sections along the stepped wall is shown in Fig. 9. It is noticed that the buoyancy force on the flow profile has an insignificant influence at the step edge section and the velocity profiles seem to be the same as the forced convective flow as shown in Fig. 9(a). It is found that the size of the recirculation region increases as the heat flux increases between the heated stepped wall and the nanofluid as shown in Figs. 9(b) and 9(c). The buoyancy force is not more significant to push up the high velocity flow with lower heat flux, but it affected the flow by increasing the reattachment length. The size of the recirculation region decreases as the distance between the step and the stepped wall increases until the flow reaches the reattachment point as shown in Figs. 9(b) and 9(c). Then, the flow starts to redevelop and approach fully developed flow as the fluid flows towards the exit as shown in Fig. 9(d).

###### Skin Friction Coefficient.

The skin friction of SiO2 nanofluid with $∅$ = 4% and dp = 25 nm for different heat fluxes in the range of 500 ≤ qw ≤ 4500 W/m2 at Re = 100 along the downstream wall is shown in Fig. 10(a). The trend of skin friction coefficient along the heated stepped wall is more sensitive to vortices and recirculation flows. It is found that the skin friction coefficient decreases slightly to a minimum point due to the contact of the upward flow with the step wall. Then, it increases gradually along the downstream wall until it reaches a maximum point upstream the exit. This is because the skin friction coefficient is inversely proportional to the velocity. It is found that the heat flux does not have significant effect on the skin friction coefficient.

###### Nusselt Number.

The Nusselt number of SiO2 nanofluid with $∅$ = 4% and dp = 25 nm for different heat fluxes in the range of 500 ≤ qw ≤ 4500 W/m2 at Re = 100 along the downstream wall is shown in Fig. 10(b). It is found that the Nusselt number increases from downstream the step wall until it reaches step wall. After that, it decreases gradually along the heated downstream wall and becomes steeper in the vicinity of the exit. The results show that the flow with high heat flux is found to have the highest minimum peak in Nu at the mixing region. Along this area, it is found that the SiO2 nanofluid has higher Nusselt number with higher heat fluxes. It is found that the heat flux does not have significant effect on the Nusselt number.

###### The Effect of Different Steps Heights.

The Nusselt number of SiO2 nanofluid with $∅$ = 4% and dp = 25 nm for different step heights of qw = 500 W/m2 at Re = 500 along the downstream wall is shown in Fig. 11. The Nusselt number increases as the step height increases due to increment of the recirculation region. The bulk temperature increases more rapidly as the step height increases. Increasing the step height causes the magnitude of the maximum kinetic energy to increase. It is noticed that the Nusselt number is higher inside the recirculation region as the step height decreases until it reaches its optimum at the reattachment point. The maximum Nusselt number is found at 5.8 mm followed by 4.8 mm and 3 mm as shown in Fig. 11(a). In this section, the total temperature vectors of SiO2 nanofluid of mixed convective flow along the downstream wall as shown in Figs. 11(b)11(d) which shows the development of a primary recirculation straight downstream the stepped wall.

###### The Effect of Flow Direction.

In this study, buoyancy assisting and opposing laminar mixed convection flow over a FFS in a 2D channel subjected to a UHF is investigated as shown in Fig. 12. This figure shows that the Nusselt number of assisting flow is higher than Nusselt number of opposing flow. This is because the body force for assisting flow has the axial flow direction, which accelerates the fluid resulting in an increase in the heat transfer coefficient. However, for opposing flow, the body force acts opposite to the axial flow direction, thus retarding the flow and possibly causing flow reversal in the upper part of the cross section. Therefore, the Nusselt number results for opposed flow were lower than the assisted flow.

## Conclusions

Numerical simulations for laminar mixed convection flow over 2D vertical FFS were reported. The emphasis is given on the heat transfer enhancement resulting from various parameters, which include the type of nanofluids (Al2O3, CuO, ZnO, and SiO2 with H2O), nanoparticles diameter in the range of 25 ≤ dp ≤ 80 nm, and volume fraction (concentration) of nanoparticles in the range of 1% ≤ $∅$ ≤ 4%. The Reynolds number of the flow over FFS was in the range of 100 < Re < 900. The downstream wall of the FFS was fixed at UHF boundary condition in the range of 500 ≤ qw ≤ 4500 W/m2 and the channel has different step heights in the range of 3 ≤ S ≤ 5.8 mm. The governing equations were solved utilizing FVM with certain assumptions and appropriate boundary conditions. In addition, the current study examined the assisting and opposing flow conditions on the heat transfer characteristics. The following conclusions are drawn from this study:

• The results show that SiO2 gives the highest Nusselt number followed by Al2O3, CuO, and ZnO, respectively, while pure water gives the lowest Nusselt number.

• The Nusselt number increased with increasing the volume fraction of nanoparticles, Reynolds number, and the step height.

• The Nusselt number increases gradually when decreasing the nanoparticles diameter.

• The Nusselt number increases as the heat fluxes increases. However, it increases or decreases based on the flow direction for different heat fluxes.

• The primary recirculation region is developed straight downstream the stepped wall for mixed convection flow at different Reynolds numbers and heat fluxes. The primary recirculation region increases as Reynolds number increases.

• The flow direction affects the heat transfer phenomena. Assisting flow condition gives better heat transfer rate than opposing flow condition.

Nomenclature
• A =

area (m2)

• AR =

aspect ratio

• Al2O3 =

aluminum oxide

• BFS =

backward facing step

• Cp =

specific heat (kJ/kg·K)

• dp =

diameter of nanofluid particles (nm)

• Dh =

hydraulic diameter at inlet, 2 h, (m)

• e =

expansion successive ratio

• ER =

expansion ratio (H/S)

• f =

elliptic relaxation function

• FFS =

forward facing step

• g =

gravitational acceleration (m/s2)

• Gr =

Grashof number (g β q S4/v2)

• h =

inlet channel height (m)

• h =

convective heat transfer coefficient (W/m2·K)

• H =

total channel height (m)

• k =

thermal conductivity (W/m·K)

• k =

turbulent kinetic energy (m2/s2)

• L =

total length channel (m)

• Nu =

Nusselt number (h s/k)

• Nus,max =

maximum Nusselt number (h s/k)

• P =

nodal point, number of processors

• P =

pressure (pa)

• Pr =

Prandtl number (μ Cp/k)

• Po =

pressure at the outlet (pa)

• qw =

Wall heat flux (W/m2)

• Re =

Reynolds number (ρ uoDh/μ)

• Ri =

Richardson number (Gr/Re2)

• S =

step height (m)

• T =

Temperature (K)

• To =

temperature at the inlet, outlet, or top wall (K)

• Tw =

heat wall temperature (K)

• u =

velocity component x-direction (m/s)

• U =

dimensionless streamwise velocity component (u/uo)

• uo =

average velocity for inlet flow (m/s)

• u =

free-stream velocity (m/s)

• UHF =

uniform heat flux

• v =

velocity component y-direction (m/s)

• V =

dimensionless transverse velocity component (v/uo)

• x =

x-coordinate direction (m)

• X =

dimensionless length at x-coordinate (x/S)

• Xe =

downstream wall length (m)

• Xi =

upstream wall length (m)

• Xn =

Nusselt number peak length (m)

• Xo =

secondary recirculation length (m)

• Xr =

reattachment length (m)

• Y =

dimensionless length at y-coordinate (y/S)

• y =

y-coordinate direction (m)

Greek Symbols
• α =

thermal diffusivity of the fluid (m2/s)

• β =

thermal expansion coefficient (1/K)

• Δ =

amount of difference

• θ =

dimensionless temperature [(T − To)/(Tw − To)]

• μ =

dynamic viscosity (N·m/s)

• υ =

kinematic viscosity (m2/s)

• ρ =

density (kg/m3)

• $φ$ =

volume fraction (%)

Subscripts
• bf =

base fluid

• f =

fluid

• i =

inlet

• nf =

nanofluid

• o =

outlet

• s =

solid (nanoparticles)

• W =

wall

## References

Luzhanskiy, B. Y., and Solntsev, V., 1972, “An Experimental Study of Flow in the Separation Zones of a Turbulent Boundary Layer Upstream of a Two-Dimensional Step,” Akad Nauk Sssr Mekh zhidk Gaza.
Baker, S., 1977, “Regions of Recirculating Flow Associated With Two-Dimensional Steps,” Ph.D. thesis, University of Surrey, Surrey, UK.
Daungthongsuk, W., and Wongwises, S., 2007, “A Critical Review of Convective Heat Transfer of Nanofluids,” Renewable Sustainable Energy Rev., 11(5), pp. 797–817.
Wang, X. Q., and Mujumdar, A. S., 2007, “Heat Transfer Characteristics of Nanofluids: A Review,” Int. J. Therm. Sci., 46(1), pp. 1–19.
Al-aswadi, A. A., Mohammed, H. A., Shuaib, N. H., and Campo, A., 2010, “Laminar Forced Convection Flow Over a Backward Facing Step Using Nanofluids,” Int. Commun. Heat Mass Transfer, 37(8), pp. 950–957.
Mohammed, H. A., Al-aswadi, A. A., Yusoff, M. Z., and Saidur, R., 2012, “Buoyancy-Assisted Mixed Convective Flows Over Backward Facing Step in a Vertical Duct Using Various Nanofluids,” Thermophys. Aeromech., 42(1), pp. 33–60.
Mohammed, H. A., Al-aswadi, A. A., Shuaib, N. H., and Saidur, R., 2011, “Convective Heat Transfer and Fluid Flow Study Over a Step Using Nanofluids: A Review,” Renewable Sustainable Energy Rev., 15(6), pp. 2921–2939.
Lin, J. T., Armaly, B. F., and Chen, T. S., 1991, “Mixed Convection Heat Transfer in Inclined Backward-Facing Step Flows,” Int. J. Heat Mass Transfer, 34, pp. 1568–1571.
Hong, B., Armaly, B. F., and Chen, T. S., 1993, “Laminar Mixed Convection in a Duct With a Backward-Facing Step: The Effects of Inclination Angle and Prandtl Number,” Int. J. Heat Mass Transfer, 36(12), pp. 3059–3067.
Abu-Mulaweh, H. I., Armaly, B. F., and Chen, T. S., 1995, “Laminar Natural Convection Flow Over a Vertical Backward-Facing Step,” ASME J. Heat Transfer, 117, pp. 895–901.
Abu-Mulaweh, H. I., Armaly, B. F., and Chen, T. S., 2001, “Turbulent Mixed-Convection Flow Over a Backward-Facing Step,” Int. J. Heat Mass Transfer, 44(14), pp. 2661–2669.
Wilhelm, D., and Kleiser, L., 2002, “Application of a Spectral Element Method to Two-Dimensional Forward-Facing Step Flow,” J. Sci. Comput., 17(1–4), pp. 619–627.
Largeau, J., and Moriniere, V., 2007, “Wall Pressure Fluctuations and Topology in Separated Flows Over a Forward-Facing Step,” Exp. Fluids, 42(1), pp. 21–40.
Gandjalikhan Nassab, S., Moosavi, R., and Hosseini Sarvari, S., 2009, “Turbulent Forced Convection Flow Adjacent to Inclined Forward Step in a Duct,” Int. J. Therm. Sci., 48(7), pp. 1319–1326.
Barbosa Saldana, J. G., Anand, N. K., and Sarin, V., 2005, “Numerical Simulation for Mixed Convective Flow Over a Three-Dimensional Horizontal Backward Facing Step,” ASME Heat Transfer/Fluids Engineering Summer Conference, Charlotte, NC, July 11–15, pp. 1031–1042.
Barbosa Saldana, J. G., and Anand, N., 2007, “Flow Over a Three-Dimensional Horizontal Forward-Facing Step,” Numerical Heat Transfer, Part A: Applications, 53(1), pp. 1–17.
Abu-Mulaweh, H., 2003, “A Review of Research on Laminar Mixed Convection Flow Over Backward-and Forward-Facing Steps,” Int. J. Therm. Sci., 42(9), pp. 897–909.
Stuer, H., Gyr, A., and Kinzelbach, W., 1999, “Laminar Separation on a Forward Facing Step,” Eur. J. Mech. B, 18(4), pp. 675–692.
Asseban, A., Lallemand, M., Saulnier, J. B., Fomin, N., Lavinskaja, E., Merzkirch, W., and Vitkin, D., 2000, “Digital Speckle Photography and Speckle Tomography in Heat Transfer Studies,” Opt. Laser Technol., 32(7–8), pp. 583–592.
Abu-Mulaweh, H. I., Armaly, B. F., Chen, T. S., and Hong, B., 1994, “Mixed Convection Adjacent to a Vertical Forward-Facing Step,” Proceedings of the 10th International Heat Transfer Conference, 5, pp. 423–428.
Abu-Mulaweh, H. I., Armaly, B. F., and Chen, T. S., 1993, “Measurements of Laminar Mixed Convection Flow Over a Horizontal Forward-Facing Step,” J. Thermophys. Heat Transfer, 7(4), pp. 569–573.
Abu-Mulaweh, H. I., Armaly, B. F., and Chen, T. S., 1993, “Measurements of Laminar Mixed Convection in a Boundary-Layer Flow Over Horizontal and Inclined Backward-Facing Steps,” Int. J. Heat Mass Transfer, 36(7), pp. 1883–1895.
Abu-Mulaweh, H. I., Chen, T. S., and Armaly, B. F., 2002, “Turbulent Mixed-Convection Flow Over a Backward-Facing Step—The Effect of Step Heights,” Int. J. Heat Fluid Flow, 23(6), pp. 758–765.
Abu-Mulaweh, H. I., Armaly, B. F., and Chen, T. S., 2003, “Measurements of Turbulent Mixed Convection Flow Over a Vertical Forward-Facing Step,” ASME Proceedings of the Summer Heat Transfer Conference, Las Vegas, NV, July 21–23, pp. 755–763, Paper No. HT2003-47088.
Abu-Nada, E., 2008, “Application of Nanofluids for Neat Transfer Enhancement of Separated Flows Encountered in a Backward Facing Step,” Int. J. Heat Fluid Flow, 29(1) pp. 242–249.
Bianco, V., Chiacchio, F., Manca, O., and Nardini, S., 2009, “Numerical Investigation of Nanofluids Forced Convection in Circular Tubes,” Appl. Therm. Eng., 29(17), pp. 3632–3642.
Patankar, S. V., 1980, “Numerical Heat Transfer and Fluid Flow,” Hemisphere Publishing Corporation, Washington, DC.
Ghasemi, B., and Aminossadati, S. M., 2010, “Brownian Motion of Nanoparticles in a Triangular Enclosure With Natural Convection,” Int. J. Therm. Sci., 49(6), pp. 931–940.
Vajjha, R. S., and Das, D. K., 2009, “Experimental Determination of Thermal Conductivity of Three Nanofluids and Development of New Correlations,” Int. J. Heat Mass Transfer, 52(21–22), pp. 4675–4682.
Corcione, M., 2010, “Heat Transfer Features of Buoyancy-Driven Nanofluids Inside Rectangular Enclosures Differentially Heated at the Sidewalls,” Int. J. Therm. Sci., 49(9), pp. 1536–1546.
Kherbeet, A., Sh., Mohammed, H. A., and Salman, B. H., 2012, “The Effect of Nanofluids on Mixed Convection Flow Over Microscale Backward-Facing Step,” Int. J. Heat Mass Transfer, 55(21–22), pp. 5870–5881.
Heshmati, A., Mohammed, H. A., and Darus, A. N., 2014, “Mixed Convection Heat Transfer of Nanofluids Over Backward Facing Step Having a Slotted Baffle,” Appl. Math. Comput., 240, pp. 368–386.
View article in PDF format.

## References

Luzhanskiy, B. Y., and Solntsev, V., 1972, “An Experimental Study of Flow in the Separation Zones of a Turbulent Boundary Layer Upstream of a Two-Dimensional Step,” Akad Nauk Sssr Mekh zhidk Gaza.
Baker, S., 1977, “Regions of Recirculating Flow Associated With Two-Dimensional Steps,” Ph.D. thesis, University of Surrey, Surrey, UK.
Daungthongsuk, W., and Wongwises, S., 2007, “A Critical Review of Convective Heat Transfer of Nanofluids,” Renewable Sustainable Energy Rev., 11(5), pp. 797–817.
Wang, X. Q., and Mujumdar, A. S., 2007, “Heat Transfer Characteristics of Nanofluids: A Review,” Int. J. Therm. Sci., 46(1), pp. 1–19.
Al-aswadi, A. A., Mohammed, H. A., Shuaib, N. H., and Campo, A., 2010, “Laminar Forced Convection Flow Over a Backward Facing Step Using Nanofluids,” Int. Commun. Heat Mass Transfer, 37(8), pp. 950–957.
Mohammed, H. A., Al-aswadi, A. A., Yusoff, M. Z., and Saidur, R., 2012, “Buoyancy-Assisted Mixed Convective Flows Over Backward Facing Step in a Vertical Duct Using Various Nanofluids,” Thermophys. Aeromech., 42(1), pp. 33–60.
Mohammed, H. A., Al-aswadi, A. A., Shuaib, N. H., and Saidur, R., 2011, “Convective Heat Transfer and Fluid Flow Study Over a Step Using Nanofluids: A Review,” Renewable Sustainable Energy Rev., 15(6), pp. 2921–2939.
Lin, J. T., Armaly, B. F., and Chen, T. S., 1991, “Mixed Convection Heat Transfer in Inclined Backward-Facing Step Flows,” Int. J. Heat Mass Transfer, 34, pp. 1568–1571.
Hong, B., Armaly, B. F., and Chen, T. S., 1993, “Laminar Mixed Convection in a Duct With a Backward-Facing Step: The Effects of Inclination Angle and Prandtl Number,” Int. J. Heat Mass Transfer, 36(12), pp. 3059–3067.
Abu-Mulaweh, H. I., Armaly, B. F., and Chen, T. S., 1995, “Laminar Natural Convection Flow Over a Vertical Backward-Facing Step,” ASME J. Heat Transfer, 117, pp. 895–901.
Abu-Mulaweh, H. I., Armaly, B. F., and Chen, T. S., 2001, “Turbulent Mixed-Convection Flow Over a Backward-Facing Step,” Int. J. Heat Mass Transfer, 44(14), pp. 2661–2669.
Wilhelm, D., and Kleiser, L., 2002, “Application of a Spectral Element Method to Two-Dimensional Forward-Facing Step Flow,” J. Sci. Comput., 17(1–4), pp. 619–627.
Largeau, J., and Moriniere, V., 2007, “Wall Pressure Fluctuations and Topology in Separated Flows Over a Forward-Facing Step,” Exp. Fluids, 42(1), pp. 21–40.
Gandjalikhan Nassab, S., Moosavi, R., and Hosseini Sarvari, S., 2009, “Turbulent Forced Convection Flow Adjacent to Inclined Forward Step in a Duct,” Int. J. Therm. Sci., 48(7), pp. 1319–1326.
Barbosa Saldana, J. G., Anand, N. K., and Sarin, V., 2005, “Numerical Simulation for Mixed Convective Flow Over a Three-Dimensional Horizontal Backward Facing Step,” ASME Heat Transfer/Fluids Engineering Summer Conference, Charlotte, NC, July 11–15, pp. 1031–1042.
Barbosa Saldana, J. G., and Anand, N., 2007, “Flow Over a Three-Dimensional Horizontal Forward-Facing Step,” Numerical Heat Transfer, Part A: Applications, 53(1), pp. 1–17.
Abu-Mulaweh, H., 2003, “A Review of Research on Laminar Mixed Convection Flow Over Backward-and Forward-Facing Steps,” Int. J. Therm. Sci., 42(9), pp. 897–909.
Stuer, H., Gyr, A., and Kinzelbach, W., 1999, “Laminar Separation on a Forward Facing Step,” Eur. J. Mech. B, 18(4), pp. 675–692.
Asseban, A., Lallemand, M., Saulnier, J. B., Fomin, N., Lavinskaja, E., Merzkirch, W., and Vitkin, D., 2000, “Digital Speckle Photography and Speckle Tomography in Heat Transfer Studies,” Opt. Laser Technol., 32(7–8), pp. 583–592.
Abu-Mulaweh, H. I., Armaly, B. F., Chen, T. S., and Hong, B., 1994, “Mixed Convection Adjacent to a Vertical Forward-Facing Step,” Proceedings of the 10th International Heat Transfer Conference, 5, pp. 423–428.
Abu-Mulaweh, H. I., Armaly, B. F., and Chen, T. S., 1993, “Measurements of Laminar Mixed Convection Flow Over a Horizontal Forward-Facing Step,” J. Thermophys. Heat Transfer, 7(4), pp. 569–573.
Abu-Mulaweh, H. I., Armaly, B. F., and Chen, T. S., 1993, “Measurements of Laminar Mixed Convection in a Boundary-Layer Flow Over Horizontal and Inclined Backward-Facing Steps,” Int. J. Heat Mass Transfer, 36(7), pp. 1883–1895.
Abu-Mulaweh, H. I., Chen, T. S., and Armaly, B. F., 2002, “Turbulent Mixed-Convection Flow Over a Backward-Facing Step—The Effect of Step Heights,” Int. J. Heat Fluid Flow, 23(6), pp. 758–765.
Abu-Mulaweh, H. I., Armaly, B. F., and Chen, T. S., 2003, “Measurements of Turbulent Mixed Convection Flow Over a Vertical Forward-Facing Step,” ASME Proceedings of the Summer Heat Transfer Conference, Las Vegas, NV, July 21–23, pp. 755–763, Paper No. HT2003-47088.
Abu-Nada, E., 2008, “Application of Nanofluids for Neat Transfer Enhancement of Separated Flows Encountered in a Backward Facing Step,” Int. J. Heat Fluid Flow, 29(1) pp. 242–249.
Bianco, V., Chiacchio, F., Manca, O., and Nardini, S., 2009, “Numerical Investigation of Nanofluids Forced Convection in Circular Tubes,” Appl. Therm. Eng., 29(17), pp. 3632–3642.
Patankar, S. V., 1980, “Numerical Heat Transfer and Fluid Flow,” Hemisphere Publishing Corporation, Washington, DC.
Ghasemi, B., and Aminossadati, S. M., 2010, “Brownian Motion of Nanoparticles in a Triangular Enclosure With Natural Convection,” Int. J. Therm. Sci., 49(6), pp. 931–940.
Vajjha, R. S., and Das, D. K., 2009, “Experimental Determination of Thermal Conductivity of Three Nanofluids and Development of New Correlations,” Int. J. Heat Mass Transfer, 52(21–22), pp. 4675–4682.
Corcione, M., 2010, “Heat Transfer Features of Buoyancy-Driven Nanofluids Inside Rectangular Enclosures Differentially Heated at the Sidewalls,” Int. J. Therm. Sci., 49(9), pp. 1536–1546.
Kherbeet, A., Sh., Mohammed, H. A., and Salman, B. H., 2012, “The Effect of Nanofluids on Mixed Convection Flow Over Microscale Backward-Facing Step,” Int. J. Heat Mass Transfer, 55(21–22), pp. 5870–5881.
Heshmati, A., Mohammed, H. A., and Darus, A. N., 2014, “Mixed Convection Heat Transfer of Nanofluids Over Backward Facing Step Having a Slotted Baffle,” Appl. Math. Comput., 240, pp. 368–386.

## Figures

Fig. 1

Schematic diagram for 2D FFS in a vertical channel

Fig. 2

Comparison of velocity distribution with the results of Al-aswadi et al. [5] in the recirculation region for S = 4.8 mm, and ER = 2 at Re = 175 for Al2O3 at different X/s, (a) 1.04, (b) 1.92, (c) 2.6, and (d) 32.8

Fig. 3

Comparison of velocity distribution with the results of Al-aswadi et al. [5] in the recirculation region for S = 4.8 mm, and ER = 2 at Re = 175 for CuO at different X/s (a) 1.04, (b) 1.92, (c) 2.6, and (d) 32.8

Fig. 4

Comparison of skin friction coefficient of SiO2 nanofluids for different Reynolds numbers (a) Re = 50 and (b) Re = 175 at the bottom wall downstream of the step

Fig. 5

Comparison of the present results with the results of Hong et al. [9] (S = 4.8 mm, and ER = 2) for Re = 100 and qw = 200 W/m2 (X = 3) at 0 deg angle (a) velocity distribution and (b) Nusselt number

Fig. 6

The effect of nanofluids parameters along the downstream wall at Re = 300, qw = 500 W/m2, ∅ = 4%, and dp = 25 nm for (a) different nanoparticle types, (b) different volume fractions, and (c) different nanoparticle diameters

Fig. 7

Velocity distributions of SiO2 nanofluid with ∅ = 4% and dp = 25 nm at different Reynolds numbers for qw = 500 W/m2 at (a) x/S = 0, (b) x/S = 1.8, (c) x/S = 12.82, and (d) exit

Fig. 8

The effect of Reynolds number of SiO2 nanofluid with ∅ = 4%, dp = 25, and qw = 500 W/m2 along the downstream wall (a) Nusselt number and (b) skin friction coefficient

Fig. 9

Velocity distributions of SiO2 nanofluid with φ = 4% and dp = 25 nm at different heat fluxes for Re = 100 at (a) x/S = 0, (b) x/S = 1.8, (c) x/S = 12.82, and (d) exit

Fig. 10

The effect of heat flux of SiO2 nanofluid with ∅ = 4% and dp = 25 at Re = 100 along the downstream wall (a) skin friction coefficient and (b) Nusselt number

Fig. 11

The effect of step heights of SiO2 nanofluid with ∅ = 4%, dp = 25, and Re = 500 along the downstream wall on (a) Nusselt number, (b) isotherms at S = 5.8 mm, (c) isotherms at S = 4.8 mm, and (d) isotherms at S = 3 mm

Fig. 12

The effect of assisting and opposing flows on Nusselt number of SiO2 nanofluid with ∅ = 4% and dp = 25 nm at qw = 500 W/m2 and Re = 100 along the downstream wall of FFS

## Tables

Table 1 Grid tests for the Nusselt number at Re = 100
Table 2 Thermophysical properties for pure water and different nanofluids at T = 300 K

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